1.
Ancient Greek
–
Ancient Greek includes the forms of Greek used in ancient Greece and the ancient world from around the 9th century BC to the 6th century AD. It is often divided into the Archaic period, Classical period. It is antedated in the second millennium BC by Mycenaean Greek, the language of the Hellenistic phase is known as Koine. Koine is regarded as a historical stage of its own, although in its earliest form it closely resembled Attic Greek. Prior to the Koine period, Greek of the classic and earlier periods included several regional dialects, Ancient Greek was the language of Homer and of fifth-century Athenian historians, playwrights, and philosophers. It has contributed many words to English vocabulary and has been a subject of study in educational institutions of the Western world since the Renaissance. This article primarily contains information about the Epic and Classical phases of the language, Ancient Greek was a pluricentric language, divided into many dialects. The main dialect groups are Attic and Ionic, Aeolic, Arcadocypriot, some dialects are found in standardized literary forms used in literature, while others are attested only in inscriptions. There are also several historical forms, homeric Greek is a literary form of Archaic Greek used in the epic poems, the Iliad and Odyssey, and in later poems by other authors. Homeric Greek had significant differences in grammar and pronunciation from Classical Attic, the origins, early form and development of the Hellenic language family are not well understood because of a lack of contemporaneous evidence. Several theories exist about what Hellenic dialect groups may have existed between the divergence of early Greek-like speech from the common Proto-Indo-European language and the Classical period and they have the same general outline, but differ in some of the detail. The invasion would not be Dorian unless the invaders had some relationship to the historical Dorians. The invasion is known to have displaced population to the later Attic-Ionic regions, the Greeks of this period believed there were three major divisions of all Greek people—Dorians, Aeolians, and Ionians, each with their own defining and distinctive dialects. Often non-west is called East Greek, Arcadocypriot apparently descended more closely from the Mycenaean Greek of the Bronze Age. Boeotian had come under a strong Northwest Greek influence, and can in some respects be considered a transitional dialect, thessalian likewise had come under Northwest Greek influence, though to a lesser degree. Most of the dialect sub-groups listed above had further subdivisions, generally equivalent to a city-state and its surrounding territory, Doric notably had several intermediate divisions as well, into Island Doric, Southern Peloponnesus Doric, and Northern Peloponnesus Doric. The Lesbian dialect was Aeolic Greek and this dialect slowly replaced most of the older dialects, although Doric dialect has survived in the Tsakonian language, which is spoken in the region of modern Sparta. Doric has also passed down its aorist terminations into most verbs of Demotic Greek, by about the 6th century AD, the Koine had slowly metamorphosized into Medieval Greek
Ancient Greek
–
Inscription about the construction of the statue of
Athena Parthenos in the
Parthenon, 440/439 BC
Ancient Greek
–
Ostracon bearing the name of
Cimon,
Stoa of Attalos
Ancient Greek
–
The words ΜΟΛΩΝ ΛΑΒΕ as they are inscribed on the marble of the 1955
Leonidas Monument at
Thermopylae
2.
Number
–
A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1,2,3, a notational symbol that represents a number is called a numeral. In addition to their use in counting and measuring, numerals are used for labels, for ordering. In common usage, number may refer to a symbol, a word, calculations with numbers are done with arithmetical operations, the most familiar being addition, subtraction, multiplication, division, and exponentiation. Their study or usage is called arithmetic, the same term may also refer to number theory, the study of the properties of numbers. Besides their practical uses, numbers have cultural significance throughout the world, for example, in Western society the number 13 is regarded as unlucky, and a million may signify a lot. Though it is now regarded as pseudoscience, numerology, the belief in a significance of numbers, permeated ancient. Numerology heavily influenced the development of Greek mathematics, stimulating the investigation of problems in number theory which are still of interest today. During the 19th century, mathematicians began to develop many different abstractions which share certain properties of numbers, among the first were the hypercomplex numbers, which consist of various extensions or modifications of the complex number system. Numbers should be distinguished from numerals, the used to represent numbers. Boyer showed that Egyptians created the first ciphered numeral system, Greeks followed by mapping their counting numbers onto Ionian and Doric alphabets. The number five can be represented by digit 5 or by the Roman numeral Ⅴ, notations used to represent numbers are discussed in the article numeral systems. The Roman numerals require extra symbols for larger numbers, different types of numbers have many different uses. Numbers can be classified into sets, called number systems, such as the natural numbers, the same number can be written in many different ways. For different methods of expressing numbers with symbols, such as the Roman numerals, each of these number systems may be considered as a proper subset of the next one. This is expressed, symbolically, by writing N ⊂ Z ⊂ Q ⊂ R ⊂ C, the most familiar numbers are the natural numbers,1,2,3, and so on. Traditionally, the sequence of numbers started with 1 However, in the 19th century, set theorists. Today, different mathematicians use the term to both sets, including 0 or not
Number
–
The number 605 in
Khmer numerals, from an inscription from 683 AD. An early use of zero as a decimal figure.
Number
–
Subsets of the
complex numbers.
3.
Mathematics
–
Mathematics is the study of topics such as quantity, structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope, Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof, when mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry, rigorous arguments first appeared in Greek mathematics, most notably in Euclids Elements. Galileo Galilei said, The universe cannot be read until we have learned the language and it is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth, carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. Benjamin Peirce called mathematics the science that draws necessary conclusions, David Hilbert said of mathematics, We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules, rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise. Albert Einstein stated that as far as the laws of mathematics refer to reality, they are not certain, Mathematics is essential in many fields, including natural science, engineering, medicine, finance and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics, Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, the history of mathematics can be seen as an ever-increasing series of abstractions. The earliest uses of mathematics were in trading, land measurement, painting and weaving patterns, in Babylonian mathematics elementary arithmetic first appears in the archaeological record. Numeracy pre-dated writing and numeral systems have many and diverse. Between 600 and 300 BC the Ancient Greeks began a study of mathematics in its own right with Greek mathematics. Mathematics has since been extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made today, the overwhelming majority of works in this ocean contain new mathematical theorems and their proofs. The word máthēma is derived from μανθάνω, while the modern Greek equivalent is μαθαίνω, in Greece, the word for mathematics came to have the narrower and more technical meaning mathematical study even in Classical times
Mathematics
–
Euclid (holding
calipers), Greek mathematician, 3rd century BC, as imagined by
Raphael in this detail from
The School of Athens.
Mathematics
–
Greek mathematician
Pythagoras (c. 570 – c. 495 BC), commonly credited with discovering the
Pythagorean theorem
Mathematics
–
Leonardo Fibonacci, the
Italian mathematician who established the Hindu–Arabic numeral system to the Western World
Mathematics
–
Carl Friedrich Gauss, known as the prince of mathematicians
4.
Operation (mathematics)
–
In mathematics, an operation is a calculation from zero or more input values to an output value. The number of operands is the arity of the operation, the most commonly studied operations are binary operations of arity 2, such as addition and multiplication, and unary operations of arity 1, such as additive inverse and multiplicative inverse. An operation of arity zero, or 0-ary operation is a constant, the mixed product is an example of an operation of arity 3, or ternary operation. Generally, the arity is supposed to be finite, but infinitary operations are sometimes considered, in this context, the usual operations, of finite arity are also called finitary operations. There are two types of operations, unary and binary. Unary operations involve only one value, such as negation and trigonometric functions, binary operations, on the other hand, take two values, and include addition, subtraction, multiplication, division, and exponentiation. Operations can involve mathematical objects other than numbers, the logical values true and false can be combined using logic operations, such as and, or, and not. Vectors can be added and subtracted, rotations can be combined using the function composition operation, performing the first rotation and then the second. Operations on sets include the binary operations union and intersection and the operation of complementation. Operations on functions include composition and convolution, operations may not be defined for every possible value. For example, in the numbers one cannot divide by zero or take square roots of negative numbers. The values for which an operation is defined form a set called its domain, the set which contains the values produced is called the codomain, but the set of actual values attained by the operation is its range. For example, in the numbers, the squaring operation only produces non-negative numbers. A vector can be multiplied by a scalar to form another vector, and the inner product operation on two vectors produces a scalar. An operation may or may not have certain properties, for example it may be associative, commutative, anticommutative, idempotent, the values combined are called operands, arguments, or inputs, and the value produced is called the value, result, or output. Operations can have fewer or more than two inputs, an operation is like an operator, but the point of view is different. An operation ω is a function of the form ω, V → Y, where V ⊂ X1 × … × Xk. The sets Xk are called the domains of the operation, the set Y is called the codomain of the operation, thus a unary operation has arity one, and a binary operation has arity two
Operation (mathematics)
–
– plus (addition) – minus (subtraction) – times (multiplication) – obelus (division)
5.
Addition
–
Addition is one of the four basic operations of arithmetic, with the others being subtraction, multiplication and division. The addition of two numbers is the total amount of those quantities combined. For example, in the picture on the right, there is a combination of three apples and two together, making a total of five apples. This observation is equivalent to the mathematical expression 3 +2 =5 i. e.3 add 2 is equal to 5, besides counting fruits, addition can also represent combining other physical objects. In arithmetic, rules for addition involving fractions and negative numbers have been devised amongst others, in algebra, addition is studied more abstractly. It is commutative, meaning that order does not matter, and it is associative, repeated addition of 1 is the same as counting, addition of 0 does not change a number. Addition also obeys predictable rules concerning related operations such as subtraction and multiplication, performing addition is one of the simplest numerical tasks. Addition of very small numbers is accessible to toddlers, the most basic task,1 +1, can be performed by infants as young as five months and even some members of other animal species. In primary education, students are taught to add numbers in the system, starting with single digits. Mechanical aids range from the ancient abacus to the modern computer, Addition is written using the plus sign + between the terms, that is, in infix notation. The result is expressed with an equals sign, for example, 3½ =3 + ½ =3.5. This notation can cause confusion since in most other contexts juxtaposition denotes multiplication instead, the sum of a series of related numbers can be expressed through capital sigma notation, which compactly denotes iteration. For example, ∑ k =15 k 2 =12 +22 +32 +42 +52 =55. The numbers or the objects to be added in addition are collectively referred to as the terms, the addends or the summands. This is to be distinguished from factors, which are multiplied, some authors call the first addend the augend. In fact, during the Renaissance, many authors did not consider the first addend an addend at all, today, due to the commutative property of addition, augend is rarely used, and both terms are generally called addends. All of the above terminology derives from Latin, using the gerundive suffix -nd results in addend, thing to be added. Likewise from augere to increase, one gets augend, thing to be increased, sum and summand derive from the Latin noun summa the highest, the top and associated verb summare
Addition
–
Part of Charles Babbage's
Difference Engine including the addition and carry mechanisms
Addition
–
The plus sign
Addition
–
A circular slide rule
6.
Subtraction
–
Subtraction is a mathematical operation that represents the operation of removing objects from a collection. It is signified by the minus sign, for example, in the picture on the right, there are 5 −2 apples—meaning 5 apples with 2 taken away, which is a total of 3 apples. It is anticommutative, meaning that changing the order changes the sign of the answer and it is not associative, meaning that when one subtracts more than two numbers, the order in which subtraction is performed matters. Subtraction of 0 does not change a number, subtraction also obeys predictable rules concerning related operations such as addition and multiplication. All of these rules can be proven, starting with the subtraction of integers and generalizing up through the real numbers, general binary operations that continue these patterns are studied in abstract algebra. Performing subtraction is one of the simplest numerical tasks, subtraction of very small numbers is accessible to young children. In primary education, students are taught to subtract numbers in the system, starting with single digits. Subtraction is written using the minus sign − between the terms, that is, in infix notation, the result is expressed with an equals sign. This is most common in accounting, formally, the number being subtracted is known as the subtrahend, while the number it is subtracted from is the minuend. All of this terminology derives from Latin, subtraction is an English word derived from the Latin verb subtrahere, which is in turn a compound of sub from under and trahere to pull, thus to subtract is to draw from below, take away. Using the gerundive suffix -nd results in subtrahend, thing to be subtracted, likewise from minuere to reduce or diminish, one gets minuend, thing to be diminished. Imagine a line segment of length b with the left end labeled a, starting from a, it takes b steps to the right to reach c. This movement to the right is modeled mathematically by addition, a + b = c, from c, it takes b steps to the left to get back to a. This movement to the left is modeled by subtraction, c − b = a, now, a line segment labeled with the numbers 1,2, and 3. From position 3, it takes no steps to the left to stay at 3 and it takes 2 steps to the left to get to position 1, so 3 −2 =1. This picture is inadequate to describe what would happen after going 3 steps to the left of position 3, to represent such an operation, the line must be extended. To subtract arbitrary natural numbers, one begins with a line containing every natural number, from 3, it takes 3 steps to the left to get to 0, so 3 −3 =0. But 3 −4 is still invalid since it leaves the line
Subtraction
–
Placard outside shop in Bordeaux advertising subtraction of 20% from the price of a second perfume
Subtraction
–
"5 − 2 = 3" (verbally, "five minus two equals three")
Subtraction
–
1 + … = 3
7.
Multiplication
–
Multiplication is one of the four elementary, mathematical operations of arithmetic, with the others being addition, subtraction and division. Multiplication can also be visualized as counting objects arranged in a rectangle or as finding the area of a rectangle whose sides have given lengths, the area of a rectangle does not depend on which side is measured first, which illustrates the commutative property. The product of two measurements is a new type of measurement, for multiplying the lengths of the two sides of a rectangle gives its area, this is the subject of dimensional analysis. The inverse operation of multiplication is division, for example, since 4 multiplied by 3 equals 12, then 12 divided by 3 equals 4. Multiplication by 3, followed by division by 3, yields the original number, Multiplication is also defined for other types of numbers, such as complex numbers, and more abstract constructs, like matrices. For these more abstract constructs, the order that the operands are multiplied sometimes does matter, a listing of the many different kinds of products that are used in mathematics is given in the product page. In arithmetic, multiplication is often written using the sign × between the terms, that is, in infix notation, there are other mathematical notations for multiplication, Multiplication is also denoted by dot signs, usually a middle-position dot,5 ⋅2 or 5. 2 The middle dot notation, encoded in Unicode as U+22C5 ⋅ dot operator, is standard in the United States, the United Kingdom, when the dot operator character is not accessible, the interpunct is used. In other countries use a comma as a decimal mark. In algebra, multiplication involving variables is often written as a juxtaposition, the notation can also be used for quantities that are surrounded by parentheses. In matrix multiplication, there is a distinction between the cross and the dot symbols. The cross symbol generally denotes the taking a product of two vectors, yielding a vector as the result, while the dot denotes taking the dot product of two vectors, resulting in a scalar. In computer programming, the asterisk is still the most common notation and this is due to the fact that most computers historically were limited to small character sets that lacked a multiplication sign, while the asterisk appeared on every keyboard. This usage originated in the FORTRAN programming language, the numbers to be multiplied are generally called the factors. The number to be multiplied is called the multiplicand, while the number of times the multiplicand is to be multiplied comes from the multiplier. Usually the multiplier is placed first and the multiplicand is placed second, however sometimes the first factor is the multiplicand, additionally, there are some sources in which the term multiplicand is regarded as a synonym for factor. In algebra, a number that is the multiplier of a variable or expression is called a coefficient, the result of a multiplication is called a product. A product of integers is a multiple of each factor, for example,15 is the product of 3 and 5, and is both a multiple of 3 and a multiple of 5
Multiplication
–
4 × 5 = 20, the rectangle is composed of 20 squares, having dimensions of 4 by 5.
