1.
Integer
–
An integer is a number that can be written without a fractional component. For example,21,4,0, and −2048 are integers, while 9.75, 5 1⁄2, the set of integers consists of zero, the positive natural numbers, also called whole numbers or counting numbers, and their additive inverses. This is often denoted by a boldface Z or blackboard bold Z 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 integers that are also rational numbers. Like the natural numbers, Z is closed under the operations of addition and multiplication, that is, however, with the inclusion of the negative natural numbers, and, importantly,0, Z is also closed under subtraction. The integers form a ring which is the most basic one, in the following sense, for any unital ring. This universal property, namely to be an object in the category of rings. 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 properties of addition and multiplication for any integers a, b and c. In the language of algebra, the first five properties listed above for addition say that Z under addition is an abelian group. As a group under addition, Z is a cyclic group, in fact, Z under addition is the only infinite cyclic group, in the sense that any infinite cyclic group is isomorphic to Z. The first four properties listed above for multiplication say that Z under multiplication is a commutative monoid. However, not every integer has an inverse, e. g. there is no integer x such that 2x =1, because the left hand side is even. This means that Z under multiplication is not a group, all the rules from the above property table, except for the last, taken together say that Z together with addition and multiplication is a commutative ring with unity. It is the prototype of all objects of algebraic structure. Only those equalities of expressions are true in Z for all values of variables, note that certain non-zero integers map to zero in certain rings. The lack of zero-divisors in the means that the commutative ring Z is an integral domain
Integer
–
Algebraic structure → Group theory Group theory
2.
Positive number
–
In mathematics, the concept of sign originates from the property of every non-zero real number of being positive or negative. Zero itself is signless, although in some contexts it makes sense to consider a signed zero, along with its application to real numbers, change of sign is used throughout mathematics and physics to denote the additive inverse, even for quantities which are not real numbers. Also, the sign can indicate aspects of mathematical objects that resemble positivity and negativity. A real number is said to be if its value is greater than zero. The attribute of being positive or 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 numeral notation, the sign of a number is often denoted by placing a plus sign or a minus sign before the number. For example, +3 denotes positive three, and −3 denotes negative three, when no plus or minus sign is given, the default interpretation is that a number is positive. Because of this notation, as well as the definition of numbers through subtraction. In this context, it makes sense to write − = +3, any non-zero number can be changed to a positive one using the absolute value function. For example, the value of −3 and the absolute value of 3 are both equal to 3. In symbols, this would be written |−3| =3 and |3| =3, the number zero is neither positive nor negative, and therefore has no sign. In arithmetic, +0 and −0 both denote the same number 0, which is the inverse of itself. Note that this definition is culturally determined, in France and Belgium,0 is said to be both positive and negative. The positive resp. negative numbers without zero are said to be strictly positive resp, in some contexts, such as signed number representations in computing, it makes sense to consider signed versions of zero, with positive zero and negative zero being different numbers. One also sees +0 and −0 in calculus and mathematical analysis when evaluating one-sided limits and this notation refers to the behaviour of a function as the input variable approaches 0 from positive or negative values respectively, these behaviours are not necessarily the same. Because zero is positive nor negative, the following phrases are sometimes used to refer to the sign of an unknown number. A number is negative if it is less than zero, a number is non-negative if it is greater than or equal to zero. A number is non-positive if it is less than or equal to zero, thus a non-negative number is either positive or zero, while a non-positive number is either negative or zero
Positive number
–
The plus and minus symbols are used to show the sign of a number.
3.
