1.
Fraction (mathematics)
–
A fraction represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight-fifths, three-quarters. A common, simple fraction consists of a non-zero denominator, displayed below that line. Denominators are also used in fractions that are not common, including mixed numerals. The picture to the right illustrates 3 4 or ¾ of a cake. Fractional numbers can also be written by using negative exponents. An integer such as the number 7 can be thought of as having an implicit denominator of one: 7 equals 7/1. Other uses for fractions are to represent ratios and to represent division. Thus 3/4 is also used to represent 3 ÷ 4. The test for a number being a rational number is that it can be written in that form. Informally, they may be distinguished by placement alone but in formal contexts they are always separated by a fraction bar. The bar may be diagonal. These marks are respectively known as the fraction slash. The denominators of English fractions are generally expressed as ordinal numbers, in the plural if the numerator is not one. When the numerator is one, it may be omitted.
Fraction (mathematics)
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A cake with one quarter (one fourth) removed. The remaining three fourths are shown. Dotted lines indicate where the cake may be cut in order to divide it into equal parts. Each fourth of the cake is denoted by the fraction ¼.
2.
Numerator
–
A fraction represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, eight-fifths, three-quarters. A common, simple fraction consists of an integer numerator displayed above a line, a non-zero integer denominator, displayed below that line. Denominators are also used in fractions that are not common, including compound fractions, complex fractions, mixed numerals. The picture to the right illustrates 3/4 of a cake. Fractional numbers can also be written by using decimals, percent signs, or negative exponents. An integer such as the number 7 can be thought of as having an implicit denominator of one: 7 equals 7/1. Other uses for fractions are to represent division. Thus 3/4 is also used to represent the ratio 3:4 and the division 3 ÷ 4. The test for a number being a rational number is that it can be written in that form. Informally, in formal contexts they are always separated by a fraction bar. The bar may be horizontal, oblique, or diagonal. These marks are respectively known as the horizontal bar, the slash or stroke, the fraction slash. The denominators of English fractions are generally expressed in the plural if the numerator is not one. When the numerator is one, it may be omitted.
Numerator
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A cake with one quarter (one fourth) removed. The remaining three fourths are shown. Dotted lines indicate where the cake may be cut in order to divide it into equal parts. Each fourth of the cake is denoted by the fraction ¼.
3.
Denominator
–
A fraction represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, eight-fifths, three-quarters. A common, simple fraction consists of an integer numerator displayed above a line, a non-zero integer denominator, displayed below that line. Denominators are also used in fractions that are not common, including compound fractions, complex fractions, mixed numerals. The picture to the right illustrates 3/4 of a cake. Fractional numbers can also be written by using decimals, percent signs, or negative exponents. An integer such as the number 7 can be thought of as having an implicit denominator of one: 7 equals 7/1. Other uses for fractions are to represent division. Thus 3/4 is also used to represent the ratio 3:4 and the division 3 ÷ 4. The test for a number being a rational number is that it can be written in that form. Informally, in formal contexts they are always separated by a fraction bar. The bar may be horizontal, oblique, or diagonal. These marks are respectively known as the horizontal bar, the slash or stroke, the fraction slash. The denominators of English fractions are generally expressed in the plural if the numerator is not one. When the numerator is one, it may be omitted.
Denominator
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A cake with one quarter (one fourth) removed. The remaining three fourths are shown. Dotted lines indicate where the cake may be cut in order to divide it into equal parts. Each fourth of the cake is denoted by the fraction ¼.
4.
Integer
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An integer is a number that can be written without a fractional component. For example, 21, − 2048 are integers, while 9.75, 5 1⁄2, √ 2 are not. The set of integers consists of the natural numbers, also called their additive inverses. This is often denoted by a boldface Z or bold standing for the German word Zahlen. ℤ is a subset of the sets of rational and real numbers and, like the natural numbers, is countably infinite. The integers form the smallest group and the smallest ring containing the natural numbers. In algebraic number theory, the integers are sometimes called rational integers to distinguish them from the more general algebraic integers. In fact, the integers are the algebraic integers that are also rational numbers. However, with the inclusion of the negative natural numbers, and, importantly, 0, Z is also closed under subtraction. This universal property, namely to be an initial object in the category of rings, characterizes the ring Z. Z is not closed under division, since the quotient of two integers, need not be an integer. Although the natural numbers are closed under exponentiation, the integers are not. The following lists some of the basic properties of multiplication for any integers a, c. In the language of abstract algebra, the first five properties listed above for addition say that Z under addition is an abelian group. In fact, Z under addition is the only infinite cyclic group, in the sense that any infinite cyclic group is isomorphic to Z.
Integer
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Algebraic structure → Group theory Group theory
5.
Positive number
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In mathematics, the concept of sign originates from the property of every non-zero real number to be positive or negative. Zero itself is signless, although in some contexts it makes sense to consider a signed zero. Also, the word "sign" can indicate aspects of mathematical objects that resemble negativity, such as the sign of a permutation. Every number has multiple attributes. A real number is said to be positive if its value is negative if it is less than zero. The attribute of being negative is called the sign of the number. Zero itself is not considered to have a sign. Also, signs are not defined for complex numbers, although the argument generalizes it in some sense. In common notation, the sign of a number is often denoted by placing a plus sign or a minus sign before the number. − 3 denotes "negative three". When no plus or sign is given, the default interpretation is that a number is positive. Likewise, "+" associates with positivity. In this context, it makes sense to write − = +3. Any non-zero number can be changed to a positive one using the absolute function. For example, the absolute value of 3 are both equal to 3.
Positive number
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The plus and minus symbols are used to show the sign of a number.
6.
Rational number
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Since q may be equal to 1, every integer is a rational number. Moreover, any repeating or terminating decimal represents a rational number. These statements hold true just for base 10, but also for any other integer base. A real number, not rational is called irrational. Irrational numbers include √ 2, π, φ. The decimal expansion of an irrational number continues without repeating. Since the set of real numbers is uncountable, almost all real numbers are irrational. In abstract algebra, the rational numbers together with certain operations of multiplication form the archetypical field of characteristic zero. As such, it is characterized as having no proper subfield or, alternatively, being the field of fractions for the ring of integers. The algebraic closure of Q is the field of algebraic numbers. In mathematical analysis, the rational numbers form a dense subset of the real numbers. The real numbers can be constructed by completion using Cauchy sequences, Dedekind cuts, or infinite decimals. Zero divided by any other integer equals zero; therefore, zero is a rational number. The term rational to the set Q refers to the fact that a rational number represents a ratio of two integers. In mathematics, "rational" is often used as a noun "rational number".
Rational number
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A diagram showing a representation of the equivalent classes of pairs of integers
7.
Summand
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Addition is one of the four basic operations of arithmetic, with the others being subtraction, multiplication and division. The addition of two whole numbers is the total amount of those quantities combined. In the picture on the right, there is a combination of three apples and two apples together, making a total of five apples. This observation is equivalent to the mathematical expression "+ 2 = 5" i.e. "3 add 2 is equal to 5". Besides counting fruits, addition can also represent combining physical objects. In arithmetic, rules for addition involving negative numbers have been devised amongst others. In algebra, addition is studied more abstractly. Addition has important properties. 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 multiplication. Performing addition is one of the simplest numerical tasks. In primary education, students are taught to add numbers in the decimal system, progressively tackling more difficult problems. Mechanical aids range from the ancient abacus to the modern computer, where research on the most efficient implementations of addition continues to this day. Addition is written using the plus sign" +" in infix notation. The result is expressed with an equals sign.
Summand
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Part of Charles Babbage's Difference Engine including the addition and carry mechanisms
Summand
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3 + 2 = 5 with apples, a popular choice in textbooks
Summand
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A circular slide rule
8.
Vulgar fraction
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A fraction represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, eight-fifths, three-quarters. A common, simple fraction consists of an integer numerator displayed above a line, a non-zero integer denominator, displayed below that line. Denominators are also used in fractions that are not common, including compound fractions, complex fractions, mixed numerals. The picture to the right illustrates 3/4 of a cake. Fractional numbers can also be written by using decimals, percent signs, or negative exponents. An integer such as the number 7 can be thought of as having an implicit denominator of one: 7 equals 7/1. Other uses for fractions are to represent division. Thus 3/4 is also used to represent the ratio 3:4 and the division 3 ÷ 4. The test for a number being a rational number is that it can be written in that form. Informally, in formal contexts they are always separated by a fraction bar. The bar may be horizontal, oblique, or diagonal. These marks are respectively known as the horizontal bar, the slash or stroke, the fraction slash. The denominators of English fractions are generally expressed in the plural if the numerator is not one. When the numerator is one, it may be omitted.
Vulgar fraction
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A cake with one quarter (one fourth) removed. The remaining three fourths are shown. Dotted lines indicate where the cake may be cut in order to divide it into equal parts. Each fourth of the cake is denoted by the fraction ¼.
9.
Decimal
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This article aims to be an accessible introduction. For the mathematical definition, see Decimal representation. The decimal system has ten as its base. It is the numerical base most widely used by modern civilizations. Other fractions have repeating decimal representations, whereas irrational numbers have infinite non-repeating decimal representations. Decimal notation is the writing of numbers in a base 10 system. Examples are Brahmi numerals, Greek numerals, Hebrew numerals, Chinese numerals, as well as the Hindu-Arabic numerals used by speakers of many European languages. Roman numerals have secondary symbols for half these values. Brahmi numerals have symbols for the nine numbers 1 -- 9, another for 1000. Chinese numerals have symbols for additional symbols for powers of 10, which in modern usage reach 1072. Decimal systems include a zero and use symbols for the ten values to represent any number, no matter how large or how small. Positional notation uses positions for each power of ten: units, tens, thousands, etc.. There were at least two presumably independent sources of decimal systems in ancient civilization: the Chinese counting rod system and the Hindu-Arabic numeral system. Ten is the number, the count of thumbs on both hands. The English digit as well as its translation in many languages is also the anatomical term for fingers and toes.