Multiplication
–
Four bags of three
marbles gives twelve marbles (4 × 3 = 12).
8.
Division (mathematics)
–
Division is one of the four basic operations of arithmetic, the others being addition, subtraction, and multiplication. The division of two numbers is the process of calculating the number of times one number is contained within one another. For example, in the picture on the right, the 20 apples are divided into groups of five apples, Division can also be thought of as the process of evaluating a fraction, and fractional notation is commonly used to represent division. Division is the inverse of multiplication, if a × b = c, then a = c ÷ b, as long as b is not zero. Division by zero is undefined for the numbers and most other contexts, because if b =0, then a cannot be deduced from b and c. In some contexts, division by zero can be defined although to a limited extent, in division, the dividend is divided by the divisor to get a quotient. In the above example,20 is the dividend, five is the divisor, in some cases, the divisor may not be contained fully by the dividend, for example,10 ÷3 leaves a remainder of one, as 10 is not a multiple of three. Sometimes this remainder is added to the quotient as a fractional part, but in the context of integer division, where numbers have no fractional part, the remainder is kept separately or discarded. Besides dividing apples, division can be applied to other physical, Division has been defined in several contexts, such as for the real and complex numbers and for more abstract contexts such as for vector spaces and fields. Division is the most mentally difficult of the four operations of arithmetic. Teaching the objective concept of dividing integers introduces students to the arithmetic of fractions, unlike addition, subtraction, and multiplication, the set of all integers is not closed under division. Dividing two integers may result in a remainder, to complete the division of the remainder, the number system is extended to include fractions or rational numbers as they are more generally called. When students advance to algebra, the theory of division intuited from arithmetic naturally extends to algebraic division of variables, polynomials. Division is often shown in algebra and science by placing the dividend over the divisor with a line, also called a fraction bar. For example, a divided by b is written a b This can be read out loud as a divided by b, a fraction is a division expression where both dividend and divisor are integers, and there is no implication that the division must be evaluated further. A second way to show division is to use the obelus, common in arithmetic, in this manner, ISO 80000-2-9.6 states it should not be used. The obelus is also used alone to represent the operation itself. In some non-English-speaking cultures, a divided by b is written a, b and this notation was introduced in 1631 by William Oughtred in his Clavis Mathematicae and later popularized by Gottfried Wilhelm Leibniz
Division (mathematics)
–
This article is about the arithmetical operation. For other uses, see
Division (disambiguation).
9.
Number theory
–
Number theory or, in older usage, arithmetic is a branch of pure mathematics devoted primarily to the study of the integers. It is sometimes called The Queen of Mathematics because of its place in the discipline. Number theorists study prime numbers as well as the properties of objects out of integers or defined as generalizations of the integers. Integers can be considered either in themselves or as solutions to equations, questions in number theory are often best understood through the study of analytical objects that encode properties of the integers, primes or other number-theoretic objects in some fashion. One may also study real numbers in relation to rational numbers, the older term for number theory is arithmetic. By the early century, it had been superseded by number theory. The use of the arithmetic for number theory regained some ground in the second half of the 20th century. In particular, arithmetical is preferred as an adjective to number-theoretic. The first historical find of a nature is a fragment of a table. The triples are too many and too large to have been obtained by brute force, the heading over the first column reads, The takiltum of the diagonal which has been subtracted such that the width. The tables layout suggests that it was constructed by means of what amounts, in language, to the identity 2 +1 =2. If some other method was used, the triples were first constructed and then reordered by c / a, presumably for use as a table. It is not known what these applications may have been, or whether there could have any, Babylonian astronomy, for example. It has been suggested instead that the table was a source of examples for school problems. While Babylonian number theory—or what survives of Babylonian mathematics that can be called thus—consists of this single, striking fragment, late Neoplatonic sources state that Pythagoras learned mathematics from the Babylonians. Much earlier sources state that Thales and Pythagoras traveled and studied in Egypt, Euclid IX 21—34 is very probably Pythagorean, it is very simple material, but it is all that is needed to prove that 2 is irrational. Pythagorean mystics gave great importance to the odd and the even, the discovery that 2 is irrational is credited to the early Pythagoreans. This forced a distinction between numbers, on the one hand, and lengths and proportions, on the other hand, the Pythagorean tradition spoke also of so-called polygonal or figurate numbers
Number theory
–
A
Lehmer sieve, which is a primitive
digital computer once used for finding
primes and solving simple
Diophantine equations.
Number theory
–
The Plimpton 322 tablet
Number theory
–
Title page of the 1621 edition of Diophantus' Arithmetica, translated into
Latin by
Claude Gaspard Bachet de Méziriac.
Number theory
–
Leonhard Euler
10.
Algebra
–
Algebra is one of the broad parts of mathematics, together with number theory, geometry and analysis. In its most general form, algebra is the study of mathematical symbols, as such, it includes everything from elementary equation solving to the study of abstractions such as groups, rings, and fields. The more basic parts of algebra are called elementary algebra, the abstract parts are called abstract algebra or modern algebra. Elementary algebra is generally considered to be essential for any study of mathematics, science, or engineering, as well as such applications as medicine, abstract algebra is a major area in advanced mathematics, studied primarily by professional mathematicians. Elementary algebra differs from arithmetic in the use of abstractions, such as using letters to stand for numbers that are unknown or allowed to take on many values. For example, in x +2 =5 the letter x is unknown, in E = mc2, the letters E and m are variables, and the letter c is a constant, the speed of light in a vacuum. Algebra gives methods for solving equations and expressing formulas that are easier than the older method of writing everything out in words. The word algebra is used in certain specialized ways. A special kind of object in abstract algebra is called an algebra. A mathematician who does research in algebra is called an algebraist, the word algebra comes from the Arabic الجبر from the title of the book Ilm al-jabr wal-muḳābala by Persian mathematician and astronomer al-Khwarizmi. The word entered the English language during the century, from either Spanish, Italian. It originally referred to the procedure of setting broken or dislocated bones. The mathematical meaning was first recorded in the sixteenth century, the word algebra has several related meanings in mathematics, as a single word or with qualifiers. As a single word without an article, algebra names a broad part of mathematics, as a single word with an article or in plural, an algebra or algebras denotes a specific mathematical structure, whose precise definition depends on the author. Usually the structure has an addition, multiplication, and a scalar multiplication, when some authors use the term algebra, they make a subset of the following additional assumptions, associative, commutative, unital, and/or finite-dimensional. In universal algebra, the word refers to a generalization of the above concept. With a qualifier, there is the distinction, Without an article, it means a part of algebra, such as linear algebra, elementary algebra. With an article, it means an instance of some abstract structure, like a Lie algebra, sometimes both meanings exist for the same qualifier, as in the sentence, Commutative algebra is the study of commutative rings, which are commutative algebras over the integers
Algebra
–
A page from
Al-Khwārizmī 's
al-Kitāb al-muḫtaṣar fī ḥisāb al-ğabr wa-l-muqābala
Algebra
–
Italian mathematician
Girolamo Cardano published the solutions to the
cubic and
quartic equations in his 1545 book
Ars magna.
11.
Geometry
–
Geometry is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer, Geometry arose independently in a number of early cultures as a practical way for dealing with lengths, areas, and volumes. Geometry began to see elements of mathematical science emerging in the West as early as the 6th century BC. By the 3rd century BC, geometry was put into a form by Euclid, whose treatment, Euclids Elements. Geometry arose independently in India, with texts providing rules for geometric constructions appearing as early as the 3rd century BC, islamic scientists preserved Greek ideas and expanded on them during the Middle Ages. By the early 17th century, geometry had been put on a solid footing by mathematicians such as René Descartes. Since then, and into modern times, geometry has expanded into non-Euclidean geometry and manifolds, while geometry has evolved significantly throughout the years, there are some general concepts that are more or less fundamental to geometry. These include the concepts of points, lines, planes, surfaces, angles, contemporary geometry has many subfields, Euclidean geometry is geometry in its classical sense. The mandatory educational curriculum of the majority of nations includes the study of points, lines, planes, angles, triangles, congruence, similarity, solid figures, circles, Euclidean geometry also has applications in computer science, crystallography, and various branches of modern mathematics. Differential geometry uses techniques of calculus and linear algebra to problems in geometry. It has applications in physics, including in general relativity, topology is the field concerned with the properties of geometric objects that are unchanged by continuous mappings. In practice, this often means dealing with large-scale properties of spaces, convex geometry investigates convex shapes in the Euclidean space and its more abstract analogues, often using techniques of real analysis. It has close connections to convex analysis, optimization and functional analysis, algebraic geometry studies geometry through the use of multivariate polynomials and other algebraic techniques. It has applications in areas, including cryptography and string theory. Discrete geometry is concerned mainly with questions of relative position of simple objects, such as points. It shares many methods and principles with combinatorics, Geometry has applications to many fields, including art, architecture, physics, as well as to other branches of mathematics. The earliest recorded beginnings of geometry can be traced to ancient Mesopotamia, the earliest known texts on geometry are the Egyptian Rhind Papyrus and Moscow Papyrus, the Babylonian clay tablets such as Plimpton 322. For example, the Moscow Papyrus gives a formula for calculating the volume of a truncated pyramid, later clay tablets demonstrate that Babylonian astronomers implemented trapezoid procedures for computing Jupiters position and motion within time-velocity space
Geometry
–
Visual checking of the
Pythagorean theorem for the (3, 4, 5)
triangle as in the
Chou Pei Suan Ching 500–200 BC.
Geometry
–
An illustration of
Desargues' theorem, an important result in
Euclidean and
projective geometry
Geometry
–
Geometry lessons in the 20th century
Geometry
–
A
European and an
Arab practicing geometry in the 15th century.
12.
Mathematical analysis
–
Mathematical analysis is the branch of mathematics dealing with limits and related theories, such as differentiation, integration, measure, infinite series, and analytic functions. These theories are studied in the context of real and complex numbers. Analysis evolved from calculus, which involves the elementary concepts and techniques of analysis, analysis may be distinguished from geometry, however, it can be applied to any space of mathematical objects that has a definition of nearness or specific distances between objects. Mathematical analysis formally developed in the 17th century during the Scientific Revolution, early results in analysis were implicitly present in the early days of ancient Greek mathematics. For instance, a geometric sum is implicit in Zenos paradox of the dichotomy. The explicit use of infinitesimals appears in Archimedes The Method of Mechanical Theorems, in Asia, the Chinese mathematician Liu Hui used the method of exhaustion in the 3rd century AD to find the area of a circle. Zu Chongzhi established a method that would later be called Cavalieris principle to find the volume of a sphere in the 5th century, the Indian mathematician Bhāskara II gave examples of the derivative and used what is now known as Rolles theorem in the 12th century. In the 14th century, Madhava of Sangamagrama developed infinite series expansions, like the power series and his followers at the Kerala school of astronomy and mathematics further expanded his works, up to the 16th century. The modern foundations of analysis were established in 17th century Europe. During this period, calculus techniques were applied to approximate discrete problems by continuous ones, in the 18th century, Euler introduced the notion of mathematical function. Real analysis began to emerge as an independent subject when Bernard Bolzano introduced the definition of continuity in 1816. In 1821, Cauchy began to put calculus on a firm logical foundation by rejecting the principle of the generality of algebra widely used in earlier work, instead, Cauchy formulated calculus in terms of geometric ideas and infinitesimals. Thus, his definition of continuity required a change in x to correspond to an infinitesimal change in y. He also introduced the concept of the Cauchy sequence, and started the theory of complex analysis. Poisson, Liouville, Fourier and others studied partial differential equations, the contributions of these mathematicians and others, such as Weierstrass, developed the -definition of limit approach, thus founding the modern field of mathematical analysis. In the middle of the 19th century Riemann introduced his theory of integration, the last third of the century saw the arithmetization of analysis by Weierstrass, who thought that geometric reasoning was inherently misleading, and introduced the epsilon-delta definition of limit. Then, mathematicians started worrying that they were assuming the existence of a continuum of numbers without proof. Around that time, the attempts to refine the theorems of Riemann integration led to the study of the size of the set of discontinuities of real functions, also, monsters began to be investigated
Mathematical analysis
–
A
strange attractor arising from a
differential equation. Differential equations are an important area of mathematical analysis with many applications to
science and
engineering.
13.
Ishango bone
–
The Ishango bone is a bone tool, dated to the Upper Paleolithic era. It is a dark brown length of bone, the fibula of a baboon, with a piece of quartz affixed to one end. It was first thought to be a stick, as it has a series of what has been interpreted as tally marks carved in three columns running the length of the tool. It has also suggested that the scratches might have been to create a better grip on the handle or for some other non-mathematical reason. The Ishango bone was found in 1960 by Belgian Jean de Heinzelin de Braucourt while exploring what was then the Belgian Congo and it was discovered in the area of Ishango near the Semliki River. Lake Edward empties into the Semliki which forms part of the headwaters of the Nile River, the bone was found among the remains of a small community that fished and gathered in this area of Africa. The settlement had been buried in a volcanic eruption, the artifact was first estimated to have originated between 9,000 BC and 6,500 BC. However, the dating of the site where it was discovered was re-evaluated, the Ishango bone is on permanent exhibition at the Royal Belgian Institute of Natural Sciences, Brussels, Belgium. Some mathematicians, scientists and archaeolgists believe the three columns of asymmetrically grouped notches imply that the implement was used to construct a numeral system, the central column begins with three notches and then doubles to 6 notches. The process is repeated for the number 4, which doubles to 8 notches, and then reversed for the number 10 and these numbers may not be purely random and instead suggest some understanding of the principle of multiplication and division by two. The bone may therefore have been used as a tool for simple mathematical procedures. He also writes that no attempt has been made to explain why a tally of something should exhibit multiples of two, prime numbers between 10 and 20, and some numbers that are almost multiples of 10. Claudia Zaslavsky has suggested that this may indicate that the creator of the tool was a woman, during earlier excavations at the Ishango site in 1959, another bone was also found. It is lighter in color and was scraped, thinned, polished, the artifact possibly held a piece of quartz like the more well-known bone or it could have been a tool handle. The 14-cm long bone has 90 notches on six sides, which are categorized as major or minor according to their length, Jean de Heinzelin interpreted the major notches as being units or multiples and the minor notches as fractions or subsidiary. He believed the bone to be an interchange rule between bases 10 and 12, lebombo bone History of mathematics Paleolithic tally sticks Shurkin, J. Engines of the mind, a history of the computer, W. W. Norton & Co.1984. P21 Bogoshi, J. Naidoo, K. and Webb, Africa, The true cradle of mathematical sciences Ishango,22000 and 50 years later, the cradle of mathematics. The On-Line Encyclopedia of Integer Sequences
Ishango bone
–
The Ishango bone on exhibition at the
Royal Belgian Institute of Natural Sciences
14.