Rational number
–
In mathematics, a rational number is any number that can be expressed as the quotient or fraction p/q of two integers, a numerator p and a non-zero denominator q. Since q may be equal to 1, every integer is a rational number. The set of all numbers, often referred to as the rationals, is usually denoted by a boldface Q, it was thus denoted in 1895 by Giuseppe Peano after quoziente. The decimal expansion of a rational number always either terminates after a number of digits or begins to repeat the same finite sequence of digits over and over. Moreover, any repeating or terminating decimal represents a rational number and these statements hold true not just for base 10, but also for any other integer base. A real number that is not rational is called irrational, irrational numbers include √2, π, e, and φ. The decimal expansion of an irrational number continues without repeating, since the set of rational numbers is countable, and the set of real numbers is uncountable, almost all real numbers are irrational. Rational numbers can be defined as equivalence classes of pairs of integers such that q ≠0, for the equivalence relation defined by ~ if. In abstract algebra, the numbers together with certain operations of addition and multiplication form the archetypical field of characteristic zero. As such, it is characterized as having no proper subfield or, alternatively, finite extensions of Q are called algebraic number fields, and the algebraic closure of Q is the field of algebraic numbers. In mathematical analysis, the numbers form a dense subset of the real numbers. The real numbers can be constructed from the numbers by completion, using Cauchy sequences, Dedekind cuts. The term rational in reference to the set Q refers to the fact that a number represents a ratio of two integers. In mathematics, rational is often used as a noun abbreviating rational number, the adjective rational sometimes means that the coefficients are rational numbers. However, a curve is not a curve defined over the rationals. Any integer n can be expressed as the rational number n/1, a b = c d if and only if a d = b c. Where both denominators are positive, a b < c d if and only if a d < b c. If either denominator is negative, the fractions must first be converted into equivalent forms with positive denominators, through the equations, − a − b = a b, two fractions are added as follows, a b + c d = a d + b c b d
Rational number
–
A diagram showing a representation of the equivalent classes of pairs of integers
4.
Summand
–
Addition is one of the four basic operations of arithmetic, with the others being subtraction, multiplication and division. The addition of two numbers is the total amount of those quantities combined. For example, in the picture on the right, there is a combination of three apples and two together, making a total of five apples. This observation is equivalent to the mathematical expression 3 +2 =5 i. e.3 add 2 is equal to 5, besides counting fruits, addition can also represent combining other physical objects. In arithmetic, rules for addition involving fractions and negative numbers have been devised amongst others, in algebra, addition is studied more abstractly. It is commutative, meaning that order does not matter, and it is associative, repeated addition of 1 is the same as counting, addition of 0 does not change a number. Addition also obeys predictable rules concerning related operations such as subtraction and multiplication, performing addition is one of the simplest numerical tasks. Addition of very small numbers is accessible to toddlers, the most basic task,1 +1, can be performed by infants as young as five months and even some members of other animal species. In primary education, students are taught to add numbers in the system, starting with single digits. Mechanical aids range from the ancient abacus to the modern computer, Addition is written using the plus sign + between the terms, that is, in infix notation. The result is expressed with an equals sign, for example, 3½ =3 + ½ =3.5. This notation can cause confusion since in most other contexts juxtaposition denotes multiplication instead, the sum of a series of related numbers can be expressed through capital sigma notation, which compactly denotes iteration. For example, ∑ k =15 k 2 =12 +22 +32 +42 +52 =55. The numbers or the objects to be added in addition are collectively referred to as the terms, the addends or the summands. This is to be distinguished from factors, which are multiplied, some authors call the first addend the augend. In fact, during the Renaissance, many authors did not consider the first addend an addend at all, today, due to the commutative property of addition, augend is rarely used, and both terms are generally called addends. All of the above terminology derives from Latin, using the gerundive suffix -nd results in addend, thing to be added. Likewise from augere to increase, one gets augend, thing to be increased, sum and summand derive from the Latin noun summa the highest, the top and associated verb summare
Summand
–
Part of Charles Babbage's Difference Engine including the addition and carry mechanisms
Summand
–
3 + 2 = 5 with apples, a popular choice in textbooks
Summand
–
A circular slide rule
5.