Decimal
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The world's earliest decimal multiplication table was made from bamboo slips, dating from 305 BC, during the Warring States period in China.
Decimal
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Numeral systems
Decimal
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Ten fingers on two hands, the possible starting point of the decimal counting.
Decimal
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Diagram of the world's earliest decimal multiplication table (c. 305 BC) from the Warring States period
10.
Number theory
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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 foundational place in the discipline. Number theorists study prime numbers as well as the properties of objects made out of integers or defined as generalizations of the integers. Integers can be considered either in themselves or as solutions to equations. One may also study real numbers in relation to rational numbers, e.g. as approximated by the latter. The older term for number theory is arithmetic. By the early twentieth century, it had been superseded by "number theory". In particular, arithmetical is preferred as an adjective to number-theoretic. 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, subtracted such that the width..." It is not known what these applications may have been, or whether there could have been any; Babylonian astronomy, for example, truly flowered only later. It has been suggested instead that the table was a source of numerical examples for school problems. While Babylonian theory -- or what survives of Babylonian mathematics that can be called thus -- consists of this striking fragment, Babylonian algebra was well developed. Late Neoplatonic sources state that Pythagoras learned mathematics from the Babylonians. Much earlier sources state that Thales and Pythagoras traveled and studied in Egypt.
Number theory
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A Lehmer sieve, which is a primitive digital computer once used for finding primes and solving simple Diophantine equations.
Number theory
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The Plimpton 322 tablet
Number theory
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Title page of the 1621 edition of Diophantus' Arithmetica, translated into Latin by Claude Gaspard Bachet de Méziriac.
Number theory
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Leonhard Euler
11.
History of mathematics
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Before the worldwide spread of knowledge, written examples of mathematical developments have come to light only in a few locales. The most mathematical texts available are Plimpton 322, the Moscow Mathematical Papyrus. All of these texts concern the so-called Pythagorean theorem, which seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry. Greek mathematics greatly expanded the matter of mathematics. Chinese mathematics made early contributions, including a system. Islamic mathematics, in turn, developed and expanded the mathematics known to these civilizations. Arabic texts on mathematics were then translated into Latin, which led to further development of mathematics in medieval Europe. From ancient times through the Middle Ages, periods of mathematical discovery were often followed by centuries of stagnation. The origins of mathematical thought lie in the concepts of form. Modern studies of animal cognition have shown that these concepts are not unique to humans. Such concepts would have been part of everyday life in hunter-gatherer societies. Prehistoric artifacts discovered in Africa, dated 20,000 years old or more suggest early attempts to quantify time. Common interpretations are that the Ishango bone shows either the earliest known demonstration of sequences of a six-month calendar. Predynastic Egyptians of the 5th millennium BC pictorially represented geometric designs. Undisputed mathematical documents are from Babylonian and dynastic Egyptian sources.
History of mathematics
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A proof from Euclid 's Elements, widely considered the most influential textbook of all time.
History of mathematics
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The Babylonian mathematical tablet Plimpton 322, dated to 1800 BC.
History of mathematics
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Image of Problem 14 from the Moscow Mathematical Papyrus. The problem includes a diagram indicating the dimensions of the truncated pyramid.
History of mathematics
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One of the oldest surviving fragments of Euclid's Elements, found at Oxyrhynchus and dated to circa AD 100. The diagram accompanies Book II, Proposition 5.
12.
Egyptian numerals
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The system of ancient Egyptian numerals was used in Ancient Egypt around 3000 BC until the early first millennium AD. The hieratic form of numerals stressed an exact finite series notation, ciphered one to one onto the Egyptian alphabet. For instance, a stone carving from Karnak shows the number 4622 as Egyptian hieroglyphs could be written in both directions. Rational numbers could also be expressed, but only as sums of unit fractions, i.e. sums of reciprocals of positive integers, except for 2⁄3 and 3⁄4. Instances of numerals written in hieratic can be found as far back as the Early Dynastic Period. The Old Kingdom Abusir Papyri are a particularly important corpus of texts that utilize hieratic numerals. A large number like 9999 could thus be written with only four signs—combining the signs for 9000, 900, 90, 9—as opposed to 36 hieroglyphs. 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 clearly 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, repeated as Roman numerals practiced. 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; this process continued into Demotic as well. 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 vowel remains uncertain: Ancient Egypt Egyptian language Egyptian mathematics Allen, James Paul.
Egyptian numerals
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Numeral systems
13.
Eye of Horus
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The Eye of Horus is an ancient Egyptian symbol of protection, royal power and good health. The eye is personified in the Wadjet. The Eye of Horus is similar to the Eye of Ra, which represents many of the same concepts. Wadjet was one of the earliest of Egyptian deities who later became associated with other goddesses such as Bast, Sekhmet, Mut, Hathor. She was the tutelary deity of the major Delta shrine the "per-nu" was under her protection. Hathor is also depicted with this eye. Funerary amulets were often made in the shape of the Eye of Horus. The Wadjet or Eye of Horus is "the central element" of seven "gold, faience, lapis lazuli" bracelets found on the mummy of Shoshenq II. The Wedjat "was intended to ward off evil. Ancient Middle-Eastern sailors would frequently paint the symbol on the bow of their vessel to ensure safe sea travel. Horus was the Egyptian sky god, usually depicted as a falcon, most likely a lanner or peregrine falcon. His right eye was associated with Ra. The symbol represents the marking around the eye of the falcon, including the "teardrop" marking sometimes found below the eye. Left eye, sometimes represented the moon and the god Djehuti. In one myth, when Set and Horus were fighting after Osiris's death, Set gouged out Horus's left eye.
Eye of Horus
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An Eye of Horus or Wedjat pendant
Eye of Horus
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The Wedjat, later called The Eye of Horus
Eye of Horus
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The crown of a Nubian king
Eye of Horus
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Wooden case decorated with bronze, silver, ivory and gold
14.
Egyptian mathematics
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Ancient Egyptian mathematics is the mathematics, 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. These labels appear to have been used as tags for grave goods and some are inscribed with numbers. 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. 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 them work with these fractions. The Egyptian Mathematical Leather Roll for instance is a table of unit fractions which are expressed as sums of other unit fractions. The Rhind Mathematical Papyrus and some of the other texts contain 2 n tables. These tables allowed the scribes to rewrite any fraction of the form 1 n as a sum of unit fractions. In the worker's village of Deir el-Medina several ostraca have been found that record volumes of dirt removed while quarrying the tombs.
Egyptian mathematics
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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
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Image of Problem 14 from the Moscow Mathematical Papyrus. The problem includes a diagram indicating the dimensions of the truncated pyramid.
15.
Middle Kingdom of Egypt
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Some scholars also include the Thirteenth Dynasty of Egypt wholly into this period as well, in which case the Middle Kingdom would finish c. 1650, while others only include it until Merneferre Ay c. 1700 BC, last king of this dynasty to be attested in both Upper and Lower Egypt. During the Middle Kingdom period, Osiris became the most important deity in popular religion. The period comprises the 11th Dynasty, which ruled from Thebes and the 12th Dynasty onwards, centered on el-Lisht. After the collapse of the Old Kingdom, Egypt entered a period of decentralization called the First Intermediate Period. Towards the end of this period, two rival dynasties, known as the Tenth and Eleventh, fought for power over the entire country. The Theban 11th Dynasty only ruled southern Egypt to the Tenth Nome of Upper Egypt. To the north, Lower Egypt was ruled by the 10th Dynasty from Herakleopolis. The struggle was to be concluded by Mentuhotep II, who ascended the Theban throne in 2055 B.C. During Mentuhotep II's fourteenth he took advantage of a revolt in the Thinite Nome to launch an attack on Herakleopolis, which met little resistance. For this reason, Mentuhotep II is regarded as the founder of the Middle Kingdom. Mentuhotep II commanded military campaigns south far as the Second Cataract in Nubia, which had gained its independence during the First Intermediate Period. He also restored Egyptian hegemony over the Sinai region, lost since the end of the Old Kingdom. Mentuhotep III was succeeded by Mentuhotep IV, whose name significantly is omitted from all Egyptian king lists. The Turin Papyrus claims after Mentuhotep III came "seven kingless years."
Middle Kingdom of Egypt
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An Osiride statue of the first pharaoh of the Middle Kingdom, Mentuhotep II
Middle Kingdom of Egypt
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The head of a statue of Senusret I.
Middle Kingdom of Egypt
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Statue head of Senusret III
16.
Moscow Mathematical Papyrus
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Golenishchev bought the papyrus in 1892 or 1893 in Thebes. It later entered the collection of the Pushkin State Museum of Fine Arts in Moscow, where it remains today. It is a mathematical papyrus along with the Rhind Mathematical Papyrus. The Moscow Mathematical Papyrus is older than the Rhind Mathematical Papyrus, while the latter is the larger of the two. The papyrus is well known for some of its geometry problems. Problems 10 and 14 compute a surface area and the volume of a frustum respectively. The remaining problems are more common in nature. 3 are ship's part problems. Aha problems involve finding unknown quantities if the sum of the part of it are given. The Rhind Mathematical Papyrus also contains four of these type of problems. Problems 1, 25 of the Moscow Papyrus are Aha problems. For problem 19 asks one to calculate a quantity taken 1 and 1/2 times and added to 4 to make 10. The pefsu number is mentioned in many offering lists. Calculate 1/2 of the result will be 2 1/2 Take this 2 1/2 four times The result is 10. Then you say to him: "Behold!