Democratic Republic of the Congo
–
The Democratic Republic of the Congo, also known as DR Congo, DRC, DROC, East Congo, Congo-Kinshasa, or simply the Congo is a country located in Central Africa. From 1971 to 1997 it was named, and is still called, Zaire. It is the second-largest country in Africa by area and eleventh largest in the world, the Congolese Civil Wars, which began in 1996, brought about the end of Mobutu Sese Sekos 32-year reign and devastated the country. These wars ultimately involved nine African nations, multiple groups of UN peacekeepers and twenty armed groups, besides the capital, Kinshasa, the other major cities, Lubumbashi and Mbuji-Mayi, are both mining communities. DR Congos largest export is raw minerals, with China accepting over 50% of DRCs exports in 2012, as of 2015, according to the Human Development Index, DR Congo has a low level of human development, ranking 176 out of 187 countries. The country was known officially as the Democratic Republic of the Congo from 1965 to 27 October 1971, in 1992, the Sovereign National Conference voted to change the name of the country to the Democratic Republic of the Congo, but the change was not put into practice. The countrys name was restored by former president Laurent-Désiré Kabila following the fall of longtime dictator Mobutu Sese Seko in 1997, some historians think that Bantu peoples began settling in the extreme northwest of Central Africa at the beginning of the 5th century and then gradually started to expand southward. Their propagation was accelerated by the transition from Stone Age to Iron Age techniques, the people living in the south and southwest were mostly San Bushmen and hunter-gatherer groups, whose technology involved only minimal use of metal technologies. The development of tools during this time period revolutionized agriculture. This led to the displacement of the groups in the east and southeast. The 10th century marked the expansion of the Bantu in West-Central Africa. Rising populations soon made intricate local, regional and foreign commercial networks that traded mostly in salt, iron. Belgian exploration and administration took place from the 1870s until the 1920s and it was first led by Sir Henry Morton Stanley, who undertook his explorations under the sponsorship of King Leopold II of Belgium. The eastern regions of the precolonial Congo were heavily disrupted by constant slave raiding, mainly from Arab–Swahili slave traders such as the infamous Tippu Tip, Leopold had designs on what was to become the Congo as a colony. Leopold formally acquired rights to the Congo territory at the Conference of Berlin in 1885 and he named it the Congo Free State. Leopolds rėgime began various infrastructure projects, such as construction of the railway ran from the coast to the capital of Leopoldville. Nearly all such projects were aimed at making it easier to increase the assets which Leopold. In the Free State, colonists brutalized the local population into producing rubber, for which the spread of automobiles, rubber sales made a fortune for Leopold, who built several buildings in Brussels and Ostend to honor himself and his country
Democratic Republic of the Congo
–
Village attacked by Arab-Swahili slavers near
Nyangwe, end of 19th century
Democratic Republic of the Congo
–
Flag
Democratic Republic of the Congo
–
View of
Leopoldville Station and Port in 1884.
Democratic Republic of the Congo
–
Force Publique soldiers in the Belgian Congo in 1918. At its peak, the Force Publique had around 19,000 African soldiers, led by 420 white officers.
15.
Egyptian mathematics
–
Ancient Egyptian mathematics is the mathematics that was developed and used in Ancient Egypt c.3000 to c.300 BC. Written evidence of the use of mathematics dates back to at least 3000 BC with the ivory labels found in Tomb U-j at Abydos and these labels appear to have been used as tags for grave goods and some are inscribed with numbers. Further evidence of the use of the base 10 number system can be found on the Narmer Macehead which depicts offerings of 400,000 oxen,1,422,000 goats and 120,000 prisoners. The evidence of the use of mathematics in the Old Kingdom is scarce, the lines in the diagram are spaced at a distance of one cubit and show the use of that unit of measurement. The earliest true mathematical documents date to the 12th dynasty, the Rhind Mathematical Papyrus which dates to the Second Intermediate Period is said to be based on an older mathematical text from the 12th dynasty. The Moscow Mathematical Papyrus and Rhind Mathematical Papyrus are so-called mathematical problem texts and they consist of a collection of problems with solutions. These texts may have been written by a teacher or a student engaged in solving typical mathematics problems, an interesting feature of Ancient Egyptian mathematics is the use of unit fractions. Scribes used tables to help work with these fractions. The Egyptian Mathematical Leather Roll for instance is a table of unit fractions which are expressed as sums of unit fractions. The Rhind Mathematical Papyrus and some of the other texts contain 2 n tables and these tables allowed the scribes to rewrite any fraction of the form 1 n as a sum of unit fractions. During the New Kingdom mathematical problems are mentioned in the literary Papyrus Anastasi I, in the workers village of Deir el-Medina several ostraca have been found that record volumes of dirt removed while quarrying the tombs. Our understanding of ancient Egyptian mathematics is impeded by the paucity of available sources. The Reisner Papyrus dates to the early Twelfth dynasty of Egypt and was found in Nag el-Deir, the Rhind Mathematical Papyrus dates from the Second Intermediate Period, but its author, Ahmes, identifies it as a copy of a now lost Middle Kingdom papyrus. The RMP is the largest mathematical text, from the New Kingdom we have a handful of mathematical texts and inscription related to computations, The Papyrus Anastasi I is a literary text from the New Kingdom. It is written as a written by a scribe named Hori. A segment of the letter describes several mathematical problems, ostracon Senmut 153 is a text written in hieratic. Ostracon Turin 57170 is a written in hieratic. Ostraca from Deir el-Medina contain computations, ostracon IFAO1206 for instance shows the calculations of volumes, presumably related to the quarrying of a tomb
Egyptian mathematics
–
Slab stela of
Old Kingdom princess
Neferetiabet (dated 2590–2565 BC) from her tomb at Giza, painting on limestone, now in the
Louvre.
Egyptian mathematics
–
Image of Problem 14 from the
Moscow Mathematical Papyrus. The problem includes a diagram indicating the dimensions of the truncated pyramid.
16.
Babylonian mathematics
–
Babylonian mathematics was any mathematics developed or practiced by the people of Mesopotamia, from the days of the early Sumerians to the fall of Babylon in 539 BC. Babylonian mathematical texts are plentiful and well edited, in respect of time they fall in two distinct groups, one from the Old Babylonian period, the other mainly Seleucid from the last three or four centuries BC. In respect of content there is any difference between the two groups of texts. Thus Babylonian mathematics remained constant, in character and content, for two millennia. In contrast to the scarcity of sources in Egyptian mathematics, our knowledge of Babylonian mathematics is derived from some 400 clay tablets unearthed since the 1850s. Written in Cuneiform script, tablets were inscribed while the clay was moist, the majority of recovered clay tablets date from 1800 to 1600 BCE, and cover topics that include fractions, algebra, quadratic and cubic equations and the Pythagorean theorem. The Babylonian tablet YBC7289 gives an approximation to 2 accurate to three significant sexagesimal digits, Babylonian mathematics is a range of numeric and more advanced mathematical practices in the ancient Near East, written in cuneiform script. Study has historically focused on the Old Babylonian period in the second millennium BC due to the wealth of data available. There has been debate over the earliest appearance of Babylonian mathematics, Babylonian mathematics was primarily written on clay tablets in cuneiform script in the Akkadian or Sumerian languages. Babylonian mathematics is perhaps an unhelpful term since the earliest suggested origins date to the use of accounting devices, such as bullae and tokens, the Babylonian system of mathematics was sexagesimal numeral system. From this we derive the modern day usage of 60 seconds in a minute,60 minutes in an hour, the Babylonians were able to make great advances in mathematics for two reasons. Firstly, the number 60 is a highly composite number, having factors of 1,2,3,4,5,6,10,12,15,20,30,60. Additionally, unlike the Egyptians and Romans, the Babylonians had a true place-value system, the ancient Sumerians of Mesopotamia developed a complex system of metrology from 3000 BC. From 2600 BC onwards, the Sumerians wrote multiplication tables on clay tablets and dealt with geometrical exercises, the earliest traces of the Babylonian numerals also date back to this period. Most clay tablets that describe Babylonian mathematics belong to the Old Babylonian, some clay tablets contain mathematical lists and tables, others contain problems and worked solutions. The Babylonians used pre-calculated tables to assist with arithmetic, for example, two tablets found at Senkerah on the Euphrates in 1854, dating from 2000 BC, give lists of the squares of numbers up to 59 and the cubes of numbers up to 32. The Babylonians used the lists of squares together with the formulae a b =2 − a 2 − b 22 a b =2 −24 to simplify multiplication, the Babylonians did not have an algorithm for long division. Instead they based their method on the fact that a b = a ×1 b together with a table of reciprocals
Babylonian mathematics
–
Babylonian clay tablet YBC 7289 with annotations. The diagonal displays an approximation of the
square root of 2 in four
sexagesimal figures, 1 24 51 10, which is good to about six
decimal digits. 1 + 24/60 + 51/60 2 + 10/60 3 = 1.41421296... The tablet also gives an example where one side of the square is 30, and the resulting diagonal is 42 25 35 or 42.4263888...
17.
Elementary arithmetic
–
Elementary arithmetic is the simplified portion of arithmetic that includes the operations of addition, subtraction, multiplication, and division. It should not be confused with elementary function arithmetic, elementary arithmetic starts with the natural numbers and the written symbols that represent them. Elementary arithmetic also includes fractions and negative numbers, which can be represented on a number line, digits are the entire set of symbols used to represent numbers. In a particular system, a single digit represents a different amount than any other digit. In modern usage, the Arabic numerals are the most common set of symbols, each single digit matches the following amounts,0, zero. Used in the absence of objects to be counted, for example, a different way of saying there are no sticks here, is to say the number of sticks here is 0.1, one. For example, here is one stick, I2, two, applied to a pair of items. Here are two sticks, I I3, three, here are three sticks, I I I4, four. Here are four sticks, I I I I5, five, here are five sticks, I I I I I6, six. Here are six sticks, I I I I I I7, here are seven sticks, I I I I I I I8, eight. Here are eight sticks, I I I I I I I I9, nine. Here are nine sticks, I I I I I I I I I Any numeral system defines the value of all numbers that more than one digit. The Hindu–Arabic numeral system includes positional notation to determine the value for any numeral, in this type of system, the increase in value for an additional digit includes one or more multiplications with the radix value and the result is added to the value of an adjacent digit. With Arabic numerals, the value of ten produces a value of twenty-one for the numeral 21. An additional multiplication with the radix value occurs for each additional digit, when two numbers are added together, the result is called a sum. The two numbers being added together are called addends, suppose you have two bags, one bag holding five apples and a second bag holding three apples. Grabbing a third, empty bag, move all the apples from the first, the third bag now holds eight apples. This illustrates the combination of three apples and five apples is eight apples, or more generally, three plus five is eight or three plus five equals eight or eight is the sum of three and five
Elementary arithmetic
–
The basic elementary arithmetic symbols.
18.
Numeral system
–
A numeral system is a writing system for expressing numbers, that is, a mathematical notation for representing numbers of a given set, using digits or other symbols in a consistent manner. It can be seen as the context that allows the symbols 11 to be interpreted as the symbol for three, the decimal symbol for eleven, or a symbol for other numbers in different bases. The number the numeral represents is called its value, ideally, a numeral system will, Represent a useful set of numbers Give every number represented a unique representation Reflect the algebraic and arithmetic structure of the numbers. For example, the decimal representation of whole numbers gives every nonzero whole number a unique representation as a finite sequence of digits. Etc. all of which have the same meaning except for some scientific, such systems are, however, not the topic of this article. The most commonly used system of numerals is the Hindu–Arabic numeral system, two Indian mathematicians are credited with developing it. Aryabhata of Kusumapura developed the notation in the 5th century. The numeral system and the concept, developed by the Hindus in India, slowly spread to other surrounding countries due to their commercial. The Arabs adopted and modified it, even today, the Arabs call the numerals which they use Rakam Al-Hind or the Hindu numeral system. The Arabs translated Hindu texts on numerology and spread them to the world due to their trade links with them. The Western world modified them and called them the Arabic numerals, hence the current western numeral system is the modified version of the Hindu numeral system developed in India. It also exhibits a great similarity to the Sanskrit–Devanagari notation, which is used in India. The simplest numeral system is the numeral system, in which every natural number is represented by a corresponding number of symbols. If the symbol / is chosen, for example, then the seven would be represented by ///////. Tally marks represent one such system still in common use, the unary system is only useful for small numbers, although it plays an important role in theoretical computer science. Elias gamma coding, which is used in data compression. The unary notation can be abbreviated by introducing different symbols for new values. The ancient Egyptian numeral system was of type, and the Roman numeral system was a modification of this idea
Numeral system
–
Numeral systems
19.
Egyptian numerals
–
The system of ancient Egyptian numerals was used in Ancient Egypt around 3000 BC until the early first millennium AD. It was a system of numeration based on the scale of ten, often rounded off to the power, written in hieroglyphs. The hieratic form of numerals stressed an exact finite series notation, the Ancient Egyptian system used bases of ten. The following hieroglyphics were used to denote powers of ten, Multiples of these values were expressed by repeating the symbol as many times as needed, for instance, a stone carving from Karnak shows the number 4622 as Egyptian hieroglyphs could be written in both directions. The symbol nfr, meaning beautiful, was used to indicate the base level in drawings of tombs and pyramids. Rational numbers could also be expressed, but only as sums of fractions, i. e. sums of reciprocals of positive integers, except for 2⁄3. The hieroglyph indicating a fraction looked like a mouth, which meant part, Fractions were written with this fractional solidus, i. e. the numerator 1, and the positive denominator below. As with most modern day languages, the ancient Egyptian language could also write out numerals as words phonetically, just like one can write thirty instead of 30 in English. The word, for instance, was written as while the numeral was This was, however, uncommon for most numbers other than one, instances of numerals written in hieratic can be found as far back as the Early Dynastic Period. The Old Kingdom Abusir Papyri are an important corpus of texts that utilize hieratic numerals. A large number like 9999 could thus be written only four signs—combining the signs for 9000,900,90. Boyer saw the new hieratic numerals as ciphered, mapping one number onto one Egyptian letter for the first time in human history, greeks adopted the new system, mapping their counting numbers onto two of their alphabets, the Doric and Ionian. In the oldest hieratic texts the individual numerals were written in a ciphered relationship to the Egyptian alphabet. But during the Old Kingdom a series of standardized writings had developed for sign-groups containing more than one numeral, however, repetition of the same numeral for each place-value was not allowed in the hieratic script. As the hieratic writing system developed over time, these sign-groups were further simplified for quick writing, two famous mathematical papyri using hieratic script are the Moscow Mathematical Papyrus and the Rhind Mathematical Papyrus. The majuscule letter A in some reconstructed forms means that the quality of that remains uncertain, Ancient Egypt Egyptian language Egyptian mathematics Allen. Middle Egyptian, An Introduction to the Language and Culture of Hieroglyphs, Egyptian Grammar, Being an Introduction to the Study of Hieroglyphs. Hieratische Paläographie, Die aegyptische Buchschrift in ihrer Entwicklung von der Fünften Dynastie bis zur römischen Kaiserzeit, Introduction Egyptian numerals Numbers and dates http, //egyptianmath. blogspot. com
Egyptian numerals
–
Numeral systems
20.
Roman numerals
–
The numeric system represented by Roman numerals originated in ancient Rome and remained the usual way of writing numbers throughout Europe well into the Late Middle Ages. Numbers in this system are represented by combinations of letters from the Latin alphabet, Roman numerals, as used today, are based on seven symbols, The use of Roman numerals continued long after the decline of the Roman Empire. The numbers 1 to 10 are usually expressed in Roman numerals as follows, I, II, III, IV, V, VI, VII, VIII, IX, Numbers are formed by combining symbols and adding the values, so II is two and XIII is thirteen. Symbols are placed left to right in order of value. Named after the year of its release,2014 as MMXIV, the year of the games of the XXII Olympic Winter Games The standard forms described above reflect typical modern usage rather than a universally accepted convention. Usage in ancient Rome varied greatly and remained inconsistent in medieval, Roman inscriptions, especially in official contexts, seem to show a preference for additive forms such as IIII and VIIII instead of subtractive forms such as IV and IX. Both methods appear in documents from the Roman era, even within the same document, double subtractives also occur, such as XIIX or even IIXX instead of XVIII. Sometimes V and L are not used, with such as IIIIII. Such variation and inconsistency continued through the period and into modern times. Clock faces that use Roman numerals normally show IIII for four o’clock but IX for nine o’clock, however, this is far from universal, for example, the clock on the Palace of Westminster in London uses IV. Similarly, at the beginning of the 20th century, different representations of 900 appeared in several inscribed dates. For instance,1910 is shown on Admiralty Arch, London, as MDCCCCX rather than MCMX, although Roman numerals came to be written with letters of the Roman alphabet, they were originally independent symbols. The Etruscans, for example, used
Roman numerals
–
Entrance to section LII (52) of the
Colosseum, with numerals still visible
Roman numerals
–
Numeral systems
Roman numerals
–
A typical
clock face with Roman numerals in
Bad Salzdetfurth, Germany
Roman numerals
–
An inscription on
Admiralty Arch, London. The number is 1910, for which MCMX would be more usual
21.