Vulgar fraction
–
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, vulgar, or simple fraction consists of an integer numerator displayed above a line, numerators and denominators are also used in fractions that are not common, including compound fractions, complex fractions, and mixed numerals. The numerator represents a number of parts, and the denominator. For example, in the fraction 3/4, the numerator,3, tells us that the fraction represents 3 equal parts, the picture to the right illustrates 34 or ¾ of a cake. Fractional numbers can also be written without using explicit numerators or denominators, by using decimals, percent signs, 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 the fraction ¾ is also used to represent the ratio 3,4 and the division 3 ÷4. The test for a number being a number is that it can be written in that form. In a fraction, the number of parts being described is the numerator. Informally, they may be distinguished by placement alone but in formal contexts they are separated by a fraction bar. The fraction bar may be horizontal, oblique, or diagonal and these marks are respectively known as the horizontal bar, the slash or stroke, the division slash, and the fraction slash. In typography, horizontal fractions are known as en or nut fractions and diagonal fractions as em fractions. The denominators of English fractions are expressed as ordinal numbers. When the denominator is 1, it may be expressed in terms of wholes but is commonly ignored. When the numerator is one, it may be omitted, a fraction may be expressed as a single composition, in which case it is hyphenated, or as a number of fractions with a numerator of one, in which case they are not. Fractions should always be hyphenated when used as adjectives, alternatively, a fraction may be described by reading it out as the numerator over the denominator, with the denominator expressed as a cardinal number. The term over is used even in the case of solidus fractions, Fractions with large denominators that are not powers of ten are often rendered in this fashion while those with denominators divisible by ten are typically read in the normal ordinal fashion. A simple fraction is a number written as a/b or a b
Vulgar fraction
–
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.
6.
Number theory
–
Number theory or, in older usage, arithmetic is a branch of pure mathematics devoted primarily to the study of the integers. It is sometimes called The Queen of Mathematics because of its place in the discipline. Number theorists study prime numbers as well as the properties of objects out of integers or defined as generalizations of the integers. Integers can be considered either in themselves or as solutions to equations, questions in number theory are often best understood through the study of analytical objects that encode properties of the integers, primes or other number-theoretic objects in some fashion. One may also study real numbers in relation to rational numbers, the older term for number theory is arithmetic. By the early century, it had been superseded by number theory. The use of the arithmetic for number theory regained some ground in the second half of the 20th century. In particular, arithmetical is preferred as an adjective to number-theoretic. The first historical find of a nature is a fragment of a table. The triples are too many and too large to have been obtained by brute force, the heading over the first column reads, The takiltum of the diagonal which has been subtracted such that the width. The tables layout suggests that it was constructed by means of what amounts, in language, to the identity 2 +1 =2. If some other method was used, the triples were first constructed and then reordered by c / a, presumably for use as a table. It is not known what these applications may have been, or whether there could have any, Babylonian astronomy, for example. It has been suggested instead that the table was a source of examples for school problems. While Babylonian number theory—or what survives of Babylonian mathematics that can be called thus—consists of this single, striking fragment, late Neoplatonic sources state that Pythagoras learned mathematics from the Babylonians. Much earlier sources state that Thales and Pythagoras traveled and studied in Egypt, Euclid IX 21—34 is very probably Pythagorean, it is very simple material, but it is all that is needed to prove that 2 is irrational. Pythagorean mystics gave great importance to the odd and the even, the discovery that 2 is irrational is credited to the early Pythagoreans. This forced a distinction between numbers, on the one hand, and lengths and proportions, on the other hand, the Pythagorean tradition spoke also of so-called polygonal or figurate numbers
Number theory
–
A Lehmer sieve, which is a primitive digital computer once used for finding primes and solving simple Diophantine equations.
Number theory
–
The Plimpton 322 tablet
Number theory
–
Title page of the 1621 edition of Diophantus' Arithmetica, translated into Latin by Claude Gaspard Bachet de Méziriac.
Number theory
–
Leonhard Euler
7.