Moscow Mathematical Papyrus
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14th problem of the Moscow Mathematical Papyrus (V. Struve, 1930)
Moscow Mathematical Papyrus
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The neutrality of this article is disputed. Relevant discussion may be found on the talk page. Please do not remove this message until the dispute is resolved. (July 2015)
17.
Kahun Papyrus
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The Kahun Papyri are a collection of ancient Egyptian texts discussing administrative, mathematical and medical topics. Its many fragments were discovered by Flinders Petrie in 1889 and are kept at the University College London. This collection of papyri is one of the largest ever found. Most of the texts are dated to ca. 1825 BC, to the reign of Amenemhat III. In general the collection spans the Middle Kingdom of Egypt. The texts span a variety of topics: Business papers of the cult of Senusret II Hymns to king Senusret III. The Kahun Gynaecological Papyrus, which deals with gynaecological illnesses and conditions. Legon PlanetMath: Kahun Papyrus and Arithmetic Progressions
Kahun Papyrus
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Fragments of the Kahun Papyrus on veterinary medicine
18.
Rhind Mathematical Papyrus
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The Rhind Mathematical Papyrus is one of the best known examples of Egyptian mathematics. It dates to around BC. It is one of the two well-known Mathematical Papyri along with the Moscow Mathematical Papyrus. The Rhind Papyrus is larger than the Moscow Mathematical Papyrus, while the latter is older than the former. The Rhind Mathematical Papyrus dates to the Second Intermediate Period of Egypt. It was copied from a now-lost text from the reign of king Amenemhat III. Written in the hieratic script, this Egyptian manuscript consists of multiple parts which in total make it over 5m long. The papyrus began to be mathematically translated in the late 19th century. The mathematical aspect remains incomplete in several respects. The Ahmose writes this copy. A handful of these stand out. A more recent overview of the Rhind Papyrus was published by Robins and Shute. The first part of the Rhind papyrus consists of a collection of 21 arithmetic and 20 algebraic problems. The problems start out followed by completion problems and more involved linear equations. The first part of the papyrus is taken up by the 2/n table.
Rhind Mathematical Papyrus
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A portion of the Rhind Papyrus
Rhind Mathematical Papyrus
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Building
19.
Ahmes
Ahmes
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A portion of the Rhind Mathematical Papyrus
20.
Second Intermediate Period
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It is best known as the period when the Hyksos made their appearance in Egypt and whose reign comprised the Fifteenth dynasty. The Twelfth Dynasty of Egypt came to an end at the end of the 19th century BCE with the death of Queen Sobekneferu. Retaining the seat of the twelfth dynasty, the thirteenth dynasty ruled from Itjtawy near Memphis and Lisht, just south of the apex of the Nile Delta. The Thirteenth Dynasty is notable for the accession of the first formally recognised Semitic-speaking king, Khendjer. The Fifteenth Dynasty dates approximately from 1650 to 1550 BC. Known rulers of the Fifteenth Dynasty are as follows: Salitis Sakir-Har Khyan Apophis, c. 1590? BC-1550 BC Khamudi, c. 1550-1540 BC The Fifteenth Dynasty of Egypt was the first Hyksos dynasty, ruled from Avaris, without control of the entire land. The Hyksos preferred to stay in northern Egypt since they infiltrated from the north-east. The names and order of kings is uncertain. The Turin King list indicates that there were six Hyksos kings, with an obscure Khamudi listed as the final king of the Fifteenth Dynasty. This is also supported by the fact that this king employed a third prenomen during his reign: Nebkhepeshre. Apepi likely employed several different prenomens throughout various periods of his reign. The Sixteenth Dynasty ruled the Theban region in Upper Egypt for 70 years. Of the two chief versions of Manetho's Aegyptiaca, Dynasty XVI is described as Theban.
Second Intermediate Period
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The political situation in the Second Intermediate Period of Egypt (circa 1650 B.C.E. — circa 1550 B.C.E.) Thebes was briefly conquered by the Hyksos circa 1580 B.C.E.
Second Intermediate Period
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Thebes (Luxor Temple pictured) was the capital of many of the Dynasty XVI pharaohs.
21.
Egyptian hieroglyphs
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Egyptian hieroglyphs were the formal writing system used in Ancient Egypt. It combined alphabetic elements, with a total of some 1,000 distinct characters. Cursive hieroglyphs were used for religious literature on papyrus and wood. Egyptian scripts are derived from hieroglyphic writing; Meroitic was a late derivation from Demotic. The system continued to be used throughout the Late Period, well as the Persian and Ptolemaic periods. Late survivals of hieroglyphic use are found well into the Roman period, extending into the 4th century AD. The decipherment of hieroglyphs would only be solved with the help of the Rosetta Stone. The hieroglyph comes from a compound of ἱερός and γλύφω, supposedly a calque of an Egyptian phrase mdw · w-nṯr "god's words". The glyphs themselves were called τὰ ἱερογλυφικὰ γράμματα "the sacred engraved letters". The hieroglyph has become a noun in English, standing for an hieroglyphic character. As used in the previous sentence, the word hieroglyphic is an adjective, but hieroglyphic has also become a noun in English, at least in non-academic usage. Hieroglyphs emerged from the preliterate artistic traditions of Egypt. For example, symbols on Gerzean pottery from c. 4000 BC have been argued to resemble hieroglyphic writing. There are around 800 hieroglyphs dating back to New Kingdom Eras. By the Greco-Roman period, there are more than 5,000.
Egyptian hieroglyphs
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A section of the Papyrus of Ani showing cursive hieroglyphs.
Egyptian hieroglyphs
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Hieroglyphs on a funerary stela in Manchester Museum
Egyptian hieroglyphs
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The Rosetta Stone in the British Museum
Egyptian hieroglyphs
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Hieroglyphs typical of the Graeco-Roman period
22.
Multiplicative inverse
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The multiplicative inverse of a a/b is b/a. Of a real number divide 1 by the number. For example, the reciprocal of 0.25 is 1 divided by 0.25, or 4. The function f that maps x to 1/x, is one of the simplest examples of a function, its own inverse. In the phrase inverse, the qualifier multiplicative is often omitted and then tacitly understood. Multiplicative inverses can be defined over mathematical domains as well as numbers. In these cases it can happen that ab ba; then "inverse" typically implies that an element is both a left and right inverse. The notation f 1 is sometimes also used for the inverse function of the function f, not in general equal to the multiplicative inverse. Only for linear maps are they strongly related. The terminology reciprocal versus inverse is not sufficient to make this distinction, since many authors prefer the opposite naming convention, probably for historical reasons. In the real numbers, zero does not have a reciprocal because no real number multiplied by 0 produces 1. The property that every element other than zero has a multiplicative inverse is part of the definition of a field, of which these are all examples. On the other hand, no integer other than 1 and 1 has an integer reciprocal, so the integers are not a field. In modular arithmetic, the modular inverse of a is also defined: it is the number x such that ax ≡ 1. This multiplicative inverse exists if and only if a and n are coprime.
Multiplicative inverse
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The reciprocal function: y = 1/ x. For every x except 0, y represents its multiplicative inverse. The graph forms a rectangular hyperbola.
23.
Dyadic rational
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These are precisely the numbers whose binary expansion is finite. The inch is customarily subdivided in dyadic rather than decimal fractions; similarly, the customary divisions of the gallon into pints are dyadic. The ancient Egyptians also used dyadic fractions in measurement, with denominators up to 64. However, the result of dividing one dyadic fraction by another is not necessarily a dyadic fraction. Addition modulo 1 forms a group; this is the Prüfer 2-group. Considering subtraction operations of the dyadic rationals gives the structure of an additive abelian group. It is called the dyadic solenoid and is an example of a solenoid group and of a protorus. The group operation on these elements multiplies any two sequences componentwise. Each element of the dyadic solenoid corresponds to a character of the dyadic rationals that maps a/2b to the complex number qba. Conversely, every character χ of the dyadic rationals corresponds to the element of the dyadic solenoid given by qi = χ. As a topological space the dyadic solenoid is a solenoid, an indecomposable continuum. The binary van der Corput sequence is an equidistributed permutation of the rational numbers. Time signatures in Western musical notation traditionally consist of dyadic fractions, although non-dyadic time signatures have been introduced by composers in the twentieth century. One example appears for Player Piano. The same is true for the majority of fixed-point datatypes, which also uses powers of two implicitly in the majority of cases.
Dyadic rational
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Dyadic rationals in the interval from 0 to 1.
24.
Prime number
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A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1, not a prime number is called a composite number. The property of being prime is called primality. A slow method of verifying the primality of a given number n is known as trial division. It consists of testing whether n is a multiple of any integer between 2 and n. Algorithms much more efficient than trial division have been devised to test the primality of large numbers. Particularly fast methods are available for numbers such as Mersenne numbers. As of January 2016, the largest known prime number has 22,338,618 decimal digits. There are infinitely many primes, as demonstrated around 300 BC. There is no known simple formula that separates prime numbers from composite numbers. However, the statistical behaviour of primes in the large, can be modelled. Many questions regarding prime numbers remain open, such as the twin prime conjecture. Such questions spurred the development of various branches of theory, focusing on analytic or algebraic aspects of numbers. Prime numbers give rise to various generalizations in mainly algebra, such as prime elements and prime ideals. A natural number is called a prime number if it has itself.
Prime number
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The number 12 is not a prime, as 12 items can be placed into 3 equal-size columns of 4 each (among other ways). 11 items cannot be all placed into several equal-size columns of more than 1 item each without some extra items leftover (a remainder). Therefore, the number 11 is a prime.
25.