Tally marks
–
Tally marks, also called hash marks, are a unary numeral system. They are a form of used for counting. They are most useful in counting or tallying ongoing results, such as the score in a game or sport, however, because of the length of large numbers, tallies are not commonly used for static text. Notched sticks, known as tally sticks, were historically used for this purpose. Counting aids other than body parts appear in the Upper Paleolithic, the oldest tally sticks date to between 35,000 and 25,000 years ago, in the form of notched bones found in the context of the European Aurignacian to Gravettian and in Africas Late Stone Age. The so-called Wolf bone is an artifact discovered in 1937 in Czechoslovakia during excavations at Vestonice, Moravia. Dated to the Aurignacian, approximately 30,000 years ago, the head of an ivory Venus figurine was excavated close to the bone. The Ishango bone, found in the Ishango region of the present-day Democratic Republic of Congo, is dated to over 20,000 years old, upon discovery, it was thought to portray a series of prime numbers. He also writes that no attempt has been made to explain why a tally of something should exhibit multiples of two, prime numbers between 10 and 20, and some numbers that are almost multiples of 10. Alexander Marshack examined the Ishango bone microscopically, and concluded that it may represent a lunar calendar. Tally marks are typically clustered in groups of five for legibility, roman numerals, the Chinese numerals for one through three, and rod numerals were derived from tally marks, as possibly was the ogham script
Tally marks
–
Counting using tally marks at
Hanakapiai Beach. The number shown is 82.
Tally marks
–
In the dot and line (or dot-dash) tally, dots represent counts from 1 to 4, lines 5 to 8, and diagonal lines 9 and 10. This method is commonly used in forestry and related fields.
22.
Decimal
–
This article aims to be an accessible introduction. For the mathematical definition, see Decimal representation, the decimal numeral system has ten as its base, which, in decimal, is written 10, as is the base in every positional numeral system. It is the base most widely used by modern civilizations. Decimal fractions have terminating decimal representations and other fractions have repeating decimal representations, Decimal notation is the writing of numbers in a base-ten numeral system. Examples are Brahmi numerals, Greek numerals, Hebrew numerals, Roman numerals, Roman numerals have symbols for the decimal powers and secondary symbols for half these values. Brahmi numerals have symbols for the nine numbers 1–9, the nine decades 10–90, plus a symbol for 100, Chinese numerals have symbols for 1–9, and additional symbols for powers of ten, which in modern usage reach 1072. Positional decimal systems include a zero and use symbols for the ten values to represent any number, positional notation uses positions for each power of ten, units, tens, hundreds, thousands, etc. The position of each digit within a number denotes the multiplier multiplied with that position has a value ten times that of the position to its right. There were at least two independent sources of positional decimal systems in ancient civilization, the Chinese counting rod system. Ten is the number which is the count of fingers and thumbs on both hands, the English word digit as well as its translation in many languages is also the anatomical term for fingers and toes. In English, decimal means tenth, decimate means reduce by a tenth, however, the symbols used in different areas are not identical, for instance, Western Arabic numerals differ from the forms used by other Arab cultures. A decimal fraction is a fraction the denominator of which is a power of ten. g, Decimal fractions 8/10, 1489/100, 24/100000, and 58900/10000 are expressed in decimal notation as 0.8,14.89,0.00024,5.8900 respectively. In English-speaking, some Latin American and many Asian countries, a period or raised period is used as the separator, in many other countries, particularly in Europe. The integer part, or integral part of a number is the part to the left of the decimal separator. The part from the separator to the right is the fractional part. It is usual for a number that consists only of a fractional part to have a leading zero in its notation. Any rational number with a denominator whose only prime factors are 2 and/or 5 may be expressed as a decimal fraction and has a finite decimal expansion. 1/2 =0.5 1/20 =0.05 1/5 =0.2 1/50 =0.02 1/4 =0.25 1/40 =0.025 1/25 =0.04 1/8 =0.125 1/125 =0.008 1/10 =0
Decimal
–
The world's earliest decimal multiplication table was made from bamboo slips, dating from 305 BC, during the
Warring States period in China.
Decimal
–
Numeral systems
Decimal
–
Ten fingers on two hands, the possible starting point of the decimal counting.
Decimal
–
Diagram of the world's earliest decimal multiplication table (c. 305 BC) from the Warring States period
23.
Positional notation
–
Positional notation or place-value notation is a method of representing or encoding numbers. Positional notation is distinguished from other notations for its use of the symbol for the different orders of magnitude. This greatly simplified arithmetic, leading to the spread of the notation across the world. With the use of a point, the notation can be extended to include fractions. The Hindu–Arabic numeral system, base-10, is the most commonly used system in the world today for most calculations, today, the base-10 system, which is likely motivated by counting with the ten fingers, is ubiquitous. Other bases have been used in the past however, and some continue to be used today, for example, the Babylonian numeral system, credited as the first positional numeral system, was base-60, but it lacked a real 0 value. Zero was indicated by a space between sexagesimal numerals, by 300 BC, a punctuation symbol was co-opted as a placeholder in the same Babylonian system. In a tablet unearthed at Kish, the scribe Bêl-bân-aplu wrote his zeros with three hooks, rather than two slanted wedges, the Babylonian placeholder was not a true zero because it was not used alone. Nor was it used at the end of a number, thus numbers like 2 and 120,3 and 180,4 and 240, looked the same because the larger numbers lacked a final sexagesimal placeholder. Counting rods and most abacuses have been used to represent numbers in a numeral system. This approach required no memorization of tables and could produce practical results quickly, for four centuries there was strong disagreement between those who believed in adopting the positional system in writing numbers and those who wanted to stay with the additive-system-plus-abacus. Although electronic calculators have largely replaced the abacus, the continues to be used in Japan. After the French Revolution, the new French government promoted the extension of the decimal system, some of those pro-decimal efforts—such as decimal time and the decimal calendar—were unsuccessful. Other French pro-decimal efforts—currency decimalisation and the metrication of weights and measures—spread widely out of France to almost the whole world. According to Joseph Needham and Lam Lay Yong, decimal fractions were first developed and used by the Chinese in the 1st century BC, the written Chinese decimal fractions were non-positional. However, counting rod fractions were positional, the Jewish mathematician Immanuel Bonfils used decimal fractions around 1350, anticipating Simon Stevin, but did not develop any notation to represent them. A forerunner of modern European decimal notation was introduced by Simon Stevin in the 16th century. A key argument against the system was its susceptibility to easy fraud by simply putting a number at the beginning or end of a quantity, thereby changing 100 into 5100
Positional notation
–
Numeral systems
24.
Counting board
–
The counting board is the precursor of the abacus, and the earliest known form of a counting device. Counting boards were made of stone or wood, and the counting was done on the board with beads, not many boards survive because of the perishable materials used in their construction. The oldest known counting board, the Salamis Tablet was discovered on the Greek island of Salamis in 1899 and it is thought to have been used by the Babylonians in about 300 BC and is more of a gaming board than a calculating device. It is marble, about 150 x 75 x 4.5 cm and it has carved Greek letters and parallel grooves. The German mathematicican Adam Ries described the use of counting boards in Rechenbuch auf Linien und Ziphren in allerlei Handthierung / geschäfften und Kaufmanschafft, in the novel Wolf Hall, Hilary Mantel refers to Thomas Cromwell using a counting board in 16th-century England
Counting board
–
Rechentisch/Counting board (engraving probably from
Strasbourg)
25.
Roman abacus
–
The Ancient Romans developed the Roman hand abacus, a portable, but less capable, base-10 version of the previous Babylonian abacus. It was the first portable calculating device for engineers, merchants and it greatly reduced the time needed to perform the basic operations of arithmetic using Roman numerals. But language, the most reliable and conservative guardian of a past culture, has come to our rescue once more, above all, it has preserved the fact of the unattached counters so faithfully that we can discern this more clearly than if we possessed an actual counting board. What the Greeks called psephoi, the Romans called calculi, the Latin word calx means pebble or gravel stone, calculi are thus little stones. Both the Roman abacus and the Chinese suanpan have been used since ancient times, the rightmost two grooves were for fractional counting. The abacus was made of a metal plate where the ran in slots. The size was such that it could fit in a shirt pocket. The beads in the shorter grooves denote fives—five units, five tens. Computations are made by means of beads which would probably have been slid up and these latter two slots are for mixed-base math, a development unique to the Roman hand abacus described in following sections. The longer slot with five beads below the Ө position allowed for the counting of 1/12 of a unit called an uncia, making the abacus useful for Roman measures. The first column was either a single slot with 4 beads or 3 slots with one, one, in either case, three symbols were included beside the single slot version or one symbol per slot for the three slot version. Many measures were aggregated by twelfths, thus the Roman pound, consisted of 12 ounces. A measure of volume, congius, consisted of 12 heminae, the Roman foot, was 12 inches. The actus, the furrow length when plowing, was 120 pedes. There were however other measures in common use - for example the sextarius was two heminae, the as, the principal copper coin in Roman currency, was also divided into 12 unciae. Again, the abacus was ideally suited for counting currency, the first column was arranged either as a single slot with three different symbols or as three separate slots with one, one and two beads or counters respectively and a distinct symbol for each slot. It is most likely that the rightmost slot or slots were used to enumerate fractions of an uncia, the upper character in this slot is the character most closely resembling that used to denote a semuncia or 1/24. The name semuncia denotes 1/2 of an uncia or 1/24 of the base unit, likewise, the next character is that used to indicate a sicilicus or 1/48 of an As, which is 1/4 of an uncia
Roman abacus
–
A reconstruction of a Roman hand abacus, made by the RGZ Museum in Mainz, 1977. The original is bronze and is held by the Bibliothèque nationale de France, in Paris.
26.
Babylonian numerals
–
Babylonian numerals were written in cuneiform, using a wedge-tipped reed stylus to make a mark on a soft clay tablet which would be exposed in the sun to harden to create a permanent record. The Babylonians, who were famous for their observations and calculations. Neither of the predecessors was a positional system and this system first appeared around 2000 BC, its structure reflects the decimal lexical numerals of Semitic languages rather than Sumerian lexical numbers. However, the use of a special Sumerian sign for 60 attests to a relation with the Sumerian system. The Babylonian system is credited as being the first known positional numeral system and this was an extremely important development, because non-place-value systems require unique symbols to represent each power of a base, which can make calculations more difficult. Only two symbols were used to notate the 59 non-zero digits and these symbols and their values were combined to form a digit in a sign-value notation quite similar to that of Roman numerals, for example, the combination represented the digit for 23. A space was left to indicate a place value, similar to the modern-day zero. Babylonians later devised a sign to represent this empty place and they lacked a symbol to serve the function of radix point, so the place of the units had to be inferred from context, could have represented 23 or 23×60 or 23×60×60 or 23/60, etc. A common theory is that 60, a highly composite number, was chosen due to its prime factorization, 2×2×3×5, which makes it divisible by 1,2,3,4,5,6,10,12,15,20. Integers and fractions were represented identically — a radix point was not written, the Babylonians did not technically have a digit for, nor a concept of, the number zero. Although they understood the idea of nothingness, it was not seen as a number—merely the lack of a number, what the Babylonians had instead was a space to mark the nonexistence of a digit in a certain place value. Babylon Babylonia History of zero Numeral system Menninger, Karl W. Number Words and Number Symbols, Number, From Ancient Civilisations to the Computer. CESCNC - a handy and easy-to use numeral converter
Babylonian numerals
–
Numeral systems
Babylonian numerals
–
Babylonian numerals
27.
Vigesimal
–
The vigesimal or base 20 numeral system is based on twenty. In a vigesimal system, twenty individual numerals are used. One modern method of finding the extra needed symbols is to write ten as the letter A20, to write nineteen as J20, and this is similar to the common computer-science practice of writing hexadecimal numerals over 9 with the letters A–F. Another method skips over the letter I, in order to avoid confusion between I20 as eighteen and one, so that the number eighteen is written as J20, the number twenty is written as 1020. According to this notation,2020 means forty in decimal = + D020 means two hundred and sixty in decimal = +10020 means four hundred in decimal = + +, in the rest of this article below, numbers are expressed in decimal notation, unless specified otherwise. For example,10 means ten,20 means twenty, in decimal, dividing by three twice only gives one digit periods because 9 is the number below ten. 21, however, the adjacent to 20 that is divisible by 3, is not divisible by 9. Ninths in vigesimal have six-digit periods, the prime factorization of twenty is 22 ×5, so it is not a perfect power. However, its part,5, is congruent to 1. Thus, according to Artins conjecture on primitive roots, vigesimal has infinitely many cyclic primes, but the fraction of primes that are cyclic is not necessarily ~37. 395%. An UnrealScript program that computes the lengths of recurring periods of various fractions in a set of bases found that, of the first 15,456 primes. In many European languages,20 is used as a base, vigesimal systems are common in Africa, for example in Yoruba. Ogún,20, is the basic numeric block, ogójì,40, =20 multiplied by 2. Ogota,60, =20 multiplied by 3, ogorin,80, =20 multiplied by 4. Ogorun,100, =20 multiplied by 5, twenty was a base in the Maya and Aztec number systems. The Maya used the names for the powers of twenty, kal, bak, pic, calab, kinchil. See also Maya numerals and Maya calendar, Mayan languages, Yucatec, the Aztec called them, cempoalli, centzontli, cenxiquipilli, cempoalxiquipilli, centzonxiquipilli and cempoaltzonxiquipilli. Note that the ce prefix at the beginning means one and is replaced with the number to get the names of other multiples of the power
Vigesimal
–
Numeral systems
Vigesimal
–
The
Maya numerals are a base-20 system.
28.