Egyptian numerals
–
The system of ancient Egyptian numerals was used in Ancient Egypt around 3000 BC until the early first millennium AD. It was a system of numeration based on the scale of ten, often rounded off to the power, written in hieroglyphs. The hieratic form of numerals stressed an exact finite series notation, the Ancient Egyptian system used bases of ten. The following hieroglyphics were used to denote powers of ten, Multiples of these values were expressed by repeating the symbol as many times as needed, for instance, a stone carving from Karnak shows the number 4622 as Egyptian hieroglyphs could be written in both directions. The symbol nfr, meaning beautiful, was used to indicate the base level in drawings of tombs and pyramids. Rational numbers could also be expressed, but only as sums of fractions, i. e. sums of reciprocals of positive integers, except for 2⁄3. The hieroglyph indicating a fraction looked like a mouth, which meant part, Fractions were written with this fractional solidus, i. e. the numerator 1, and the positive denominator below. As with most modern day languages, the ancient Egyptian language could also write out numerals as words phonetically, just like one can write thirty instead of 30 in English. The word, for instance, was written as while the numeral was This was, however, uncommon for most numbers other than one, instances of numerals written in hieratic can be found as far back as the Early Dynastic Period. The Old Kingdom Abusir Papyri are an important corpus of texts that utilize hieratic numerals. A large number like 9999 could thus be written only four signs—combining the signs for 9000,900,90. Boyer saw the new hieratic numerals as ciphered, mapping one number onto one Egyptian letter for the first time in human history, greeks adopted the new system, mapping their counting numbers onto two of their alphabets, the Doric and Ionian. In the oldest hieratic texts the individual numerals were written in a ciphered relationship to the Egyptian alphabet. But during the Old Kingdom a series of standardized writings had developed for sign-groups containing more than one numeral, however, repetition of the same numeral for each place-value was not allowed in the hieratic script. As the hieratic writing system developed over time, these sign-groups were further simplified for quick writing, two famous mathematical papyri using hieratic script are the Moscow Mathematical Papyrus and the Rhind Mathematical Papyrus. The majuscule letter A in some reconstructed forms means that the quality of that remains uncertain, Ancient Egypt Egyptian language Egyptian mathematics Allen. Middle Egyptian, An Introduction to the Language and Culture of Hieroglyphs, Egyptian Grammar, Being an Introduction to the Study of Hieroglyphs. Hieratische Paläographie, Die aegyptische Buchschrift in ihrer Entwicklung von der Fünften Dynastie bis zur römischen Kaiserzeit, Introduction Egyptian numerals Numbers and dates http, //egyptianmath. blogspot. com
Egyptian numerals
–
Numeral systems
8.
Eye of Horus
–
The Eye of Horus is an ancient Egyptian symbol of protection, royal power and good health. The eye is personified in the goddess Wadjet, the Eye of Horus is similar to the Eye of Ra, which belongs to a different god, Ra, but represents many of the same concepts. Wadjet was one of the earliest of Egyptian deities who later associated with other goddesses such as Bast, Sekhmet, Mut. She was the deity of Lower Egypt and 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 element of seven gold, faience, carnelian. The Wedjat was intended to protect the pharaoh in the afterlife, Ancient Egyptian and Middle-Eastern sailors would frequently paint the symbol on the bow of their vessel to ensure safe sea travel. Horus was the ancient Egyptian sky god who was depicted as a falcon. His right eye was associated with the sun god, Ra, the eye symbol represents the marking around the eye of the falcon, including the teardrop marking sometimes found below the eye. The mirror image, or left eye, sometimes represented the moon, in one myth, when Set and Horus were fighting for the throne after Osiriss death, Set gouged out Horuss left eye. The majority of the eye was restored by either Hathor or Thoth, when Horuss eye was recovered, he offered it to his father, Osiris, in hopes of restoring his life. Hence, the eye of Horus was often used to sacrifice, healing, restoration. There are seven different hieroglyphs used to represent the eye, most commonly ir. t in Egyptian, in Egyptian myth the eye was not the passive organ of sight but more an agent of action, protection or wrath. The Eye of Horus was represented as a hieroglyph, designated D10 in Gardiners sign list and it is represented in the Unicode character block for Egyptian hieroglyphs as U+13080. In Ancient Egyptian most fractions were written as the sum of two or more unit fractions, with scribes possessing tables of answers, thus instead of 3⁄4, one would write 1⁄2 + 1⁄4. Studies from the 1970s to this day in Egyptian mathematics have clearly shown this theory was fallacious, the evolution of the symbols used in mathematics, although similar to the different parts of the Eye of Horus, is now known to be distinct. Wadjet eye tatoos associated with Hathor depicted on 3, 000-year-old mummy
Eye of Horus
–
An Eye of Horus or Wedjat pendant
Eye of Horus
–
The Wedjat, later called The Eye of Horus
Eye of Horus
–
The crown of a Nubian king
Eye of Horus
–
Wooden case decorated with bronze, silver, ivory and gold
9.