Composite number
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A composite number is a positive integer that can be formed by multiplying together two smaller positive integers. Equivalently, it is a positive integer that has at least one divisor other than itself. Every positive integer is composite, the unit 1, so the composite numbers are exactly the numbers that are not prime and not a unit. For example, 14 is a composite number because it is the product of the two smaller integers 2 × 7. Likewise, the integers 3 are not composite numbers because each of them can only be divided by one and itself. Every composite number can be written as the product of two or more primes. This fact is called the fundamental theorem of arithmetic. There are several known primality tests that can determine whether a number is composite, without necessarily revealing the factorization of a composite input. One way to classify composite numbers is by counting the number of prime factors. A composite number with two prime factors is a 2-almost prime. A composite number with three prime factors is a sphenic number. For the latter μ = 2 x = 1, while for the former μ = 2 x 1 = − 1. However, for prime numbers, the function also returns − μ = 1. For a number n with one or more repeated prime factors, μ = 0. If all the prime factors of a number are repeated it is called a powerful number.
Composite number
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Overview
26.
Practical number
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Practical numbers were used with the problem of representing rational numbers as Egyptian fractions. He gives a table of Egyptian fraction expansions for fractions with practical denominators. The name "practical number" is due to Srinivasan. This characterization makes it possible to determine whether a number is practical by examining its prime factorization. Every power of two is also a practical number. Practical numbers have also been shown to be analogous with prime numbers in many of their properties. If the ordered set of all divisors of the practical number n is d 1, d 2. . . In other words the ordered sequence of all divisors d 1 < d 2 <. . . < d j of a practical number has to be a complete sub-sequence. A positive integer greater than one with prime n = p 1 α 1.. .
Practical number
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Overview
27.
Semiprime
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In mathematics, a semiprime is a natural number, the product of two prime numbers. . Semiprimes that are not perfect squares are called distinct, semiprimes. By definition, semiprime numbers have no composite factors other than themselves. For example, its only factors are 1, 2, 13, 26. The total number of prime factors Ω for a semiprime n is two, by definition. A semiprime is either a square of a square-free. It is unlikely, that a way could be found to prove a larger number is a semiprime without knowing the two factors. A composite non-divisible by primes ≤ n 3 is semiprime. Various methods, such as the Goldwasser-Kilian ECPP theorem have been used to create provable, unfactored semiprimes with hundreds of digits. These are considered novelties, because it is simpler to multiply two primes together. If otherwise q are the same, φ = φ = p = p2 − p = n − p. In the RSA Factoring Challenge, RSA Security offered prizes for the factoring of several prizes were awarded. The most recent such challenge closed in 2007. Else the number can be quickly factored by Fermat's factorization method.
Semiprime
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Overview
28.
Square number
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For example, 9 is a square number, since it can be written as × 3. The equivalent exponentiation n2, usually pronounced as "n squared". The name square number comes from the name of the shape; see below. Square numbers are non-negative. Another way of saying that a integer is a square number, is that its square root is again an integer. For √ 9 = 3, so 9 is a square number. A positive integer that has no square divisors except 1 is called square-free. For a non-negative n, the nth square number is n2, with 02 = 0 being the zeroth one. The concept of square can be extended to some other number systems. Starting with 1, there are ⌊√m⌋ square numbers up to and including m, where the expression ⌊x⌋ represents the floor of the number x. Hence, a square with side length n has area n2. The expression for the square number is n2. The formula follows: n 2 = ∑ k = 1 n. So for example, 52 25 = 1 + 3 + 5 + 7 + 9. There are several recursive methods for computing square numbers.
Square number
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m = 1 2 = 1
29.
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 Western books to describe Hindu -- Arabic numbers traditionally described as "Arabic Numerals". By addressing the applications of 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 system, including methods for converting between different representation systems. The second section presents calculations of profit and interest. The fourth section derives approximations, both geometrical, of irrational numbers such as square roots. The book also includes proofs in Euclidean geometry. Fibonacci's method of solving algebraic equations shows the influence of Abū Kāmil Shujāʿ ibn Aslam. There are three key differences between Fibonacci's notation and modern notation. We generally write a fraction to the right of the whole number to which it is added, for instance 2 3 for 7/3. Fibonacci instead would write the same fraction to i.e. 1 3 2. The notation was read to left.
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.
30.
Ptolemy
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Claudius Ptolemy was a Greek writer, known as a mathematician, astronomer, geographer, astrologer, poet of a single epigram in the Greek Anthology. He lived in the city of Alexandria in the Roman province of Egypt, held Roman citizenship. Beyond that, reliable details of his life are known. His birthplace has been given as Ptolemais Hermiou in the Thebaid by the 14th-century astronomer Theodore Meliteniotes. Ptolemy wrote several scientific treatises, three of which were of importance to later Byzantine, Islamic and European science. The first is the astronomical treatise now known as the Almagest, although it was originally then known as the "Great Treatise". The second is the Geography, a thorough discussion of the geographic knowledge of the Greco-Roman world. The third is the astrological treatise in which he attempted to adapt horoscopic astrology to the natural philosophy of his day. This is sometimes more commonly known as the Tetrabiblos from the Greek meaning "Four Books" or by the Latin Quadripartitum. If, as was common, this was the emperor, citizenship would have been granted between AD 68. The astronomer would also have had a praenomen, which remains unknown. Ptolemaeus is a Greek name. It is of Homeric form. All the kings after him, until Egypt became a Roman province in 30 BC, were also Ptolemies. Abu Ma ` shar recorded a belief that a different member of this royal line "attributed it to Ptolemy".
Ptolemy
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Engraving of a crowned Ptolemy being guided by the muse Astronomy, from Margarita Philosophica by Gregor Reisch, 1508. Although Abu Ma'shar believed Ptolemy to be one of the Ptolemies who ruled Egypt after the conquest of Alexander the title ‘King Ptolemy’ is generally viewed as a mark of respect for Ptolemy's elevated standing in science.
Ptolemy
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Early Baroque artist's rendition
Ptolemy
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A 15th-century manuscript copy of the Ptolemy world map, reconstituted from Ptolemy's Geography (circa 150), indicating the countries of " Serica " and "Sinae" (China) at the extreme east, beyond the island of "Taprobane" (Sri Lanka, oversized) and the "Aurea Chersonesus" (Malay Peninsula).
Ptolemy
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Prima Europe tabula. A C15th copy of Ptolemy's map of Britain
31.
Almagest
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The Almagest is the critical source of information on ancient Greek astronomy. It has also been valuable to students of mathematics because it documents Hipparchus's work, lost. Ptolemy set up a public inscription in 147 or 148. The late N. T. Hamilton found that the version of Ptolemy's models set out in the Canopic Inscription was earlier than the version in the Almagest. Hence it can not have been completed before about 150, a century after Ptolemy began observing. The Syntaxis Mathematica or Almagest consists of thirteen sections, called books. An example illustrating how the Syntaxis was organized is given below. It is a Latin edition printed in 1515 at Venice by Petrus Lichtenstein. Then follows an explanation of chords with table of chords; observations of the obliquity of the ecliptic; and an introduction to spherical trigonometry. There is also a study of the angles made by the ecliptic with tables. Book III covers the motion of the Sun. Ptolemy begins explaining the theory of epicycles. Book VI covers solar and lunar eclipses. Books VII and VIII cover the motions of the fixed stars, including precession of the equinoxes. They also contain a catalogue of 1022 stars, described by their positions in the constellations.
Almagest
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Ptolemy's Almagest became an authoritative work for many centuries.
Almagest
Almagest
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Picture of George Trebizond's Latin translation of Almagest
32.
Babylonian mathematics
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Babylonian mathematical texts are plentiful and well edited. In respect of content there is scarcely any difference between the two groups of texts. Thus Babylonian mathematics remained constant, in character and content, for nearly 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, baked hard in an oven or by the heat of the sun. The Babylonian tablet YBC 7289 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 early second millennium BC due to the wealth of data available. There has been debate over the earliest appearance of Babylonian mathematics, with historians suggesting a range of dates between the 5th and 3rd millennia BC. Babylonian mathematics was primarily written on clay tablets in cuneiform script in the Akkadian or Sumerian languages. 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, 360 degrees in a circle. The Babylonians were able to make great advances in mathematics for two reasons. Additionally, unlike the Egyptians and Romans, the Babylonians had a true place-value system, where digits written in the left column represented larger values. The ancient Sumerians of Mesopotamia developed a complex system of metrology from 3000 BC.
Babylonian mathematics
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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...
33.
Leonardo of Pisa
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Fibonacci popularized the Hindu–Arabic numeral system to the Western World primarily through his composition in 1202 of Liber Abaci. He also introduced Europe to the sequence of Fibonacci numbers, which he used in Liber Abaci. Fibonacci was born around 1175 for Pisa. Guglielmo directed a trading post in North Africa. It was in Bugia that he learned about the Hindu -- Arabic numeral system. Fibonacci travelled extensively around the Mediterranean coast, learning about their systems of doing arithmetic. 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. It has been estimated to be between 1240 and 1250, most likely in Pisa. In the Liber Abaci, Fibonacci introduced today known as Hindu-Arabic numerals. The book advocated 0 -- 9 and place value. The book had a profound impact on European thought. No copies of the 1202 edition are known to exist. The book also discusses prime numbers.
Leonardo of Pisa
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Portrait by an unknown artist
Leonardo of Pisa
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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.
Leonardo of Pisa
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19th century statue of Fibonacci in Camposanto, Pisa.
34.
Mixed radix
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Mixed radix numeral systems are non-standard positional numeral systems in which the numerical base varies from position to position. In numeral format, the radix point is marked by a full stop or period. The base for each digit is the number of corresponding units that make up the next larger unit. As a consequence there is no base for the first digit, since here the "next larger unit" does not exist. The most familiar example of mixed radix systems is in timekeeping and calendars. Western time radices include decimal centuries, years as well as duodecimal months, trigesimal days, overlapped with base 52 weeks and septenary days. One variant uses tridecimal months, septenary days. Time is further divided by quadrivigesimal hours, then decimal fractions thereof. A mixed radix system can often benefit from a tabular summary. Hoc notations for mixed radix numeral systems are commonplace. The Maya calendar consists of several overlapping cycles of different radices. A short tzolk ` in overlaps vigesimal named days with tridecimal numbered days. A haab' consists of vigesimal days, base-52 years forming a round. In addition, a long count of vigesimal days, then vigesimal tun, k ` atun, b ` ak ` tun, etc. tracks historical dates. Mixed-radix numbers of the same base can be manipulated using a generalization of manual arithmetic algorithms.