Maya numerals
–
The Maya numeral system is a vigesimal positional notation used in the Maya civilization to represent numbers. The numerals are made up of three symbols, zero, one and five, for example, thirteen is written as three dots in a horizontal row above two horizontal lines stacked above each other. Numbers after 19 were written vertically in powers of twenty, for example, thirty-three would be written as one dot above three dots, which are in turn atop two lines. The first dot represents one twenty or 1×20, which is added to three dots and two bars, or thirteen, upon reaching 202 or 400, another row is started. The number 429 would be written as one dot above one dot above four dots, the powers of twenty are numerals, just as the Hindu-Arabic numeral system uses powers of tens. Other than the bar and dot notation, Maya numerals can be illustrated by face type glyphs or pictures, the face glyph for a number represents the deity associated with the number. These face number glyphs were used, and are mostly seen on some of the most elaborate monumental carving. Addition and subtraction, Adding and subtracting numbers below 20 using Maya numerals is very simple, addition is performed by combining the numeric symbols at each level, If five or more dots result from the combination, five dots are removed and replaced by a bar. If four or more bars result, four bars are removed, similarly with subtraction, remove the elements of the subtrahend symbol from the minuend symbol, If there are not enough dots in a minuend position, a bar is replaced by five dots. If there are not enough bars, a dot is removed from the next higher minuend symbol in the column, the Maya/Mesoamerican Long Count calendar required the use of zero as a place-holder within its vigesimal positional numeral system. A shell glyph – – was used as a symbol for these Long Count dates. However, since the eight earliest Long Count dates appear outside the Maya homeland, it is assumed that the use of zero predated the Maya, indeed, many of the earliest Long Count dates were found within the Olmec heartland. However, the Olmec civilization had come to an end by the 4th century BC, in the Long Count portion of the Maya calendar, a variation on the strictly vigesimal numbering is used. The Long Count changes in the place value, it is not 20×20 =400, as would otherwise be expected. This is supposed to be because 360 is roughly the number of days in a year, subsequent place values return to base-twenty. In fact, every known example of large numbers uses this modified vigesimal system and it is reasonable to assume, but not proven by any evidence, that the normal system in use was a pure base-20 system. Maya Mathematics - online converter from decimal numeration to Maya numeral notation, anthropomorphic Maya numbers - online story of number representations
Maya numerals
–
Numeral systems
Maya numerals
–
Maya numerals
Maya numerals
–
Detail showing three columns of glyphs from
La Mojarra Stela 1. The left column uses Maya numerals to show a Long Count date of 8.5.16.9.7, or 156 CE.
29.
Hellenistic civilization
–
It is often considered a period of transition, sometimes even of decadence or degeneration, compared to the enlightenment of the Greek Classical era. The Hellenistic period saw the rise of New Comedy, Alexandrian poetry, the Septuagint, Greek science was advanced by the works of the mathematician Euclid and the polymath Archimedes. The religious sphere expanded to include new gods such as the Greco-Egyptian Serapis, eastern deities such as Attis and Cybele, the Hellenistic period was characterized by a new wave of Greek colonization which established Greek cities and kingdoms in Asia and Africa. This resulted in the export of Greek culture and language to new realms. Equally, however, these new kingdoms were influenced by the cultures, adopting local practices where beneficial, necessary. Hellenistic culture thus represents a fusion of the Ancient Greek world with that of the Near East, Middle East and this mixture gave rise to a common Attic-based Greek dialect, known as Koine Greek, which became the lingua franca through the Hellenistic world. Scholars and historians are divided as to what event signals the end of the Hellenistic era, Hellenistic is distinguished from Hellenic in that the first encompasses the entire sphere of direct ancient Greek influence, while the latter refers to Greece itself. The word originated from the German term hellenistisch, from Ancient Greek Ἑλληνιστής, from Ἑλλάς, Hellenistic is a modern word and a 19th-century concept, the idea of a Hellenistic period did not exist in Ancient Greece. Although words related in form or meaning, e. g, the major issue with the term Hellenistic lies in its convenience, as the spread of Greek culture was not the generalized phenomenon that the term implies. Some areas of the world were more affected by Greek influences than others. The Greek population and the population did not always mix, the Greeks moved and brought their own culture. While a few fragments exist, there is no surviving historical work which dates to the hundred years following Alexanders death. The works of the major Hellenistic historians Hieronymus of Cardia, Duris of Samos, the earliest and most credible surviving source for the Hellenistic period is Polybius of Megalopolis, a statesman of the Achaean League until 168 BC when he was forced to go to Rome as a hostage. His Histories eventually grew to a length of forty books, covering the years 220 to 167 BC, another important source, Plutarchs Parallel Lives though more preoccupied with issues of personal character and morality, outlines the history of important Hellenistic figures. Appian of Alexandria wrote a history of the Roman empire that includes information of some Hellenistic kingdoms, other sources include Justins epitome of Pompeius Trogus Historiae Philipicae and a summary of Arrians Events after Alexander, by Photios I of Constantinople. Lesser supplementary sources include Curtius Rufus, Pausanias, Pliny, in the field of philosophy, Diogenes Laertius Lives and Opinions of Eminent Philosophers is the main source. Ancient Greece had traditionally been a collection of fiercely independent city-states. After the Peloponnesian War, Greece had fallen under a Spartan hegemony, in which Sparta was pre-eminent but not all-powerful
Hellenistic civilization
–
The
Nike of Samothrace is considered one of the greatest masterpieces of
Hellenistic art.
Hellenistic civilization
–
Alexander fighting the Persian king
Darius III. From the
Alexander Mosaic,
Naples National Archaeological Museum.
Hellenistic civilization
–
Alexander's empire at the time of its maximum expansion.
30.
Euclid
–
Euclid, sometimes called Euclid of Alexandria to distinguish him from Euclides of Megara, was a Greek mathematician, often referred to as the father of geometry. He was active in Alexandria during the reign of Ptolemy I, in the Elements, Euclid deduced the principles of what is now called Euclidean geometry from a small set of axioms. Euclid also wrote works on perspective, conic sections, spherical geometry, number theory, Euclid is the anglicized version of the Greek name Εὐκλείδης, which means renowned, glorious. Very few original references to Euclid survive, so little is known about his life, the date, place and circumstances of both his birth and death are unknown and may only be estimated roughly relative to other people mentioned with him. He is rarely mentioned by name by other Greek mathematicians from Archimedes onward, the few historical references to Euclid were written centuries after he lived by Proclus c.450 AD and Pappus of Alexandria c.320 AD. Proclus introduces Euclid only briefly in his Commentary on the Elements, Proclus later retells a story that, when Ptolemy I asked if there was a shorter path to learning geometry than Euclids Elements, Euclid replied there is no royal road to geometry. This anecdote is questionable since it is similar to a story told about Menaechmus, a detailed biography of Euclid is given by Arabian authors, mentioning, for example, a birth town of Tyre. This biography is generally believed to be completely fictitious, however, this hypothesis is not well accepted by scholars and there is little evidence in its favor. The only reference that historians rely on of Euclid having written the Elements was from Proclus, although best known for its geometric results, the Elements also includes number theory. The geometrical system described in the Elements was long known simply as geometry, today, however, that system is often referred to as Euclidean geometry to distinguish it from other so-called non-Euclidean geometries that mathematicians discovered in the 19th century. In addition to the Elements, at least five works of Euclid have survived to the present day and they follow the same logical structure as Elements, with definitions and proved propositions. Data deals with the nature and implications of information in geometrical problems. On Divisions of Figures, which only partially in Arabic translation. It is similar to a first-century AD work by Heron of Alexandria, catoptrics, which concerns the mathematical theory of mirrors, particularly the images formed in plane and spherical concave mirrors. The attribution is held to be anachronistic however by J J OConnor, phaenomena, a treatise on spherical astronomy, survives in Greek, it is quite similar to On the Moving Sphere by Autolycus of Pitane, who flourished around 310 BC. Optics is the earliest surviving Greek treatise on perspective, in its definitions Euclid follows the Platonic tradition that vision is caused by discrete rays which emanate from the eye. One important definition is the fourth, Things seen under a greater angle appear greater, proposition 45 is interesting, proving that for any two unequal magnitudes, there is a point from which the two appear equal. Other works are attributed to Euclid, but have been lost
Euclid
–
Euclid by
Justus van Gent, 15th century
Euclid
–
One of the oldest surviving fragments of Euclid's Elements, found at
Oxyrhynchus and dated to circa AD 100 (
P. Oxy. 29). The diagram accompanies Book II, Proposition 5.
Euclid
–
Statue in honor of Euclid in the
Oxford University Museum of Natural History
31.
Greek mathematics
–
Greek mathematics, as the term is used in this article, is the mathematics written in Greek, developed from the 7th century BC to the 4th century AD around the shores of the Eastern Mediterranean. Greek mathematicians lived in cities spread over the entire Eastern Mediterranean from Italy to North Africa but were united by culture, Greek mathematics of the period following Alexander the Great is sometimes called Hellenistic mathematics. The word mathematics itself derives from the ancient Greek μάθημα, meaning subject of instruction, the study of mathematics for its own sake and the use of generalized mathematical theories and proofs is the key difference between Greek mathematics and those of preceding civilizations. The origin of Greek mathematics is not well documented, the earliest advanced civilizations in Greece and in Europe were the Minoan and later Mycenaean civilization, both of which flourished during the 2nd millennium BC. While these civilizations possessed writing and were capable of advanced engineering, including four-story palaces with drainage and beehive tombs, though no direct evidence is available, it is generally thought that the neighboring Babylonian and Egyptian civilizations had an influence on the younger Greek tradition. Historians traditionally place the beginning of Greek mathematics proper to the age of Thales of Miletus. Little is known about the life and work of Thales, so little indeed that his date of birth and death are estimated from the eclipse of 585 BC, despite this, it is generally agreed that Thales is the first of the seven wise men of Greece. The two earliest mathematical theorems, Thales theorem and Intercept theorem are attributed to Thales. The former, which states that an angle inscribed in a semicircle is a right angle and it is for this reason that Thales is often hailed as the father of the deductive organization of mathematics and as the first true mathematician. Thales is also thought to be the earliest known man in history to whom specific mathematical discoveries have been attributed, another important figure in the development of Greek mathematics is Pythagoras of Samos. Like Thales, Pythagoras also traveled to Egypt and Babylon, then under the rule of Nebuchadnezzar, Pythagoras established an order called the Pythagoreans, which held knowledge and property in common and hence all of the discoveries by individual Pythagoreans were attributed to the order. And since in antiquity it was customary to give all credit to the master, aristotle for one refused to attribute anything specifically to Pythagoras as an individual and only discussed the work of the Pythagoreans as a group. One of the most important characteristics of the Pythagorean order was that it maintained that the pursuit of philosophical and mathematical studies was a basis for the conduct of life. Indeed, the philosophy and mathematics are said to have been coined by Pythagoras. From this love of knowledge came many achievements and it has been customarily said that the Pythagoreans discovered most of the material in the first two books of Euclids Elements. The reason it is not clear exactly what either Thales or Pythagoras actually did is that almost no documentation has survived. The only evidence comes from traditions recorded in such as Proclus’ commentary on Euclid written centuries later. Some of these works, such as Aristotle’s commentary on the Pythagoreans, are themselves only known from a few surviving fragments
Greek mathematics
–
Statue of Euclid in the
Oxford University Museum of Natural History
Greek mathematics
–
An illustration of
Euclid 's proof of the
Pythagorean Theorem
Greek mathematics
–
The
Antikythera mechanism, an ancient mechanical calculator.
32.
Pythagoreanism
–
Later revivals of Pythagorean doctrines led to what is now called Neopythagoreanism or Neoplatonism. Pythagorean ideas exercised an influence on Aristotle, and Plato. According to tradition, pythagoreanism developed at some point into two schools of thought, the mathēmatikoi and the akousmatikoi. There is the inner and outer circle John Burnet noted Lastly, we have one admitted instance of a philosophic guild, that of the Pythagoreans. And it will be found that the hypothesis, if it is to be called by that name, of a regular organisation of scientific activity will alone explain all the facts. The development of doctrine in the hands of Thales, Anaximander, according to Iamblichus in The life of Pythagoras, by Thomas Taylor There were also two forms of philosophy, for the two genera of those that pursued it, the Acusmatici and the Mathematici. The latter are acknowledged to be Pythagoreans by the rest but the Mathematici do not admit that the Acusmatici derived their instructions from Pythagoras, memory was the most valued faculty. All these auditions were of three kinds, some signifying what a thing is, others what it especially is, others what ought or ought not to be done. By musical sounds alone unaccompanied with words they healed the passions of the soul and certain diseases, enchanting in reality and it is probable that from hence this name epode, i. e. enchantment, came to be generally used. Each of these he corrected through the rule of virtue, attempering them through appropriate melodies, therefore its function is none of what are called ‘parts of virtue’, for it is better than all of them and the end produced is always better than the knowledge that produces it. Nor is every virtue of the soul in that way a function, nor is success, for if it is to be productive, different ones will produce different things, as the skill of building produces a house. However, intelligence is a part of virtue and of success, according to historians like Thomas Gale, Thomas Taler, or Cantor, Archytas became the head of the school, about a century after the murder of Pythagoras. According to August Böckh, who cites Nicomachus, Philolaus was the successor of Pythagoras, and according to Cicero, Philolaus was teacher of Archytas of Tarentum. According to the historians from the Stanford Encyclopedia of Philosophy, Philolaus and Eurytus are identified by Aristoxenus as teachers of the last generation of Pythagoreans, a Echecrates is mentioned by Aristoxenus as a student of Philolaus and Eurytus. The mathēmatikoi were supposed to have extended and developed the more mathematical, the mathēmatikoi did think that the akousmatikoi were Pythagorean, but felt that their own group was more representative of Pythagoras. Commentary from Sir William Smith, Dictionary of Greek and Roman Biography, Aristotle states the fundamental maxim of the Pythagoreans in various forms. According to Philolaus, number is the dominant and self-produced bond of the continuance of things. But number has two forms, the even and the odd, and a third, resulting from the mixture of the two, the even-odd and this third species is one itself, for it is both even and odd
Pythagoreanism
–
Bust of
Pythagoras,
Musei Capitolini,
Rome.
Pythagoreanism
–
Pythagoreans celebrate sunrise by
Fyodor Bronnikov
Pythagoreanism
–
Excerpt from
Philolaus Pythagoras book, (Charles Peter Mason, 1870)
33.
Greek numerals
–
Greek numerals are a system of writing numbers using the letters of the Greek alphabet. These alphabetic numerals are known as Ionic or Ionian numerals, Milesian numerals. In modern Greece, they are used for ordinal numbers. For ordinary cardinal numbers, however, Greece uses Arabic numerals, attic numerals, which were later adopted as the basis for Roman numerals, were the first alphabetic set. They were acrophonic, derived from the first letters of the names of the numbers represented and they ran =1, =5, =10, =100, =1000, and =10000. 50,500,5000, and 50000 were represented by the letter with minuscule powers of ten written in the top right corner, the same system was used outside of Attica, but the symbols varied with the local alphabets, in Boeotia, was 1000. The present system probably developed around Miletus in Ionia, 19th-century classicists placed its development in the 3rd century BC, the occasion of its first widespread use. The present system uses the 24 letters adopted by Euclid as well as three Phoenician and Ionic ones that were not carried over, digamma, koppa, and sampi. The position of characters within the numbering system imply that the first two were still in use while the third was not. Greek numerals are decimal, based on powers of 10, the units from 1 to 9 are assigned to the first nine letters of the old Ionic alphabet from alpha to theta. Each multiple of one hundred from 100 to 900 was then assigned its own separate letter as well and this alphabetic system operates on the additive principle in which the numeric values of the letters are added together to obtain the total. For example,241 was represented as, in ancient and medieval manuscripts, these numerals were eventually distinguished from letters using overbars, α, β, γ, etc. In medieval manuscripts of the Book of Revelation, the number of the Beast 666 is written as χξϛ, although the Greek alphabet began with only majuscule forms, surviving papyrus manuscripts from Egypt show that uncial and cursive minuscule forms began early. These new letter forms sometimes replaced the ones, especially in the case of the obscure numerals. The old Q-shaped koppa began to be broken up and simplified, the numeral for 6 changed several times. During antiquity, the letter form of digamma came to be avoided in favor of a special numerical one. By the Byzantine era, the letter was known as episemon and this eventually merged with the sigma-tau ligature stigma. In modern Greek, a number of changes have been made
Greek numerals
–
Numeral systems
Greek numerals
–
A
Constantinopolitan map of the British Isles from
Ptolemy 's
Geography (c. 1300), using Greek numerals for its
graticule: 52–63°N of the
equator and 6–33°E from Ptolemy's
Prime Meridian at the
Fortunate Isles.