Egyptian mathematics
–
Ancient Egyptian mathematics is the mathematics that was developed and used in Ancient Egypt c.3000 to c.300 BC. Written evidence of the use of mathematics dates back to at least 3000 BC with the ivory labels found in Tomb U-j at Abydos and these labels appear to have been used as tags for grave goods and some are inscribed with numbers. Further evidence of the use of the base 10 number system can be found on the Narmer Macehead which depicts offerings of 400,000 oxen,1,422,000 goats and 120,000 prisoners. The evidence of the use of mathematics in the Old Kingdom is scarce, the lines in the diagram are spaced at a distance of one cubit and show the use of that unit of measurement. The earliest true mathematical documents date to the 12th dynasty, the Rhind Mathematical Papyrus which dates to the Second Intermediate Period is said to be based on an older mathematical text from the 12th dynasty. The Moscow Mathematical Papyrus and Rhind Mathematical Papyrus are so-called mathematical problem texts and they consist of a collection of problems with solutions. These texts may have been written by a teacher or a student engaged in solving typical mathematics problems, an interesting feature of Ancient Egyptian mathematics is the use of unit fractions. Scribes used tables to help work with these fractions. The Egyptian Mathematical Leather Roll for instance is a table of unit fractions which are expressed as sums of unit fractions. The Rhind Mathematical Papyrus and some of the other texts contain 2 n tables and these tables allowed the scribes to rewrite any fraction of the form 1 n as a sum of unit fractions. During the New Kingdom mathematical problems are mentioned in the literary Papyrus Anastasi I, in the workers village of Deir el-Medina several ostraca have been found that record volumes of dirt removed while quarrying the tombs. Our understanding of ancient Egyptian mathematics is impeded by the paucity of available sources. The Reisner Papyrus dates to the early Twelfth dynasty of Egypt and was found in Nag el-Deir, the Rhind Mathematical Papyrus dates from the Second Intermediate Period, but its author, Ahmes, identifies it as a copy of a now lost Middle Kingdom papyrus. The RMP is the largest mathematical text, from the New Kingdom we have a handful of mathematical texts and inscription related to computations, The Papyrus Anastasi I is a literary text from the New Kingdom. It is written as a written by a scribe named Hori. A segment of the letter describes several mathematical problems, ostracon Senmut 153 is a text written in hieratic. Ostracon Turin 57170 is a written in hieratic. Ostraca from Deir el-Medina contain computations, ostracon IFAO1206 for instance shows the calculations of volumes, presumably related to the quarrying of a tomb
Egyptian mathematics
–
Slab stela of Old Kingdom princess Neferetiabet (dated 2590–2565 BC) from her tomb at Giza, painting on limestone, now in the Louvre.
Egyptian mathematics
–
Image of Problem 14 from the Moscow Mathematical Papyrus. The problem includes a diagram indicating the dimensions of the truncated pyramid.
10.
Middle Kingdom of Egypt
–
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, during the Middle Kingdom period, Osiris became the most important deity in popular religion. The period comprises two phases, the 11th Dynasty, which ruled from Thebes and the 12th Dynasty onwards which was centered on el-Lisht, after the collapse of the Old Kingdom, Egypt entered a period of weak Pharaonic power and decentralization called the First Intermediate Period. Towards the end of period, two rival dynasties, known in Egyptology as the Tenth and Eleventh, fought for power over the entire country. The Theban 11th Dynasty only ruled southern Egypt from the first cataract to the Tenth Nome of Upper Egypt, to the north, Lower Egypt was ruled by the rival 10th Dynasty from Herakleopolis. The struggle was to be concluded by Mentuhotep II, who ascended the Theban throne in 2055 B. C, during Mentuhotep IIs fourteenth regnal year, he took advantage of a revolt in the Thinite Nome to launch an attack on Herakleopolis, which met little resistance. After toppling the last rulers of the 10th Dynasty, Mentuhotep began consolidating his power over all Egypt, for this reason, Mentuhotep II is regarded as the founder of the Middle Kingdom. Mentuhotep II commanded military campaigns south as far as the Second Cataract in Nubia and he also restored Egyptian hegemony over the Sinai region, which had been lost to Egypt since the end of the Old Kingdom. He also sent the first expedition to Punt during the Middle Kingdom, by means of ships constructed at the end of Wadi Hammamat, Mentuhotep III was succeeded by Mentuhotep IV, whose name significantly is omitted