Mixed radix
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Numeral systems
35.
Floor and ceiling functions
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In mathematics and computer science, the floor and ceiling functions map a real number to the largest previous or the smallest following integer, respectively. Carl Friedrich Gauss introduced the square bracket notation for the floor function in his third proof of quadratic reciprocity. Both notations are now used in mathematics; this article follows Iverson. The language APL uses ⌊x; other computer languages commonly use notations like entier, INT, or floor. In mathematics, it can also be written with boldface or double brackets. The ceiling function is usually denoted by ceil or ceiling in non-APL computer languages that have a notation for this function. The J Programming Language, a follow on to APL, designed to use standard keyboard symbols, uses >. for ceiling and <. for floor. In mathematics, there is another notation with reversed boldface or double brackets ] ] x x[\!x[. The fractional part is the sawtooth function, denoted by for real x and defined by the formula = x − ⌊ x ⌋. For all x, 0 ≤ < 1. HTML 4.0 uses the same names: ⌊, ⌋, ⌈, ⌉. Unicode contains codepoints for these symbols at U+2308–U+230B: ⌈x⌉, ⌊x⌋. In the following formulas, x and y are real numbers, k, m, n are integers, Z is the set of integers. Floor and ceiling may be defined by the set equations ⌊ x ⌋ = max, ⌈ x ⌉ = min. Then ⌊ x ⌋ = m and ⌈ x ⌉ = n may also be taken as the definition of floor and ceiling.
Floor and ceiling functions
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Floor function
36.
Sylvester's sequence
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In number theory, Sylvester's sequence is an integer sequence in which each member of the sequence is the product of the previous members, plus one. The few terms of the sequence are: 2, 3, 7, 43, 1807, 3263443, 10650056950807, 113423713055421844361000443. Sylvester's sequence is named after James Joseph Sylvester, who first investigated it in 1880. Values derived from this sequence have also been used to construct finite Egyptian fraction representations of 1, hard instances for online algorithms. Formally, Sylvester's sequence can be defined by the formula s n = 1 + ∏ i = 0 n − 1 s i. The product of an empty set is 1, so s0 = 2. Alternatively, one may define the sequence by the recurrence s i = i − 1 + 1, with s0 = 2. It is straightforward to show by induction that this is equivalent to the other definition. The Sylvester numbers grow doubly exponentially as a function of n. The sum of the first k terms of the infinite series provides the closest possible underestimate of 1 by any Egyptian fraction. For example, the first four terms add to 1805/1806, therefore any Egyptian fraction for a number in the open interval requires at least five terms. Alternatively, the terms of the sequence after the first can be viewed as the denominators of the greedy expansion of 1/2. This sequence provides an example showing that double-exponential growth is not enough to cause an sequence to be an irrationality sequence. Badea surveys progress related to this conjecture; see also Brown. If i < j, it follows from the definition that sj 1.
Sylvester's sequence
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Graphical demonstration of the convergence of the sum 1/2 + 1/3 + 1/7 + 1/43 +... to 1. Each row of k squares of side length 1/ k has total area 1/ k, and all the squares together exactly cover a larger square with area 1. Squares with side lengths 1/1807 or smaller are too small to see in the figure and are not shown.
37.
James Joseph Sylvester
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James Joseph Sylvester FRS was an English mathematician. He made fundamental contributions to matrix theory, invariant theory, number theory, combinatorics. At his death, he was professor at Oxford. Sylvester was born James Joseph in London, England. Abraham Joseph, was a merchant. At the age of 14, Sylvester was a student of Augustus De Morgan at the University of London. His family withdrew him from the University after he was accused of stabbing a fellow student with a knife. Subsequently he attended the Liverpool Royal Institution. Sylvester began his study of mathematics at Cambridge in 1831, where his tutor was John Hymers. For the same reason, he was unable to obtain a Smith's prize. In 1838 Sylvester became professor of natural philosophy at University College London. In 1841, he was awarded an MA by Trinity College, Dublin. One of his private pupils was Florence Nightingale. In 1872, he finally received his M.A. from Cambridge, having been denied the degrees due to his being a Jew. In 1876 Sylvester again crossed the Atlantic Ocean to become the inaugural professor of mathematics at the new Johns Hopkins University in Baltimore, Maryland.
James Joseph Sylvester
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James Joseph Sylvester
38.
Smooth number
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In number theory, a smooth number is an integer which factors completely into small prime numbers. The term seems to have been coined by Leonard Adleman. Smooth numbers are especially important in cryptography relying on factorization. The 2-smooth numbers are just the powers of 2. A positive integer is called B-smooth if none of its prime factors is greater than B. For example, 1,620 has prime factorization 22 × 34 × 5; therefore 1,620 is 5-smooth because none of its prime factors are greater than 5. Note that B does not have to be a prime factor. If the largest prime factor of a number is p then the number is B-smooth for any B ≥ p. Usually B is given as a prime, but composite numbers work as well. A number is B-smooth if and only if it is p-smooth, where p is the largest prime less than or equal to B. By using B-smooth numbers, one ensures that the base cases of this recursion are small primes, for which efficient algorithms exist. Regular numbers play a special role in Babylonian mathematics. The problem of generating these numbers efficiently has been used as a test problem for functional programming. Smooth numbers have a number of applications to cryptography. Although most applications involve cryptanalysis, the VSH function is one example of a constructive use of smoothness to obtain a provably secure design.
Smooth number
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Overview
39.
Ernest S. Croot, III
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Ernest S. Croot III is a mathematician and Professor at the School of Mathematics, Georgia Institute of Technology. He is known for contributlng to the solutlon of the cap problem. Ernest Croot attended Centre College at Danville, Kentucky, where he received a B.S. in Mathematics and a B.S. in Computer Science in 1994. In 2000, he completed a Ph.D. in Mathematics at the University of Georgia under the supervision of Andrew Granville. Croot's personal web page at Georgia Tech Mathematics Genealogy Project profile
Ernest S. Croot, III
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Ernest S. Croot III
40.
Primary pseudoperfect number
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The first primary pseudoperfect numbers are 2, 6, 42, 1806, 47058, 2214502422, 52495396602.... Then the two sequences diverge. It is unknown whether there are any odd primary pseudoperfect numbers. The prime factors of primary pseudoperfect numbers sometimes may provide solutions to Znám's problem, in which all elements of the set are prime. For instance, the prime factors of the pseudoperfect number 47058 form the solution set to Znám's problem. If a primary number N is one less than a prime number, then N × is also primary pseudoperfect. For instance, 47059 is prime, so 47058 × 47059 = 2214502422 is also primary pseudoperfect. Primary pseudoperfect numbers were first named by Butske, Jaje, Mayernik. Giuga number Anne, Premchand, "Egyptian fractions and the inheritance problem", The College Mathematics Journal, The College Mathematics Journal, Vol. 29, No. 4, 29: 296–300, doi:10.2307/2687685, JSTOR 2687685. Primary Pseudoperfect Number at PlanetMath.org. Weisstein, Eric W. "Primary Pseudoperfect Number". MathWorld.
Primary pseudoperfect number
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Graphical demonstration that 1 = 1/2 + 1/3 + 1/11 + 1/23 + 1/31 + 1/(2×3×11×23×31). Therefore the product, 47058, is primary pseudoperfect.
41.
Irrational number
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In mathematics, an irrational number is a real number that cannot be expressed as a ratio of integers, i.e. as a fraction. Therefore, irrational numbers, when written as decimal numbers, do not terminate, nor do they repeat. The same can be said for any irrational number. As a consequence of Cantor's proof that the rationals countable, it follows that almost all real numbers are irrational. The first proof of the existence of irrational numbers is usually attributed to a Pythagorean, who probably discovered them while identifying sides of the pentagram. His reasoning is as follows: Start with an isosceles triangle with side lengths of integers a, b, c. The ratio of the hypotenuse to a leg is represented by c:b. Assume a, b, c are in the smallest possible terms. By the Pythagorean theorem: c2 = a2+b2 = b2+b2 = 2b2. . Since c2 = 2b2, c2 is divisible by 2, therefore even. Since c2 is even, c must be even. Since c is even, b must be odd. Since c is even, dividing c by 2 yields an integer. Let y be this integer.
Irrational number
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The number is irrational.
42.
Abelian group
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That is, these are the groups that obey the axiom of commutativity. Abelian groups generalize the arithmetic of addition of integers. They are named after Niels Henrik Abel. Finite abelian groups are very well understood. On the other hand, the theory of infinite abelian groups is an area of current research. The • is a general placeholder for a concretely given operation. Commutativity For all a, b in a • b = b • a. A group in which the operation is not commutative is called a "non-abelian group" or "non-commutative group". There are two notational conventions for abelian groups -- additive and multiplicative. Generally, the multiplicative notation is the usual notation for groups, while the additive notation is the usual notation for rings. If the group is G = under the ⋅, the th entry of this table contains the product gi ⋅ gj. The group is only if this table is symmetric about the main diagonal. This is true since if the group is abelian, then gi ⋅ gj = ⋅ gi. This implies that the entry of the table equals the th entry, thus the table is symmetric about the main diagonal. Thus Z, form an abelian group under addition, as do the integers modulo n, Z/nZ.
Abelian group
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Algebraic structure → Group theory Group theory
43.