34.
Archimedes
–
Archimedes of Syracuse was a Greek mathematician, physicist, engineer, inventor, and astronomer. Although few details of his life are known, he is regarded as one of the scientists in classical antiquity. He was also one of the first to apply mathematics to physical phenomena, founding hydrostatics and statics and he is credited with designing innovative machines, such as his screw pump, compound pulleys, and defensive war machines to protect his native Syracuse from invasion. Archimedes died during the Siege of Syracuse when he was killed by a Roman soldier despite orders that he should not be harmed. Cicero describes visiting the tomb of Archimedes, which was surmounted by a sphere and a cylinder, unlike his inventions, the mathematical writings of Archimedes were little known in antiquity. Archimedes was born c.287 BC in the city of Syracuse, Sicily, at that time a self-governing colony in Magna Graecia. The date of birth is based on a statement by the Byzantine Greek historian John Tzetzes that Archimedes lived for 75 years, in The Sand Reckoner, Archimedes gives his fathers name as Phidias, an astronomer about whom nothing is known. Plutarch wrote in his Parallel Lives that Archimedes was related to King Hiero II, a biography of Archimedes was written by his friend Heracleides but this work has been lost, leaving the details of his life obscure. It is unknown, for instance, whether he married or had children. During his youth, Archimedes may have studied in Alexandria, Egypt and he referred to Conon of Samos as his friend, while two of his works have introductions addressed to Eratosthenes. Archimedes died c.212 BC during the Second Punic War, according to the popular account given by Plutarch, Archimedes was contemplating a mathematical diagram when the city was captured. A Roman soldier commanded him to come and meet General Marcellus but he declined, the soldier was enraged by this, and killed Archimedes with his sword. Plutarch also gives an account of the death of Archimedes which suggests that he may have been killed while attempting to surrender to a Roman soldier. According to this story, Archimedes was carrying mathematical instruments, and was killed because the thought that they were valuable items. General Marcellus was reportedly angered by the death of Archimedes, as he considered him a valuable asset and had ordered that he not be harmed. Marcellus called Archimedes a geometrical Briareus, the last words attributed to Archimedes are Do not disturb my circles, a reference to the circles in the mathematical drawing that he was supposedly studying when disturbed by the Roman soldier. This quote is given in Latin as Noli turbare circulos meos. The phrase is given in Katharevousa Greek as μὴ μου τοὺς κύκλους τάραττε
Archimedes
–
Archimedes Thoughtful by
Fetti (1620)
Archimedes
–
Cicero Discovering the Tomb of Archimedes by
Benjamin West (1805)
Archimedes
–
Artistic interpretation of Archimedes' mirror used to burn Roman ships. Painting by
Giulio Parigi.
Archimedes
–
A sphere has 2/3 the volume and surface area of its circumscribing cylinder including its bases. A
sphere and
cylinder were placed on the tomb of Archimedes at his request. (see also:
Equiareal map)
35.
Diophantus
–
Diophantus of Alexandria, sometimes called the father of algebra, was an Alexandrian Greek mathematician and the author of a series of books called Arithmetica, many of which are now lost. These texts deal with solving algebraic equations and this led to tremendous advances in number theory, and the study of Diophantine equations and of Diophantine approximations remain important areas of mathematical research. Diophantus coined the term παρισότης to refer to an approximate equality and this term was rendered as adaequalitas in Latin, and became the technique of adequality developed by Pierre de Fermat to find maxima for functions and tangent lines to curves. Diophantus was the first Greek mathematician who recognized fractions as numbers, thus he allowed positive rational numbers for the coefficients, in modern use, Diophantine equations are usually algebraic equations with integer coefficients, for which integer solutions are sought. Diophantus also made advances in mathematical notation, little is known about the life of Diophantus. He lived in Alexandria, Egypt, probably from between AD200 and 214 to 284 or 298, much of our knowledge of the life of Diophantus is derived from a 5th-century Greek anthology of number games and puzzles created by Metrodorus. One of the states, Here lies Diophantus, the wonder behold. Alas, the child of master and sage After attaining half the measure of his fathers life chill fate took him. After consoling his fate by the science of numbers for four years and this puzzle implies that Diophantus age x can be expressed as x = x/6 + x/12 + x/7 +5 + x/2 +4 which gives x a value of 84 years. However, the accuracy of the information cannot be independently confirmed, the Arithmetica is the major work of Diophantus and the most prominent work on algebra in Greek mathematics. It is a collection of problems giving numerical solutions of both determinate and indeterminate equations, of the original thirteen books of which Arithmetica consisted only six have survived, though there are some who believe that four Arab books discovered in 1968 are also by Diophantus. Some Diophantine problems from Arithmetica have been found in Arabic sources and it should be mentioned here that Diophantus never used general methods in his solutions. Hermann Hankel, renowned German mathematician made the following remark regarding Diophantus, “Our author not the slightest trace of a general, comprehensive method is discernible, each problem calls for some special method which refuses to work even for the most closely related problems. The portion of the Greek Arithmetica that survived, however, was, like all ancient Greek texts transmitted to the modern world, copied by. In addition, some portion of the Arithmetica probably survived in the Arab tradition. ”Arithmetica was first translated from Greek into Latin by Bombelli in 1570, however, Bombelli borrowed many of the problems for his own book Algebra. The editio princeps of Arithmetica was published in 1575 by Xylander, the best known Latin translation of Arithmetica was made by Bachet in 1621 and became the first Latin edition that was widely available. Pierre de Fermat owned a copy, studied it, and made notes in the margins. I have a marvelous proof of this proposition which this margin is too narrow to contain. ”Fermats proof was never found
Diophantus
–
Title page of the 1621 edition of Diophantus' Arithmetica, translated into
Latin by
Claude Gaspard Bachet de Méziriac.
Diophantus
–
Problem II.8 in the Arithmetica (edition of 1670), annotated with Fermat's comment which became
Fermat's Last Theorem.
36.
Counting rods
–
Counting rods are small bars, typically 3–14 cm long, that were used by mathematicians for calculation in ancient China, Japan, Korea, and Vietnam. They are placed horizontally or vertically to represent any integer or rational number. The written forms based on them are called rod numerals and they are a true positional numeral system with digits for 1–9 and a blank for 0, from the Warring states period to the 16th century. Counting rods were used by ancient Chinese for more two thousand years. In 1954, forty-odd counting rods of the Warring States period were found in Zuǒjiāgōngshān Chu Grave No.15 in Changsha, in 1973, archeologists unearthed a number of wood scripts from a Han dynasty tomb in Hubei. On one of the scripts was written, “当利二月定算”. This is one of the earliest examples of using counting rod numerals in writing, in 1976, a bundle of Western Han counting rods made of bones was unearthed from Qianyang County in Shaanxi. The use of counting rods must predate it, Laozi said a good calculator doesnt use counting rods, the Book of Han recorded, they calculate with bamboo, diameter one fen, length six cun, arranged into a hexagonal bundle of two hundred seventy one pieces. At first calculating rods were round in section, but by the time of the Sui dynasty triangular rods were used to represent positive numbers. After the abacus flourished, counting rods were abandoned except in Japan, counting rods represent digits by the number of rods, and the perpendicular rod represents five. To avoid confusion, vertical and horizontal forms are alternately used, generally, vertical rod numbers are used for the position for the units, hundreds, ten thousands, etc. while horizontal rod numbers are used for the tens, thousands, hundred thousands etc. It is written in Sunzi Suanjing that one is vertical, ten is horizontal, red rods represent positive numbers and black rods represent negative numbers. Ancient Chinese clearly understood negative numbers and zero, though they had no symbol for the latter, later, a go stone was sometimes used to represent zero. This alternation of vertical and horizontal rod numeral form is important to understanding written transcription of rod numerals on manuscripts correctly. In the same manuscript,405 was transcribed as, with a space in between for obvious reasons, and could in no way be interpreted as 45. In other words, transcribed rod numerals may not be positional, the value of a number depends on its physical position on the counting board. A9 at the rightmost position on the stands for 9. Moving the batch of rods representing 9 to the one position gives 9 or 90
Counting rods
–
Numeral systems
Counting rods
–
Yang Hui (Pascal's)
triangle, as depicted by
Zhu Shijie in 1303, using rod numerals.
Counting rods
–
rod numeral place value from Yongle Encyclopedia: 71,824
Counting rods
–
Japanese counting board with grids
37.
Nine Chapters on the Mathematical Art
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The Nine Chapters on the Mathematical Art is a Chinese mathematics book, composed by several generations of scholars from the 10th–2nd century BCE, its latest stage being from the 2nd century CE. This book is one of the earliest surviving texts from China. Entries in the book usually take the form of a statement of a problem, followed by the statement of the solution, and these were commented on by Liu Hui in the 3rd century. The method of chapter 7 was not found in Europe until the 13th century, there is also the mathematical proof given in the treatise for the Pythagorean theorem. The influence of The Nine Chapters greatly assisted the development of ancient mathematics in the regions of Korea and its influence on mathematical thought in China persisted until the Qing Dynasty era. Liu Hui wrote a detailed commentary on this book in 263. Lius commentary is of great mathematical interest in its own right, the Nine Chapters is an anonymous work, and its origins are not clear. This is no longer the case, the Suàn shù shū or writings on reckoning is an ancient Chinese text on mathematics approximately seven thousand characters in length, written on 190 bamboo strips. It was discovered together with writings in 1983 when archaeologists opened a tomb in Hubei province. It is among the corpus of known as the Zhangjiashan Han bamboo texts. From documentary evidence this tomb is known to have closed in 186 BCE. While its relationship to the Nine Chapters is still under discussion by scholars, the Zhoubi Suanjing, a mathematics and astronomy text, was also compiled during the Han, and was even mentioned as a school of mathematics in and around 180 CE by Cai Yong. Contents of The Nine Chapters are as follows, 方田 Fangtian - Bounding fields, areas of fields of various shapes, manipulation of vulgar fractions. Liu Huis commentary includes a method for calculation of π and the value of 3.14159. 粟米 Sumi - Millet and rice, exchange of commodities at different rates, pricing. Distribution of commodities and money at proportional rates, deriving arithmetic and geometric sums, division by mixed numbers, extraction of square and cube roots, diameter of sphere, perimeter and diameter of circle. 商功 Shanggong - Figuring for construction, volumes of solids of various shapes. 盈不足 Yingbuzu - Excess and deficit, linear problems solved using the principle known later in the West as the rule of false position
Nine Chapters on the Mathematical Art
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A page of The Nine Chapters on the Mathematical Art
Nine Chapters on the Mathematical Art
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History
38.
Liu Hui
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Liu Hui was an Ancient Chinese mathematician. He lived in the state of Cao Wei during the Three Kingdoms period of Chinese history, in 263, he edited and published a book with solutions to mathematical problems presented in the famous Chinese book of mathematics known as The Nine Chapters on the Mathematical Art. He was a descendant of the Marquis of Zixiang of the Han dynasty and he completed his commentary to the Nine Chapters in the year 263. He probably visited Luoyang, and measured the suns shadow, along with Zu Chongzhi, Liu Hui was known as one of the greatest mathematicians of ancient China. Liu Hui expressed all of his results in the form of decimal fractions. Liu provided commentary on a proof of a theorem identical to the Pythagorean theorem. In the field of plane areas and solid figures, Liu Hui was one of the greatest contributors to empirical solid geometry, for example, he found that a wedge with rectangular base and both sides sloping could be broken down into a pyramid and a tetrahedral wedge. He also found that a wedge with trapezoid base and both sides sloping could be made to give two tetrahedral wedges separated by a pyramid, in his commentaries on the Nine Chapters, he presented, An algorithm for calculation of pi in the comments to chapter 1. He calculated pi to 3.141024 < π <3.142074 with a 192 sided polygon, Archimedes used a circumscribed 96-polygon to obtain the inequality π <227, and then used an inscribed 96-gon to obtain the inequality 22371 < π. Liu Hui used only one inscribed 96-gon to obtain his π inequality, but he commented that 3.142074 was too large, and picked the first three digits of π =3.141024 ~3.14 and put it in fraction form π =15750. He later invented a method and obtained π =3.1416. Nine Chapters had used the value 3 for π, but Zhang Heng had previously estimated pi to the root of 10. Cavalieris principle to find the volume of a cylinder and the intersection of two perpendicular cylinders although this work was finished by Zu Chongzhi and Zu Gengzhi. Lius commentaries often include explanations why some methods work and why others do not, although his commentary was a great contribution, some answers had slight errors which was later corrected by the Tang mathematician and Taoist believer Li Chunfeng. Liu Hui also presented, in an appendix of 263 AD called Haidao Suanjing or The Sea Island Mathematical Manual, several problems related to surveying. This book contained many practical problems of geometry, including the measurement of the heights of Chinese pagoda towers and this smaller work outlined instructions on how to measure distances and heights with tall surveyors poles and horizontal bars fixed at right angles to them. Liu Huis information about surveying was known to his contemporaries as well, the cartographer and state minister Pei Xiu outlined the advancements of cartography, surveying, and mathematics up until his time. This included the first use of a grid and graduated scale for accurate measurement of distances on representative terrain maps
Liu Hui
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Survey of sea island
39.
0 (number)
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0 is both a number and the numerical digit used to represent that number in numerals. The number 0 fulfills a role in mathematics as the additive identity of the integers, real numbers. As a digit,0 is used as a placeholder in place value systems, names for the number 0 in English include zero, nought or naught, nil, or—in contexts where at least one adjacent digit distinguishes it from the letter O—oh or o. Informal or slang terms for zero include zilch and zip, ought and aught, as well as cipher, have also been used historically. The word zero came into the English language via French zéro from Italian zero, in pre-Islamic time the word ṣifr had the meaning empty. Sifr evolved to mean zero when it was used to translate śūnya from India, the first known English use of zero was in 1598. The Italian mathematician Fibonacci, who grew up in North Africa and is credited with introducing the system to Europe. This became zefiro in Italian, and was contracted to zero in Venetian. The Italian word zefiro was already in existence and may have influenced the spelling when transcribing Arabic ṣifr, modern usage There are different words used for the number or concept of zero depending on the context. For the simple notion of lacking, the words nothing and none are often used, sometimes the words nought, naught and aught are used. Several sports have specific words for zero, such as nil in football, love in tennis and it is often called oh in the context of telephone numbers. Slang words for zero include zip, zilch, nada, duck egg and goose egg are also slang for zero. Ancient Egyptian numerals were base 10 and they used hieroglyphs for the digits and were not positional. By 1740 BC, the Egyptians had a symbol for zero in accounting texts. The symbol nfr, meaning beautiful, was used to indicate the base level in drawings of tombs and pyramids. By the middle of the 2nd millennium BC, the Babylonian mathematics had a sophisticated sexagesimal positional numeral system, the lack of a positional value was indicated by a space between sexagesimal numerals. By 300 BC, a symbol was co-opted as a placeholder in the same Babylonian system. In a tablet unearthed at Kish, the scribe Bêl-bân-aplu wrote his zeros with three hooks, rather than two slanted wedges, the Babylonian placeholder was not a true zero because it was not used alone
0 (number)
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Example of the early Greek symbol for zero (lower right corner) from a 2nd-century papyrus
0 (number)
0 (number)
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The number 605 in Khmer numerals, from the Sambor inscription (
Saka era 605 corresponds to AD 683). The earliest known material use of zero as a decimal figure.