from all ancient Egyptian king lists. The Turin Papyrus claims that after Mentuhotep III came seven kingless years, despite this absence, his reign is attested from a few inscriptions in Wadi Hammamat that record expeditions to the Red Sea coast and to quarry stone for the royal monuments. The leader of expedition was his vizier Amenemhat, who is widely assumed to be the future pharaoh Amenemhet I. Mentuhotep IVs absence from the king lists has prompted the theory that Amenemhet I usurped his throne, while there are no contemporary accounts of this struggle, certain circumstantial evidence may point to the existence of a civil war at the end of the 11th dynasty. Inscriptions left by one Nehry, the Haty-a of Hermopolis, suggest that he was attacked at a place called Shedyet-sha by the forces of the reigning king, but his forces prevailed. Khnumhotep I, an official under Amenemhet I, claims to have participated in a flotilla of 20 ships to pacify Upper Egypt, donald Redford has suggested these events should be interpreted as evidence of open war between two dynastic claimants. What is certain is that, however he came to power, from the 12th dynasty onwards, pharaohs often kept well-trained standing armies, which included Nubian contingents. These formed the basis of larger forces which were raised for defence against invasion, however, the Middle Kingdom was basically defensive in its military strategy, with fortifications built at the First Cataract of the Nile, in the Delta and across the Sinai Isthmus. Early in his reign, Amenemhet I was compelled to campaign in the Delta region, in addition, he strengthened defenses between Egypt and Asia, building the Walls of the Ruler in the East Delta region. Perhaps in response to this perpetual unrest, Amenemhat I built a new capital for Egypt in the north, known as Amenemhet Itj Tawy, or Amenemhet, the location of this capital is unknown, but is presumably near the citys necropolis, the present-day el-Lisht
Middle Kingdom of Egypt
–
An Osiride statue of the first pharaoh of the Middle Kingdom, Mentuhotep II
Middle Kingdom of Egypt
–
The head of a statue of Senusret I.
Middle Kingdom of Egypt
–
Statue head of Senusret III
11.
Moscow Mathematical Papyrus
–
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, approximately 5½ m long and varying between 3.8 and 7.6 cm wide, its format was divided into 25 problems with solutions by the Soviet Orientalist Vasily Vasilievich Struve in 1930. It is a well-known 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 problems in the Moscow Papyrus follow no particular order, 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. Problems 2 and 3 are ships part problems, one of the problems calculates the length of a ships rudder and the other computes the length of a ships mast given that it is 1/3 + 1/5 of the length of a cedar log originally 30 cubits long. Aha problems involve finding unknown quantities if the sum of the quantity, the Rhind Mathematical Papyrus also contains four of these type of problems. Problems 1,19, and 25 of the Moscow Papyrus are Aha problems, for instance problem 19 asks one to calculate a quantity taken 1 and ½ times and added to 4 to make 10. In other words, in mathematical notation one is asked to solve 3 /2 × x +4 =10 Most of the problems are pefsu problems,10 of the 25 problems. A pefsu measures the strength of the beer made from a heqat of grain pefsu = number loaves of bread number of heqats of grain A higher pefsu number means weaker bread or beer, the pefsu number is mentioned in many offering lists. Then reckon what you need for a des-jug of beer like the beer called 1/2 1/4 malt-date beer The result is 1/2 of the heqat measure needed for des-jug of beer made from Upper-Egyptian grain. Calculate 1/2 of 5 heqat, the result will be 2 1/2 Take this 2 1/2 four times The result is 10, then you say to him, Behold. The beer quantity is found to be correct, problems 11 and 23 are Baku problems. These calculate the output of workers, problem 11 asks if someone brings in 100 logs measuring 5 by 5, then how many logs measuring 4 by 4 does this correspond to. Problem 23 finds the output of a given that he has to cut. Seven of the problems are geometry problems and range from computing areas of triangles, to finding the surface area of a hemisphere. The 10th problem of the Moscow Mathematical Papyrus asks for a calculation of the area of a hemisphere or possibly the area of a semi-cylinder. Below we assume that the problem refers to the area of a hemisphere, the text of problem 10 runs like this, Example of calculating a basket
Moscow Mathematical Papyrus
–
14th problem of the Moscow Mathematical Papyrus (V. Struve, 1930)
Moscow Mathematical Papyrus
–
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)
12.