Double exponential function
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A double exponential function is a constant raised to the power of an exponential function. The general formula is f = a b x = a, which grows much more quickly than an exponential function. Factorials grow faster than exponential functions, but much slower than doubly exponential functions. Tetration and the Ackermann function grow even faster. See Big O notation for a comparison of the rate of growth of various functions. The inverse of the exponential function is the double ln. Aho and Sloane observed that in several important integer sequences, each term is a constant plus the square of the previous term. The first few numbers, starting with 0, are 2,5,277,5195977... Additional sequences of this type include The prime numbers 2, 11, 1361... a = ⌊ A 3 n ⌋ where A ≈ 1.306377883863 is Mills' constant. In the worst case, a Gröbner basis may have a number of elements, doubly exponential in the number of variables. On the other hand, the worst-case complexity of Gröbner basis algorithms is doubly exponential in the number of variables well as in the size. Finding a complete set of associative-commutative unifiers Satisfying CTL+ Quantifier elimination on real closed fields takes doubly exponential time. Thus, the overall time for the algorithm is O where h is the actual output size. Some number theoretical bounds are double exponential. Perfect numbers with n prime factors are known to be at most 2 4 n a result of Nielsen.
Double exponential function
–
A double exponential function (red curve) compared to a single exponential function (blue curve).
44.
Dorian M. Goldfeld
–
Dorian Morris Goldfeld is an American mathematician. He received his B.S. degree from Columbia University. Since 1985, he has been a professor at Columbia University. He is a member of the editorial board of Acta Arithmetica and of The Ramanujan Journal. He is a co-founder and member of SecureRF, a corporation that has developed the world's first linear-based security solutions. Dorian Goldfeld's research interests include various topics in theory. In his thesis, he proved a version of Artin's conjecture without the use of the Riemann Hypothesis. In 1976 Goldfeld provided an ingredient for the effective solution of Gauss' class number problem for imaginary quadratic fields. . This effective lower bound then allows the determination of all imaginary fields with a given number after a finite number of computations. He has also made contributions to the understanding of Siegel zeroes, to the ABC conjecture, to cryptography. Together with his wife, father-in-law, Dr. Michael Anshel, both mathematicians, Dorian Goldfeld founded the field of Braid Group cryptography. In 1985 he received the Vaughan prize. In 1986 he was an invited speaker at the International Congress of Mathematicians in Berkeley. In April 2009 he was elected a Fellow of the American Academy of Arts and Sciences.
Dorian M. Goldfeld
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Dorian Goldfeld at The Analytic Theory of Automorphic Forms workshop, Oberwolfach, Germany (2011)
45.
Journal of Number Theory
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The Journal of Number Theory is a mathematics journal that publishes a broad spectrum of original research in number theory. The journal was established in 1969 at Ohio State University. It is currently published monthly with 6 volumes per year. The editor-in-chief is David Goss. Official website
Journal of Number Theory
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Journal of Number Theory
46.
JSTOR
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JSTOR is a digital library founded in 1995. Originally containing back issues of academic journals, it now also includes books and primary sources, current issues of journals. It provides full-text searches of almost 2,000 journals. President of Princeton University from 1972 to 1988, founded JSTOR. Most libraries found it prohibitively expensive in terms of space to maintain a comprehensive collection of journals. By digitizing many journal titles, JSTOR allowed libraries to outsource the storage of journals with the confidence that they would remain available long-term. Full-text search ability improved access dramatically. Bowen initially considered using CD-ROMs for distribution. JSTOR originally encompassed ten economics and history journals. It became a fully searchable index accessible from any ordinary web browser. Special software was put in place to make graphs clear and readable. With the success of this limited project, then-president of JSTOR, wanted to expand the number of participating journals. The work of adding these volumes to JSTOR was completed by December 2000. The Andrew W. Mellon Foundation funded JSTOR initially. Until January 2009 JSTOR operated in New York City and in Ann Arbor, Michigan.
JSTOR
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The JSTOR front page
47.
American Mathematical Monthly
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The American Mathematical Monthly is a mathematical journal founded by Benjamin Finkel in 1894. It is published each year by the Mathematical Association of America. The American Mathematical Monthly is an expository journal intended from undergraduate students to research professionals. Articles are reviewed and edited for quality of exposition as well as content. In this the American Mathematical Monthly fulfills a different role from that of mathematical research journals. The American Mathematical Monthly is the most widely read journal in the world according to records on JSTOR. Since 1997, the journal has been available online at the Mathematical Association of America's website. The MAA gives Awards annually to "authors of articles of expository excellence" published in the American Mathematical Monthly.
American Mathematical Monthly
–
American Mathematical Monthly
48.
International Standard Book Number
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The International Standard Book Number is a unique numeric commercial book identifier. An ISBN is assigned to each variation of a book. For example, an e-book, a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned after 1 January 2007, 10 digits long if assigned before 2007. The method of assigning an ISBN varies from country to country, often depending on how large the publishing industry is within a country. The initial ISBN configuration of recognition was generated based upon the 9-digit Standard Book Numbering created in 1966. The 10-digit ISBN format was published in 1970 as international standard ISO 2108. The International Standard Serial Number, identifies periodical publications such as magazines; and the International Standard Music Number covers for musical scores. The ISBN configuration of recognition was generated in 1967 in the United Kingdom by Emery Koltay. The 10-digit ISBN format was published as international standard ISO 2108. The United Kingdom continued to use the 9-digit SBN code until 1974. The ISO on-line facility only refers back to 1978. An SBN may be converted by prefixing the digit "0". This can be converted to ISBN 0-340-01381-8; the digit does not need to be re-calculated. Since 1 ISBNs have contained 13 digits, a format, compatible with "Bookland" European Article Number EAN-13s.
International Standard Book Number
–
A 13-digit ISBN, 978-3-16-148410-0, as represented by an EAN-13 bar code
49.
Ronald Graham
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He has done important work in scheduling theory, computational geometry, quasi-randomness. Graham was born in Taft, California. In 1962, he began working at Bell Labs and later AT&T Labs. He retired from AT&T in 1999 after 37 years. His 1977 paper gave a "large number" as an upper bound for its solution. Graham popularized the concept of the Erdős number, named after the highly prolific Hungarian mathematician Paul Erdős. A scientist's Erdős number is the minimum number of coauthored publications away from a publication with Erdős. Graham's Erdős number is 1. He co-authored almost 30 papers with Erdős, was also a good friend. Erdős often allowed him to look after his mathematical papers and even his income. Graham and Erdős visited the young mathematician Jon Folkman when he was hospitalized with cancer. Between 1994 Graham served as the president of the American Mathematical Society. He has published five books, including Concrete Mathematics with Donald Knuth and Oren Patashnik. He is married to Fan Chung Graham, the Akamai Professor in Internet Mathematics at the University of California, San Diego. He has a son Marc.
Ronald Graham
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Ronald Graham
Ronald Graham
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Ronald Graham juggling a four ball fountain (1986)
Ronald Graham
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Ronald Graham, his wife Fan Chung, and Paul Erdős, Japan 1986
50.
Pacific Journal of Mathematics
Pacific Journal of Mathematics
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Pacific Journal of Mathematics
51.
Richard K. Guy
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Richard Kenneth Guy is a British mathematician, professor emeritus in the Department of Mathematics at the University of Calgary. He is known for his work in number theory, geometry, recreational mathematics, graph theory. He is best known for co-authorship of authorship of Unsolved Problems in Number Theory. He has also published over 300 papers. For this paper he received the MAA Lester R. Ford Award. Guy was born Sept 1916 in Nuneaton, Warwickshire, England, to Adeline Augusta Tanner and William Alexander Charles Guy. Both of his parents were teachers, rising to the rank of headmistress and headmaster, respectively. He was not enthusiastic about most of the curriculum. He was good at sports, however, excelled in mathematics. At the age of 17 he read Dickson's History of the Theory of Numbers. He said it was better than "the whole works of Shakespeare." His future was set. By then he had also developed a passion for climbing. In 1935 Guy entered Caius College, at the University of Cambridge as a result of winning several scholarships. To win the most important of these he had to write exams for two days.
Richard K. Guy
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Richard K. Guy in June 2005
52.
Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
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Transactions of the AMS, February 2006 issue.
53.
American Journal of Mathematics
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The American Journal of Mathematics is a bimonthly mathematics journal published by the Johns Hopkins University Press. Initially W. E. Story was associate editor in charge; he was replaced by Thomas Craig in 1880. For volume 7 Simon Newcomb became chief editor until 1894. Then with volume 16 it was "Edited by Thomas Craig with the Co-operation of Simon Newcomb" until 1898. The American Journal of Mathematics is a general-interest journal covering all the major areas of contemporary mathematics. According to the Journal Citation Reports, its 2009 factor is 1.337, ranking it 22nd out of 255 journals in the category "Mathematics". As of 2012, the editors are Christopher D. Sogge, editor-in-chief, William Minicozzi II, Freydoon Shahidi, Vyacheslav Shokurov. Official website
American Journal of Mathematics
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American Journal of Mathematics
54.
Ian Stewart (mathematician)
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Ian Nicholas Stewart FRS is a British mathematician and a popular-science and science-fiction writer. He is Emeritus Professor of Mathematics at the University of Warwick, England. Stewart was born in England. While in the sixth form at school he came to the attention of the mathematics teacher. The teacher had Stewart sit A-level examinations without any preparation along with the upper-sixth students; Stewart was placed first in the examination. The teacher arranged for Stewart to be admitted on a scholarship to Churchill College, where he obtained a BA in mathematics. He is well known to catastrophe theory. While at Warwick, Stewart edited the mathematical Manifold. He also wrote a column called "Mathematical Recreations" for Scientific American magazine from 1991 to 2001. This followed the work of past columnists like Martin Gardner, Douglas Hofstadter, A.K. Dewdney. Stewart has held visiting academic positions in Germany, the US. Stewart has collaborated based on Pratchett's Discworld. In March 2014 Incredible Numbers by Professor Ian Stewart, launched in the App Store. The app was produced with Profile Books and Touch Press.