0 (number)
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The back of Olmec stela C from
Tres Zapotes, the second oldest
Long Count date discovered. The numerals 7.16.6.16.18 translate to September, 32 BC (Julian). The glyphs surrounding the date are thought to be one of the few surviving examples of
Epi-Olmec script.
40.
Aryabhata
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Aryabhata or Aryabhata I was the first of the major mathematician-astronomers from the classical age of Indian mathematics and Indian astronomy. His works include the Āryabhaṭīya and the Arya-siddhanta, furthermore, in most instances Aryabhatta would not fit the metre either. Aryabhata mentions in the Aryabhatiya that it was composed 3,600 years into the Kali Yuga and this corresponds to 499 CE, and implies that he was born in 476. Aryabhata called himself a native of Kusumapura or Pataliputra, Bhāskara I describes Aryabhata as āśmakīya, one belonging to the Aśmaka country. During the Buddhas time, a branch of the Aśmaka people settled in the region between the Narmada and Godavari rivers in central India. It has been claimed that the aśmaka where Aryabhata originated may be the present day Kodungallur which was the capital city of Thiruvanchikkulam of ancient Kerala. This is based on the belief that Koṭuṅṅallūr was earlier known as Koṭum-Kal-l-ūr, however, K. Chandra Hari has argued for the Kerala hypothesis on the basis of astronomical evidence. Aryabhata mentions Lanka on several occasions in the Aryabhatiya, but his Lanka is an abstraction and it is fairly certain that, at some point, he went to Kusumapura for advanced studies and lived there for some time. Both Hindu and Buddhist tradition, as well as Bhāskara I, identify Kusumapura as Pāṭaliputra, Aryabhata is also reputed to have set up an observatory at the Sun temple in Taregana, Bihar. Aryabhata is the author of treatises on mathematics and astronomy. His major work, Aryabhatiya, a compendium of mathematics and astronomy, was referred to in the Indian mathematical literature and has survived to modern times. The mathematical part of the Aryabhatiya covers arithmetic, algebra, plane trigonometry and it also contains continued fractions, quadratic equations, sums-of-power series, and a table of sines. This work appears to be based on the older Surya Siddhanta and uses the midnight-day reckoning, a third text, which may have survived in the Arabic translation, is Al ntf or Al-nanf. It claims that it is a translation by Aryabhata, but the Sanskrit name of work is not known. Probably dating from the 9th century, it is mentioned by the Persian scholar and chronicler of India, direct details of Aryabhatas work are known only from the Aryabhatiya. The name Aryabhatiya is due to later commentators, Aryabhata himself may not have given it a name. His disciple Bhaskara I calls it Ashmakatantra and it is also occasionally referred to as Arya-shatas-aShTa, because there are 108 verses in the text. It is written in the terse style typical of sutra literature
Aryabhata
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Statue of Aryabhata on the grounds of
IUCAA,
Pune. As there is no known information regarding his appearance, any image of Aryabhata originates from an artist's conception.
Aryabhata
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India's first satellite named after Aryabhata
41.
Brahmagupta
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Brahmagupta was an Indian mathematician and astronomer. He is the author of two works on mathematics and astronomy, the Brāhmasphuṭasiddhānta, a theoretical treatise, and the Khaṇḍakhādyaka. According to his commentators, Brahmagupta was a native of Bhinmal, Brahmagupta was the first to give rules to compute with zero. The texts composed by Brahmagupta were composed in verse in Sanskrit. As no proofs are given, it is not known how Brahmaguptas results were derived, Brahmagupta was born in 598 CE according to his own statement. He lived in Bhillamala during the reign of the Chapa dynasty ruler Vyagrahamukha and he was the son of Jishnugupta. He was a Shaivite by religion, even though most scholars assume that Brahmagupta was born in Bhillamala, there is no conclusive evidence for it. However, he lived and worked there for a part of his life. Prithudaka Svamin, a commentator, called him Bhillamalacharya, the teacher from Bhillamala. Sociologist G. S. Ghurye believed that he might have been from the Multan region or the Abu region and it was also a center of learning for mathematics and astronomy. Brahmagupta became an astronomer of the Brahmapaksha school, in the year 628, at an age of 30, he composed Brāhmasphuṭasiddhānta which is believed to be a revised version of the received siddhanta of the Brahmapaksha school. Scholars state that he has incorported a great deal of originality to his revision, the book consists of 24 chapters with 1008 verses in the ārya meter. Later, Brahmagupta moved to Ujjain, which was also a centre for astronomy. At the mature age of 67, he composed his next well known work Khanda-khādyaka and he is believed to have died in Ujjain. Brahmagupta had a plethora of criticism directed towards the work of rival astronomers, the division was primarily about the application of mathematics to the physical world, rather than about the mathematics itself. In Brahmaguptas case, the disagreements stemmed largely from the choice of astronomical parameters, the historian of science George Sarton called him one of the greatest scientists of his race and the greatest of his time. Brahmaguptas mathematical advances were carried on to further extent by Bhāskara II, a descendant in Ujjain. Prithudaka Svamin wrote commentaries on both of his works, rendering difficult verses into simpler language and adding illustrations, lalla and Bhattotpala in the 8th and 9th centuries wrote commentaries on the Khanda-khadyaka
Brahmagupta
42.
Division by 0
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In mathematics, division by zero is division where the divisor is zero. Such a division can be expressed as a/0 where a is the dividend. In ordinary arithmetic, the expression has no meaning, as there is no number which, multiplied by 0, gives a, and so division by zero is undefined. Since any number multiplied by zero is zero, the expression 0/0 also has no defined value, in computing, a program error may result from an attempt to divide by zero. When division is explained at the elementary level, it is often considered as splitting a set of objects into equal parts. As an example, consider having ten cookies, and these cookies are to be distributed equally to five people at a table, each person would receive 105 =2 cookies. Similarly, if there are ten cookies, and only one person at the table, so, for dividing by zero, what is the number of cookies that each person receives when 10 cookies are evenly distributed amongst 0 people at a table. Certain words can be pinpointed in the question to highlight the problem, the problem with this question is the when. There is no way to evenly distribute 10 cookies to nobody, in mathematical jargon, a set of 10 items cannot be partitioned into 0 subsets. So 100, at least in elementary arithmetic, is said to be either meaningless, similar problems occur if one has 0 cookies and 0 people, but this time the problem is in the phrase the number. A partition is possible, but since the partition has 0 parts, vacuously every set in our partition has a number of elements, be it 0,2,5. If there are, say,5 cookies and 2 people, in any integer partition of a 5-set into 2 parts, one of the parts of the partition will have more elements than the other. But the problem with 5 cookies and 2 people can be solved by cutting one cookie in half, the problem with 5 cookies and 0 people cannot be solved in any way that preserves the meaning of divides. Another way of looking at division by zero is that division can always be checked using multiplication. Considering the 10/0 example above, setting x = 10/0, if x equals ten divided by zero, then x times zero equals ten, but there is no x that, when multiplied by zero, gives ten. If instead of x=10/0 we have x=0/0, then every x satisfies the question what number x, multiplied by zero, the Brahmasphutasiddhanta of Brahmagupta is the earliest known text to treat zero as a number in its own right and to define operations involving zero. The author could not explain division by zero in his texts, according to Brahmagupta, A positive or negative number when divided by zero is a fraction with the zero as denominator. Zero divided by a negative or positive number is zero or is expressed as a fraction with zero as numerator
Division by 0
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Most calculators, such as this
Texas Instruments TI-86, will halt execution and display an error message when the user or a running program attempts to divide by zero.
Division by 0
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The function y = 1/ x. As x approaches 0 from the right, y approaches infinity. As x approaches 0 from the left, y approaches negative infinity.
43.
Syriac Christianity
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With a history going back to the 1st century AD, Syriac Christianity is, in modern times, represented by denominations primarily in the Middle East, Asia Minor and in Kerala, India. Christianity began in the Middle East in Jerusalem among Jewish Aramaic-speaking Semitic peoples of Judah and it quickly spread, initially to other Semitic peoples, in Parthian-ruled Assyria and Mesopotamia, Roman-ruled Syria, Phoenicia, southern and eastern Asia Minor, and northwestern Persia and Malta. Adherents sometimes identify as Syriacs or Syrians, Syriac Christian heritage is transmitted through various Neo Aramaic dialects of old Aramaic. Unlike the Greek Christian culture, Assyrian Christian culture borrowed much from early Rabbinic Judaism, Antioch was the political capital of this culture, and was the seat of the Patriarchs of the church. However, Antioch was heavily Hellenized, and the Assyrian cities of Edessa, Nisibis and this split owed just as much to the politics of the day as it did to theological orthodoxy. Ctesiphon, which was at the also the Sassanid capital. After the Council of Chalcedon in 451, many Syriac Christians within the Roman Empire rebelled against its decisions, the Patriarchate of Antioch was then divided between a Chalcedonian and non-Chalcedonian communion. The Chalcedonians were often labelled Melkites, while their opponents were labelled as Monophysites and Jacobites, the Maronite Church found itself caught between the two, but claims to have always remained faithful to the Catholic Church and in communion with the bishop of Rome, the Pope. The church has persisted as an entity under Islamic rule. The community was one of those granted autonomy in governing itself in religious, in the 19th century many left for other parts of Christendom, creating a substantial diaspora. Over time, some groups within each of these branches have entered into communion with the Church of Rome, when they lost Assyria itself to the Parthian Empire, they retained the name Syria but only applied it to what had been Aramea, which they still retained. The Neo-Assyrian kingdom of Osroene was the first Christian kingdom in history, in 431 A. D. the Council of Ephesus declared Nestorianism to be a heresy. The Nestorian priests, who were persecuted in the Byzantine Empire, sought refuge in Mesopotamia where the Church of the East was dominant, there was a synthesis between the Assyrian Church and Nestorian doctrine. From there they spread Christianity to Persia, India, China and this was the beginning of the Nestorian Church, the eastern branch of Syrian Christianity. The western branch, the Jacobite Church, appeared after the Council of Chalcedon condemned Monophysitism in 451 A. D and these people are in fact ethnic Assyrians originating from the Assyrian homeland in northern Iraq. The older Assyrian designation has almost completely replaced the word Nestorian, however, the word Nestorian continues to be used in some Western academic literature. The word Syrian has become ambiguous in English since it can refer now to a citizen of Syria regardless of ethnicity, in Arabic, however, the word for a citizen of Syria has a different form from the traditional word for an ethnic Assyrian/Syrian. The Maronites in Lebanon are divided between those who claim Lebanese-Phoenician national identity and those who claim Arab national identity, the Maronite Church, a West Syrian Rite Eastern Catholic Church
Syriac Christianity
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Celebration at a
Syriac Orthodox monastery in
Mosul,
Ottoman Syria, early 20th century.
44.
Stepped Reckoner
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The step reckoner was a digital mechanical calculator invented by the German mathematician Gottfried Wilhelm Leibniz around 1672 and completed in 1694. The name comes from the translation of the German term for its mechanism, Staffelwalze. It was the first calculator that could perform all four arithmetic operations, two prototypes were built, today only one survives in the National Library of Lower Saxony in Hanover, Germany. Several later replicas are on display, such as the one at the Deutsches Museum, despite the mechanical flaws of the stepped reckoner, it suggested possibilities to future calculator builders. The operating mechanism, invented by Leibniz, called the cylinder or Leibniz wheel, was used in many calculating machines for 200 years. The stepped reckoner was based on a mechanism that Leibniz invented. It is unclear how many different variants of the calculator were made, some sources, such as the drawing to the right, show a 12 digit version. This section describes the surviving 16 digit prototype in Hanover, the machine is about 67 cm long, made of polished brass and steel, mounted in an oak case. It consists of two attached parallel parts, a section to the rear, which can hold 16 decimal digits. The input section has 8 dials with knobs to set the number, a telephone-like dial to the right to set the multiplier digit. The result appears in the 16 windows on the rear accumulator section, there is also a tens-carry indicator and a control to set the machine to zero. Sequences of these operations can be performed on the number in the accumulator, for example it can calculate roots by a series of divisions and additions, Leibniz got the idea for a calculating machine in 1672 in Paris, from a pedometer. Later he learned about Blaise Pascals machine when he read Pascals Pensees and he concentrated on expanding Pascals mechanism so it could multiply and divide. He presented a model to the Royal Society of London on 1 February 1673. In English, a number consisting of a series of figures and his first preliminary brass machine was built between 1674 and 1685. His so-called older machine was built between 1686 and 1694, the younger machine, the surviving machine, was built from 1690 to 1720. In 1775 the younger machine was sent to the University of Göttingen for repair, in 1876 a crew of workmen found it in an attic room of a university building in Göttingen. It was returned to Hanover in 1880, from 1894 to 1896 Artur Burkhardt, founder of a major German calculator company restored it, and it has been kept at the Niedersächsische Landesbibliothek ever since
Stepped Reckoner
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Replica of Leibniz's Stepped Reckoner in the Deutsches Museum. — Gottfried Leibniz
45.
Codex Vigilanus
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The Codex Vigilanus or Codex Albeldensis is an illuminated compilation of various historical documents accounting for a period extending from antiquity to the 10th century in Hispania. The compilers were three monks of the Riojan monastery of San Martín de Albelda, Vigila, after whom it was named and who was the illustrator, Serracino, his friend, the first compilation was finished in 881, but was updated up to 976. The original manuscript is preserved in the library of El Escorial, at the time of its compilation, Albelda was the cultural and intellectual centre of the Kingdom of Pamplona. The Codex contains, among other pieces of information, the first mention and representation of Arabic numerals in the West. They were introduced by the Arabs into Spain around 900, the illuminations are stylistically unique, combining Visigothic, Mozarabic, and Carolingian elements. The interlace patterns and the drapery show Carolingian, as well Italo-Byzantine, the use of animals as decoration and for supporting columns also parallels contemporary Frankish usage. More Carolingian and less Byzantine influence is evident in the Codex Aemilianensis, interlace Decoration and the Influence of the North on Mozarabic Illumination. The Art Bulletin, Vol.42, No.3, zoomorphic Decoration and the Problem of the Sources of Mozarabic Illumination. Art Journal, Vol.25, No.1, salvus of Albelda and Frontier Monasticism in Tenth-Century Navarre. Latin text and Spanish translation of the Chronicon Albeldense
Codex Vigilanus
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The scribes: Serracino, Vigila, and García as drawn by Vigila.
Codex Vigilanus
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The people of Toledo.
Codex Vigilanus
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The first Arabic numerals in the West.
46.