Ian Stewart (mathematician)
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Ian Stewart
55.
Scientific American
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Scientific American is an American popular science magazine. Many famous scientists, including Albert Einstein, have contributed articles in the past 170 years. It is the oldest continuously published monthly magazine in the United States. Scientific American was founded by publisher Rufus M. Porter as a four-page weekly newspaper. Throughout its early years, much emphasis was placed on reports of what was going on at the U.S. Patent Office. Current issues include a "this date in history" section, featuring excerpts from articles originally published 50, 100, 150 years earlier. Topics include noteworthy advances in the history of science and technology. Porter sold the publication to Alfred Ely Beach and Orson Desaix Munn I a mere ten months after founding it. Until 1948, it remained owned by Munn & Company. Under Orson Desaix Munn III, grandson of Orson I, it had evolved into something of a "workbench" publication, similar to the twentieth-century incarnation of Popular Science. In the years after World War II, the magazine fell into decline. Thus the partners -- general manager Donald H. Miller, Jr. -- essentially created a new magazine. Miller retired in 1984 when Gerard Piel's Jonathan became president and editor; circulation had grown fifteen-fold since 1948. In 1986, it was sold to the Holtzbrinck group of Germany, which has owned it since.
Scientific American
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Cover of the March 2005 issue
Scientific American
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PDF of first issue: Scientific American Vol. 1, No. 01 published August 28, 1845
Scientific American
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Special Navy Supplement, 1898
56.
Dirk Jan Struik
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Dirk Jan Struik was a Dutch mathematician and Marxian theoretician who spent most of his life in the United States. Dirk Jan Struik was born in 1894 as a teacher's son, Struik attended the Hogere Burgerschool in The Hague. It was in this school that he was first introduced by some of his teachers. In 1912 Struik entered University of Leiden, where he showed great interest in physics, influenced by the eminent professors Paul Ehrenfest and Hendrik Lorentz. In 1917 he worked as a high school mathematics teacher for a while, after which he worked as a research assistant for J.A. Schouten. It was during this period that he developed his doctoral dissertation, "The Application of Tensor Methods to Riemannian Manifolds." In 1922 Struik obtained his doctorate from University of Leiden. He was appointed to a teaching position in 1923. He married Ruth Ramler, a Czech mathematician with a doctorate from the Charles University of Prague. In 1924, funded by a Rockefeller fellowship, Struik traveled to Rome to collaborate with the Italian mathematician Tullio Levi-Civita. It was in Rome that Struik first developed a keen interest in the history of mathematics. He also started researching Renaissance mathematics at this time. In 1926 Struik was offered positions both at the Massachusetts Institute of Technology. He decided to accept the latter, where he spent the rest of his academic career.
Dirk Jan Struik
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Dirk Jan Struik
57.
London Mathematical Society
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The London Mathematical Society is one of the United Kingdom's learned societies for mathematics. The Society was established on 16 January 1865, the first president being Augustus De Morgan. The Society soon moved into Burlington House, Piccadilly. The initial activities of the Society included talks and publication of a journal. The LMS was used in 1888. The Society was granted a royal charter in a century after its foundation. In 1998 the Society moved from rooms in Burlington House at 57 -- 58 Russell Square, Bloomsbury, to accommodate an expansion of its staff. The Society is also a member of the UK Science Council. On 4 the Joint Planning Group for the LMS and IMA proposed a merger of two societies to form a single, unified society. The councils of both societies commended the report to their members. Those in favour of the merger argued a single society would give a coherent voice when dealing with Research Councils. While accepted by the IMA membership, the proposal was rejected by the LMS membership on 29 May 2009 by 591 to 458. It also copublishes Nonlinearity with the Institute of Physics. In addition, the Society jointly with the Institute of its Applications awards the David Crighton Medal every three years. The Book of Presidents 1865–1965.
London Mathematical Society
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De Morgan House
58.
David Eppstein
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David Arthur Eppstein is an American computer scientist and mathematician. He is a Chancellor's Professor of science at University of California, Irvine. He is known for his work in computational geometry, recreational mathematics. In 2011, he was named an ACM Fellow. He was co-chair of the Computer Science Department there from 2002 to 2005. In 2014, he was named a Chancellor's Professor. Since 2007, Eppstein has been an administrator at the English Wikipedia. In 1984 he was awarded with a National Science Foundation Graduate Research Fellowship. In 1992, Eppstein received a National Science Foundation Young Investigator Award along with six UC-Irvine academics. In 2011, he was named an ACM Fellow for his contributions to computational geometry. Eppstein, David. "Finding shortest paths". SIAM Journal on Computing. 28: 652–673. Doi:10.1109/SFCS.1994.365697.
David Eppstein
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David Eppstein
59.
The Wolfram Demonstrations Project
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It is hosted by Wolfram Research, whose stated goal is to bring computational exploration to the widest possible audience. At its launch it has grown to over 10,000. The site won a Parents' Award in 2008. Each Demonstration also has a brief description about the concept being shown. Demonstrations are now easily embeddable into any blog. Each Demonstration page includes a snippet of JavaScript code in the section of the sidebar. The website is organized into topics such as science, mathematics, computer science, art, finance. They cover a variety of levels, from elementary school mathematics including quantum mechanics and models of biological organisms. The site is aimed at both students, as well as researchers who wish to present their ideas to the broadest possible audience. Wolfram Research's staff edits the Demonstrations, which may be created by any user of Mathematica, then freely published and freely downloaded. The Demonstrations are open-source, which means that they not only demonstrate itself, but also show how to implement it. The use of the web to transmit interactive programs is reminiscent of Sun's Java applets, Adobe's Flash, the open-source processing. However, those creating Demonstrations have access to the visualization capabilities of Mathematica making it more suitable for technical demonstrations. The Demonstrations Project also has similarities like Wikipedia and Flickr. Its model is similar to Adobe's Acrobat and Flash strategy of charging for development tools but providing a free reader.
The Wolfram Demonstrations Project
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Surface tangents.
The Wolfram Demonstrations Project
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Shell growth.
The Wolfram Demonstrations Project
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Legal structures.
60.
Ratio
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In mathematics, a ratio is a relationship between two numbers indicating how many times the first number contains the second. For example, if a bowl of fruit contains six lemons, then the ratio of oranges to lemons is eight to six. Thus, a ratio can be a fraction as opposed to a whole number. Also, the ratio of oranges to the total amount of fruit is 8:14. The numbers compared in a ratio can be any quantities such as objects, persons, lengths, or spoonfuls. A ratio is written "a to b" or a:b, or sometimes expressed arithmetically as a quotient of the two. When the two quantities have the same units, as is often the case, their ratio is a dimensionless number. A rate is a quotient of variables having different units. But in many applications, the ratio is often used instead for this more general notion as well. B being the consequent. The proportion expressing the equality of the ratios A:B and C:D is written A:B = C:D or A:B::C:D. B and C are called the means. The equality of three or more proportions is called a continued proportion. Ratios are sometimes used with three or more terms. The ratio of the dimensions of a "two by four", ten inches long is 2:4:10.
Ratio
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The ratio of width to height of standard-definition television.
61.
Division (mathematics)
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Division is one of the four basic operations of arithmetic, the others being addition, subtraction, multiplication. The division of two natural numbers is the process of calculating the number of times one number is contained within one another. Division can also be thought of as the process of evaluating a fraction, fractional notation is commonly used to represent division. Division is the inverse of multiplication; if a × b = c, then a = c ÷ b, as b is not zero. 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, the quotient is four. Besides dividing apples, division can be applied to other physical and abstract objects. Teaching division usually leads to the concept of fractions being introduced to school pupils. Unlike 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 system is extended to include rational numbers as they are more generally called. Division is often shown in algebra and science by placing the dividend over the divisor with a horizontal line, also called a fraction bar, between them. A fraction is a division expression where both dividend and divisor are integers, there is no implication that the division must be evaluated further. ISO 80000-2-9.6 states it should not be used. The obelus is also used alone to represent itself, for instance as a label on a key of a calculator.
Division (mathematics)
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This article is about the arithmetical operation. For other uses, see Division (disambiguation).
62.
Divisor
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An n is divisible by another integer m if m is a factor of n, so that dividing n by m leaves no remainder. Under this definition, the statement m ∣ 0 holds for every m. Before, but with the additional constraint k ≠ 0. Under this definition, the statement m ∣ 0 does not hold for m ≠ 0. In the remainder of this article, which definition is applied is indicated where this is significant. Divisors can be negative well as positive, although sometimes the term is restricted to positive divisors. For example, there are six divisors of 4; only the positive ones would usually be mentioned. − 1 divide every integer. Every integer is a divisor of itself. Every integer is a divisor of 0. Numbers not divisible by 2 are called odd. 1, − 1, − n are known as the trivial divisors of n. A divisor of n, not a trivial divisor is known as a non-trivial divisor. There are divisibility rules which allow one to recognize certain divisors of a number from the number's digits. The generalization can be said to be the concept of divisibility in any integral domain.
Divisor
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The divisors of 10 illustrated with Cuisenaire rods: 1, 2, 5, and 10
63.
Aspect ratio
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The aspect ratio of a geometric shape is the ratio of its sizes in different dimensions. The ratio is expressed as two numbers separated by a colon. The values x and y do not represent actual heights but, rather, the relationship between width and height. As 8:5, 16:10 and 1.6:1 are three ways of representing the same aspect ratio. The term is most commonly used to: Graphic / image Image aspect ratio Display aspect ratio: the aspect ratio for computer displays. A square has the smallest possible ratio of 1:1. An ellipse with an ratio of 1:1 is a circle. A circle has the minimal DWAR, 1. A square has a DWAR of sqrt. A square has the minimal CVAR, 1. A circle has a CVAR of sqrt. An axis-parallel rectangle of height H, where W > H, has a CVAR of sqrt = sqrt. If the d is fixed, then all reasonable definitions of aspect ratio are equivalent to within constant factors. Aspect ratios are mathematically expressed as x:y. In digital images there is a subtle distinction between the Storage Aspect Ratio; see Distinctions.
Aspect ratio
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This article is about shapes. For the aspect ratio of image, film and video, see Aspect ratio (image).
64.
Binary number
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The base-2 system is a positional notation with a radix of 2. Of its straightforward implementation in electronic circuitry using logic gates, the binary system is used internally by almost all modern computers and computer-based devices. Each digit is referred to as a bit. The binary system was devised by Gottfried Leibniz in 1679 and appears in his article Explication de l'Arithmétique Binaire. Systems related to binary numbers have appeared earlier in multiple cultures including India. Leibniz was specifically inspired by the Chinese I Ching. The scribes of ancient Egypt used two different systems for Horus-Eye fractions. The method used for ancient Egyptian multiplication is also closely related to binary numbers. This method can be seen in the Rhind Mathematical Papyrus, which dates to around 1650 BC. The I Ching dates from the 9th century BC in China. The binary notation in the I Ching is used to interpret its quaternary divination technique. It is based on taoistic duality of yin and yang. The Indian scholar Pingala developed a binary system for describing prosody. He used binary numbers in the form of short and long syllables, making it similar to Morse code. Pingala's Hindu classic titled Chandaḥśāstra describes the formation of a matrix in order to give a unique value to each meter.
Binary number
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Numeral systems
Binary number
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Arithmetic values represented by parts of the Eye of Horus
Binary number
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Gottfried Leibniz
Binary number
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George Boole
65.
Decimal fraction
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This article aims to be an accessible introduction. For the mathematical definition, see Decimal representation. The decimal system has ten as its base. It is the numerical base most widely used by modern civilizations. Other fractions have repeating decimal representations, whereas irrational numbers have infinite non-repeating decimal representations. Decimal notation is the writing of numbers in a base 10 system. Examples are Brahmi numerals, Greek numerals, Hebrew numerals, Chinese numerals, as well as the Hindu-Arabic numerals used by speakers of many European languages. Roman numerals have secondary symbols for half these values. Brahmi numerals have symbols for the nine numbers 1 -- 9, another for 1000. Chinese numerals have symbols for additional symbols for powers of 10, which in modern usage reach 1072. Decimal systems include a zero and use symbols for the ten values to represent any number, no matter how large or how small. Positional notation uses positions for each power of ten: units, tens, thousands, etc.. There were at least two presumably independent sources of decimal systems in ancient civilization: the Chinese counting rod system and the Hindu-Arabic numeral system. Ten is the number, the count of thumbs on both hands. The English digit as well as its translation in many languages is also the anatomical term for fingers and toes.
Decimal fraction
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The world's earliest decimal multiplication table was made from bamboo slips, dating from 305 BC, during the Warring States period in China.
Decimal fraction
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Numeral systems
Decimal fraction
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Ten fingers on two hands, the possible starting point of the decimal counting.
Decimal fraction
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Diagram of the world's earliest decimal multiplication table (c. 305 BC) from the Warring States period
66.
Golden ratio
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The figure on the right illustrates the geometric relationship. Its value is: φ = 1 + 2 = 1.6180339887.... The golden ratio is also called the golden mean or golden section. Other names mean ratio, medial section, divine proportion, divine section, golden proportion, golden cut, golden number. The golden ratio appears in some patterns including the spiral arrangement of leaves and other plant parts. Two quantities a and b are said to be in the golden φ if a + b a = a b = φ. One method for finding the value of φ is to start with the left fraction. Through substituting in b/a = 1 / φ, a + b a = 1 + b a = 1 + 1 φ. Therefore, 1 + φ = φ. Multiplying by φ gives φ + 1 = φ 2 which can be rearranged to φ 2 − φ − 1 = 0. Now the semicircle is drawn around the point B. The arising intersection E corresponds 2 φ. Up, the perpendicular on the line segment A E ¯ from the point D will be establish. The parallel F S ¯ to the line segment C M ¯, produces, as it were, the hypotenuse of the right triangle S D F. It is well recognizable, the triangle M S C are similar to each other.
Golden ratio
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Michael Maestlin, first to publish a decimal approximation of the golden ratio, in 1597
Golden ratio
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Line segments in the golden ratio
Golden ratio
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Many of the proportions of the Parthenon are alleged to exhibit the golden ratio.
Golden ratio
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The drawing of a man's body in a pentagram suggests relationships to the golden ratio.
67.
Silver ratio
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This defines the silver ratio as an mathematical constant, whose value of one plus the square root of 2 is approximately 2.4142135623. The ratio is denoted by δS. These fractions provide rational approximations of the silver ratio, analogous to the approximation of the golden ratio by ratios of consecutive Fibonacci numbers. Equivalently, 2 + b a = a b = δ S Therefore, 2 + 1 δ S = δ S. Multiplying by δS and rearranging gives δ S 2 − 2 δ S − 1 = 0. Using the quadratic formula, two solutions can be obtained. In fact it is the second smallest quadratic number after the golden ratio. This means the distance from δ n S to the nearest integer is 1/δ n S ≈ 0.41n. Thus, the sequence of fractional parts of δ n S, n = 1, 2, 3... converges. In particular, this sequence is not equidistributed 1. The paper sizes under ISO 216 are rectangles in the proportion 1:√2, sometimes called "A4 rectangles". Removing a largest square from one of these sheets leaves one again with ratio 1: √ 2. A rectangle whose aspect ratio is the silver ratio is sometimes called a silver rectangle with golden rectangles. Confusingly, "silver rectangle" can also refer to the paper sizes specified by ISO 216. The rectangle is connected to the regular octagon.
Silver ratio
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Silver ratio within the octagon
68.
Interval (music)
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In music theory, an interval is the difference between two pitches. In Western music, intervals are most commonly differences between notes of a diatonic scale. The smallest of these intervals is a semitone. Intervals smaller than a semitone are called microtones. They can be formed using the notes of various kinds of non-diatonic scales. Intervals can be arbitrarily small, even imperceptible to the human ear. In physical terms, an interval is the ratio between two sonic frequencies. For example, any two notes an octave apart have a frequency ratio of 2:1. For this reason, intervals are often measured in a unit derived from the logarithm of the ratio. In Western theory, the most common scheme for intervals describes two properties of the interval: the quality and number. Examples include the minor third or perfect fifth. These names describe not only the difference in semitones between the upper and lower notes, but also how the interval is spelled. The importance of spelling stems from the historical practice of differentiating the frequency ratios of enharmonic intervals such as G–G♯ and G–A♭. The size of an interval can be represented using two alternative and equivalently valid methods,: cents. The size of an interval between two notes may be measured by the ratio of their frequencies.
Interval (music)
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Melodic and harmonic intervals. Play (help · info)
69.
Percentage
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In mathematics, a percentage is a number or ratio expressed as a fraction of 100. It is often denoted using the percent sign, "%", or the abbreviations "pct.", "pct"; sometimes the abbreviation "pc" is also used. A percentage is a dimensionless number. For example, 45% is equal to 45⁄100, or 0.45. Percentages which are not ratios are expressed as a relationship of the part to the whole. If there were 1000 students then 500 would be male. A related system which expresses a number as a fraction of 1,000 uses the terms "per mil" and "millage". Percentages are used as ratios to express how large or small one quantity is relative to another quantity. The first quantity usually represents a part of, or a change in, the second quantity. For example, an increase of $0.15 on a price of $2.50 is an increase by a fraction of 0.15/2.50 = 0.06. Expressed as a percentage, this is therefore a 6% increase. While percentage values may often be limited to lie between 0 and 100, there is no mathematical restriction and percentages may take on other values. For example, it is common to refer to 35 %, especially for percent changes and comparisons. In Ancient Rome, long before the existence of the decimal system, computations were often made in fractions which were multiples of 1⁄100. For example, Augustus levied a tax of 1⁄100 on goods sold at auction known as venalium.
Percentage
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Contents
70.
Egyptian fraction
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An Egyptian fraction is a finite sum of distinct unit fractions, such as 1 2 + 1 3 + 1 16. The value of an expression of this type is a positive rational a/b; for instance the Egyptian fraction above sums to 43/48. Every rational number can be represented by an Egyptian fraction. In mathematical notation, Egyptian fractions have been superseded by vulgar fractions and decimal notation. Beyond their historical use, Egyptian fractions have some practical advantages over other representations of fractional numbers. For instance, Egyptian fractions can help in dividing a number of objects into equal shares. For more information on this subject, see Egyptian numerals, Egyptian mathematics. Egyptian notation was developed in the Middle Kingdom of Egypt, altering the Old Kingdom's Eye of Horus numeration system. The Rhind Mathematical Papyrus, introduced improved ways of writing Egyptian fractions. Solutions to each problem were written out with the final answers of all 84 problems being expressed in Egyptian fraction notation. 2/n tables similar to the one on the Rhind papyrus also appear on some of the other texts. However, as the Kahun Papyrus shows, vulgar fractions were also used by scribes within their calculations. Similarly in hieratic script they drew a line over the letter representing the number. These have been called "Horus-Eye fractions" after a theory that they were based on the parts of the Eye of Horus symbol. In particular, study in this area has concentrated on understanding the tables of expansions for numbers of the 2/n in the Rhind papyrus.
Egyptian fraction
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Eye of Horus