Fibonacci
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Fibonacci was an Italian mathematician, considered to be the most talented Western mathematician of the Middle Ages. The name he is called, Fibonacci, is short for figlio di Bonacci and he is also known as Leonardo Bonacci, Leonardo of Pisa, Leonardo Pisano Bigollo. Fibonacci popularized the Hindu–Arabic numeral system in the Western World primarily through his composition in 1202 of Liber Abaci and he also introduced Europe to the sequence of Fibonacci numbers, which he used as an example in Liber Abaci. Fibonacci was born around 1175 to Guglielmo Bonacci, a wealthy Italian merchant and, by some accounts, Guglielmo directed a trading post in Bugia, a port in the Almohad dynastys sultanate in North Africa. Fibonacci travelled with him as a boy, and it was in Bugia that he learned about the Hindu–Arabic numeral system. Fibonacci travelled extensively around the Mediterranean coast, meeting with many merchants and he soon realised the many advantages of the Hindu-Arabic system. In 1202, he completed the Liber Abaci which popularized Hindu–Arabic numerals in Europe, Fibonacci became a guest of Emperor Frederick II, who enjoyed mathematics and science. The date of Fibonaccis death is not known, but it has estimated to be between 1240 and 1250, most likely in Pisa. In the Liber Abaci, Fibonacci introduced the so-called modus Indorum, the book advocated numeration with the digits 0–9 and place value. The book was well-received throughout educated Europe and had a impact on European thought. No copies of the 1202 edition are known to exist, the book also discusses irrational numbers and prime numbers. Liber Abaci posed, and solved, a problem involving the growth of a population of rabbits based on idealized assumptions, the solution, generation by generation, was a sequence of numbers later known as Fibonacci numbers. Although Fibonaccis Liber Abaci contains the earliest known description of the sequence outside of India, in the Fibonacci sequence of numbers, each number is the sum of the previous two numbers. Fibonacci began the sequence not with 0,1,1,2, as modern mathematicians do but with 1,1,2, etc. He carried the calculation up to the place, that is 233. Fibonacci did not speak about the ratio as the limit of the ratio of consecutive numbers in this sequence. In the 19th century, a statue of Fibonacci was constructed and raised in Pisa, today it is located in the western gallery of the Camposanto, historical cemetery on the Piazza dei Miracoli. There are many mathematical concepts named after Fibonacci because of a connection to the Fibonacci numbers, examples include the Brahmagupta–Fibonacci identity, the Fibonacci search technique, and the Pisano period
Fibonacci
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Portrait by an unknown artist
Fibonacci
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A page of Fibonacci's
Liber Abaci from the
Biblioteca Nazionale di Firenze showing (in box on right) the Fibonacci sequence with the position in the sequence labeled in Roman numerals and the value in Hindu-Arabic numerals.
Fibonacci
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19th century statue of Fibonacci in Camposanto, Pisa.
47.
Liber Abaci
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Liber Abaci is a historic book on arithmetic by Leonardo of Pisa, known later by his nickname Fibonacci. Liber Abaci was among the first Western books to describe Hindu–Arabic numbers traditionally described as Arabic Numerals, by addressing the applications of both commercial tradesmen and mathematicians, it contributed to convincing the public of the superiority of the Hindu–Arabic numeral system. The title of Liber Abaci means The Book of Calculation, the second version of Liber Abaci was dedicated to Michael Scot in 1227 CE. No versions of the original 1202 CE book have been found, the first section introduces the Hindu–Arabic numeral system, including methods for converting between different representation systems. The second section presents examples from commerce, such as conversions of currency and measurements, another example in this chapter, describing the growth of a population of rabbits, was the origin of the Fibonacci sequence for which the author is most famous today. The fourth section derives approximations, both numerical and geometrical, of irrational numbers such as square roots, the book also includes proofs in Euclidean geometry. Fibonaccis method of solving algebraic equations shows the influence of the early 10th-century Egyptian mathematician Abū Kāmil Shujāʿ ibn Aslam, there are three key differences between Fibonaccis notation and modern fraction notation. We generally write a fraction to the right of the number to which it is added. Fibonacci instead would write the same fraction to the left, i. e.132. That is, b a d c = a c + b c d, the notation was read from right to left. For example, 29/30 could be written as 124235 and this can be viewed as a form of mixed radix notation, and was very convenient for dealing with traditional systems of weights, measures, and currency. Sigler also points out an instance where Fibonacci uses composite fractions in which all denominators are 10, Fibonacci sometimes wrote several fractions next to each other, representing a sum of the given fractions. For instance, 1/3+1/4 = 7/12, so a notation like 14132 would represent the number that would now more commonly be written as the mixed number 2712, or simply the improper fraction 3112. Notation of this form can be distinguished from sequences of numerators and denominators sharing a fraction bar by the break in the bar. If all numerators are 1 in a written in this form, and all denominators are different from each other. This notation was also combined with the composite fraction notation. The complexity of this notation allows numbers to be written in different ways. In the Liber Abaci, Fibonacci says the following introducing the Modus Indorum or the method of the Indians, today known as Hindu–Arabic numerals or traditionally, just Arabic numerals
Liber Abaci
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A page of the Liber Abaci from the
Biblioteca Nazionale di Firenze showing (on right) the numbers of the
Fibonacci sequence.
48.
Zero
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Gonzalo Barrios, known by his gamertag ZeRo, is a Chilean professional Super Smash Bros. player. He is considered one of the best Super Smash Bros. for Wii U players in the world, ZeRo had a record-breaking 56-tournament winning streak in 2015, in which he also won several high-profile tournaments like EVO2015 and The Big House 5. In the past has also been a top ranked Super Smash Bros and he mains Diddy Kong in Super Smash Bros. for Wii U, and mained Pit in Project M, Meta Knight in Brawl, and Fox in Melee. Barrios has had ZeRo as his gamertag since 2005 and he has been playing Smash since Super Smash Bros. in 1999. He started to travel and play in Melee tournaments in a local Akiba Game Store in early 2007, ZeRo quit Smash completely until December 2010 and then focused only on Brawl. ZeRo placed second in Brawl at Apex 2014, losing to Nairo and was the champion of the Smash Wii U at Apex 2015 and he defeated Dabuz, who was playing Captain Olimar, in the finals. ZeRo qualified for the MLG Anaheim 2014 championship bracket and finished 17th, ZeRo was ranked in 2014 by Melee it on Me as the 35th best Melee player in the world. On November 25,2014, he criticized Diddy Kongs repetitive playstyle in Smash Wii U, however, ZeRo later retracted this statement, and now says Diddy Kong is his favorite character to play with. Barrios streams on twitch. tv Mondays through Thursdays at 1PM PST, ZeRo attributes much of his success to training with Mew2King. He was sponsored by CLASH Tournaments for part of 2014 until resigning in November, e-Sports Earnings estimates that ZeRo has earned a career total of US$31,484.22 from tournaments. ZeRo was considered the third best Brawl player in the world by CLASH Tournaments in the 2014 SSBBRank, in early 2014, he picked up Melee again, as well as the mod Project M. Since November 2014 when he placed third at Skys Smash 4 Invitational, ZeRo won EVO2015, the largest Smash for Wii U. tournament at the time beating Mr. R in the finals. On the August 1,2015, Team SoloMid announced ZeRo as the player in their Super Smash Bros. division. At The Big House 5, ZeRo was knocked into Losers bracket very early and he qualified for the top 32. He led off by defeating Sonic main StaticManny, Sheik main top Melee Captain Falcon main Wizzrobe, Mario main Ally, in top 8, he beat Mario main 2Scoops Zenyou, Rosalina and Luma main Raquayza07, and beat Mario main ANTi in a very close set. In Losers finals, he handily beat Dabuz, who now plays Rosalina and Luma, after easily beating Nairo the first set of Grand Finals, ZeRo clutched out a win in the second set, 3-2, to win TBH5, despite suffering an early upset. In MLG World Finals 2015, ZeRo defeated Ness main Nakat, StaticManny, Ally, there, Nairo took two sets off of ZeRo, ending ZeRos reign at 56 tournaments. Kotaku named ZeRo, The Smash Bros, champ as one of the gamers of 2015, honoring his 56 tournament streak
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49.
Middle Ages
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In the history of Europe, the Middle Ages or Medieval Period lasted from the 5th to the 15th century. It began with the fall of the Western Roman Empire and merged into the Renaissance, the Middle Ages is the middle period of the three traditional divisions of Western history, classical antiquity, the medieval period, and the modern period. The medieval period is subdivided into the Early, High. Population decline, counterurbanisation, invasion, and movement of peoples, the large-scale movements of the Migration Period, including various Germanic peoples, formed new kingdoms in what remained of the Western Roman Empire. In the seventh century, North Africa and the Middle East—once part of the Byzantine Empire—came under the rule of the Umayyad Caliphate, although there were substantial changes in society and political structures, the break with classical antiquity was not complete. The still-sizeable Byzantine Empire survived in the east and remained a major power, the empires law code, the Corpus Juris Civilis or Code of Justinian, was rediscovered in Northern Italy in 1070 and became widely admired later in the Middle Ages. In the West, most kingdoms incorporated the few extant Roman institutions, monasteries were founded as campaigns to Christianise pagan Europe continued. The Franks, under the Carolingian dynasty, briefly established the Carolingian Empire during the later 8th, the Crusades, first preached in 1095, were military attempts by Western European Christians to regain control of the Holy Land from Muslims. Kings became the heads of centralised nation states, reducing crime and violence, intellectual life was marked by scholasticism, a philosophy that emphasised joining faith to reason, and by the founding of universities. Controversy, heresy, and the Western Schism within the Catholic Church paralleled the conflict, civil strife. Cultural and technological developments transformed European society, concluding the Late Middle Ages, the Middle Ages is one of the three major periods in the most enduring scheme for analysing European history, classical civilisation, or Antiquity, the Middle Ages, and the Modern Period. Medieval writers divided history into periods such as the Six Ages or the Four Empires, when referring to their own times, they spoke of them as being modern. In the 1330s, the humanist and poet Petrarch referred to pre-Christian times as antiqua, leonardo Bruni was the first historian to use tripartite periodisation in his History of the Florentine People. Bruni and later argued that Italy had recovered since Petrarchs time. The Middle Ages first appears in Latin in 1469 as media tempestas or middle season, in early usage, there were many variants, including medium aevum, or middle age, first recorded in 1604, and media saecula, or middle ages, first recorded in 1625. The alternative term medieval derives from medium aevum, tripartite periodisation became standard after the German 17th-century historian Christoph Cellarius divided history into three periods, Ancient, Medieval, and Modern. The most commonly given starting point for the Middle Ages is 476, for Europe as a whole,1500 is often considered to be the end of the Middle Ages, but there is no universally agreed upon end date. English historians often use the Battle of Bosworth Field in 1485 to mark the end of the period
Middle Ages
–
The
Cross of Mathilde, a
crux gemmata made for
Mathilde, Abbess of Essen (973–1011), who is shown kneeling before the Virgin and Child in the
enamel plaque. The body of Christ is slightly later. Probably made in
Cologne or
Essen, the cross demonstrates several medieval techniques:
cast figurative sculpture,
filigree, enamelling, gem polishing and setting, and the reuse of Classical
cameos and
engraved gems.
Middle Ages
–
A late Roman statue
depicting the four Tetrarchs, now in
Venice
Middle Ages
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Coin of
Theodoric
Middle Ages
–
Mosaic showing
Justinian with
the bishop of
Ravenna, bodyguards, and courtiers
50.
Liberal arts
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Grammar, logic, and rhetoric were the core liberal arts, while arithmetic, geometry, the theory of music, and astronomy also played a part in education. In modern times, liberal arts education term that can be interpreted in different ways and it can refer to academic subjects such as literature, philosophy, mathematics, and social and physical sciences, or it can also refer to overall studies in a liberal arts degree program. For example, Harvard University offers a Bachelor of Arts degree, for both interpretations, the term generally refers to matters not relating to the professional, vocational, or technical curriculum. The four scientific artes – music, arithmetic, geometry and astronomy – were known from the time of Boethius onwards as the Quadrivium. After the 9th century, the three arts of the humanities – grammar, logic, and rhetoric – were classed as well as the Trivium. It was in that form that the seven liberal arts were studied in the medieval Western university. During the Middle Ages, logic gradually came to take predominance over the parts of the Trivium. In the Renaissance, the Italian humanists and their Northern counterparts, despite in many respects continuing the traditions of the Middle Ages, the ideal of a liberal arts, or humanistic education grounded in classical languages and literature, persisted until the middle of the twentieth century. Some subsections of the arts are in the trivium—the verbal arts, grammar, logic, and rhetoric, and in the quadrivium—the numerical arts, arithmetic, geometry, music. Analyzing and interpreting information is also included, the liberal arts education at the secondary school level prepares the student for higher education at a university. They are thus meant for the more academically minded students, in addition to the usual curriculum, students of a liberal arts education often study Latin and Ancient Greek. Some liberal arts education provide general education, others have a specific focus, today, a number of other areas of specialization exist, such as gymnasiums specializing in economics, technology or domestic sciences. In some countries, there is a notion of progymnasium, which is equivalent to beginning classes of the full gymnasium, here, the prefix pro is equivalent to pre. In the United States, liberal arts colleges are schools emphasizing undergraduate study in the liberal arts, in most parts of Europe, liberal arts education is deeply rooted. In Germany, Austria and countries influenced by their education system, the term is not to be mixed up with some modern educational concepts that use a similar wording. Educational institutions that see themselves in that tradition are often a Gymnasium and they aim at providing their pupils with comprehensive education in order to form personality with regard to a pupils own humanity as well as his/her innate intellectual skills. Going back to the tradition of the liberal arts in Europe, education in the above sense was freed from scholastic thinking. In particular, Wilhelm von Humboldt played a key role in that regard, universities encourage students to do so and offer respective opportunities, but do not make such activities part of the universitys curriculum
Liberal arts
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Philosophia et septem artes liberales, The seven liberal arts – Picture from the
Hortus deliciarum of
Herrad of Landsberg (12th century)
Liberal arts
–
Page from
Marriage of Mercury and Philology
51.
Greek Numerals
–
Greek numerals are a system of writing numbers using the letters of the Greek alphabet. These alphabetic numerals are known as Ionic or Ionian numerals, Milesian numerals. In modern Greece, they are used for ordinal numbers. For ordinary cardinal numbers, however, Greece uses Arabic numerals, attic numerals, which were later adopted as the basis for Roman numerals, were the first alphabetic set. They were acrophonic, derived from the first letters of the names of the numbers represented and they ran =1, =5, =10, =100, =1000, and =10000. 50,500,5000, and 50000 were represented by the letter with minuscule powers of ten written in the top right corner, the same system was used outside of Attica, but the symbols varied with the local alphabets, in Boeotia, was 1000. The present system probably developed around Miletus in Ionia, 19th-century classicists placed its development in the 3rd century BC, the occasion of its first widespread use. The present system uses the 24 letters adopted by Euclid as well as three Phoenician and Ionic ones that were not carried over, digamma, koppa, and sampi. The position of characters within the numbering system imply that the first two were still in use while the third was not. Greek numerals are decimal, based on powers of 10, the units from 1 to 9 are assigned to the first nine letters of the old Ionic alphabet from alpha to theta. Each multiple of one hundred from 100 to 900 was then assigned its own separate letter as well and this alphabetic system operates on the additive principle in which the numeric values of the letters are added together to obtain the total. For example,241 was represented as, in ancient and medieval manuscripts, these numerals were eventually distinguished from letters using overbars, α, β, γ, etc. In medieval manuscripts of the Book of Revelation, the number of the Beast 666 is written as χξϛ, although the Greek alphabet began with only majuscule forms, surviving papyrus manuscripts from Egypt show that uncial and cursive minuscule forms began early. These new letter forms sometimes replaced the ones, especially in the case of the obscure numerals. The old Q-shaped koppa began to be broken up and simplified, the numeral for 6 changed several times. During antiquity, the letter form of digamma came to be avoided in favor of a special numerical one. By the Byzantine era, the letter was known as episemon and this eventually merged with the sigma-tau ligature stigma. In modern Greek, a number of changes have been made
Greek Numerals
–
Numeral systems
Greek Numerals
–
A
Constantinopolitan map of the British Isles from
Ptolemy 's
Geography (c. 1300), using Greek numerals for its
graticule: 52–63°N of the
equator and 6–33°E from Ptolemy's
Prime Meridian at the
Fortunate Isles.
Weyde, P. H. Vander (1879). "Arithmetic". The American Cyclopædia.
Weyde, P. H. Vander (1879). "