Kahun Papyrus
–
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 and this collection of papyri is one of the largest ever found. Most of the texts are dated to ca.1825 BC, 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. The Lahun Mathematical Papyri are a collection of mathematical texts A veterinarian papyrus A late Middle Kingdom account, listing festivals A Kahun Mathematical Fragment, legon PlanetMath, Kahun Papyrus and Arithmetic Progressions
Kahun Papyrus
–
Fragments of the Kahun Papyrus on veterinary medicine
13.
Second Intermediate Period
–
The Second Intermediate Period marks a period when Ancient Egypt fell into disarray for a second time, between the end of the Middle Kingdom and the start of the New Kingdom. It is best known as the period when the Hyksos made their appearance in Egypt, the Twelfth Dynasty of Egypt came to an end at the end of the 19th century BC with the death of Queen Sobekneferu. Apparently she had no heirs, causing the twelfth dynasty to come to an end, and, with it. Retaining the seat of the dynasty, the thirteenth dynasty ruled from Itjtawy near Memphis and Lisht. The Thirteenth Dynasty is notable for the accession of the first formally recognised Semitic-speaking king, the Fifteenth Dynasty dates approximately from 1650 to 1550 BC. Known rulers of the Fifteenth Dynasty are as follows, Salitis Sakir-Har Khyan Apophis, 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, the surviving traces on the X figure appears to give the figure 8 which suggests that the summation should be read as 6 kings ruling 108 years. Some scholars argue there were two Apophis kings named Apepi I and Apepi II, but this is due to the fact there are two known prenomens for this king, Awoserre and Aqenenre. However, the Danish Egyptologist Kim Ryholt maintains in his study of the Second Intermediate Period that these prenomens all refer to one man, Apepi and 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 and this scenario is not unprecedented, as later kings, including the famous Ramesses II and Seti II, are known to have used two different prenomens in their own reigns. The Sixteenth Dynasty ruled the Theban region in Upper Egypt for 70 years, of the two chief versions of Manethos Aegyptiaca, Dynasty XVI is described by the more reliable Africanus as shepherd kings, but by Eusebius as Theban. For this reason other scholars do not follow Ryholt and see only insufficient evidence for the interpretation of the Sixteenth Dynasty as Theban, the continuing war against Dynasty XV dominated the short-lived 16th dynasty. The armies of the 15th dynasty, winning town after town from their enemies, continually encroached on the 16th dynasty territory, eventually threatening. Famine, which had plagued Upper Egypt during the late 13th dynasty, from Ryholts reconstruction of the Turin canon,15 kings of the dynasty can now be named, five of whom appear in contemporary sources. While most likely based in Thebes itself, some may have been local rulers from other important Upper Egyptian towns, including Abydos, El Kab. By the reign of Nebiriau I, the controlled by the 16th dynasty extended at least as far north as Hu. Not listed in the Turin canon is Wepwawetemsaf, who left a stele at Abydos and was likely a local kinglet of the Abydos Dynasty, Ryholt gives the list of kings of the 16th dynasty as shown in the table below
Second Intermediate Period
–
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
–
Thebes (Luxor Temple pictured) was the capital of many of the Dynasty XVI pharaohs.
14.
Prime number
–
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 that is not a number is called a composite number. For example,5 is prime because 1 and 5 are its only positive integer factors, the property of being prime is called primality. A simple but slow method of verifying the primality of a 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 of forms, 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 by Euclid around 300 BC. There is no simple formula that separates prime numbers from composite numbers. However, the distribution of primes, that is to say, many questions regarding prime numbers remain open, such as Goldbachs conjecture, and the twin prime conjecture. Such questions spurred the development of branches of number theory. Prime numbers give rise to various generalizations in other domains, mainly algebra, such as prime elements. A natural number is called a number if it has exactly two positive divisors,1 and the number itself. Natural numbers greater than 1 that are not prime are called composite, among the numbers 1 to 6, the numbers 2,3, and 5 are the prime numbers, while 1,4, and 6 are not prime. 1 is excluded as a number, for reasons explained below. 2 is a number, since the only natural numbers dividing it are 1 and 2. Next,3 is prime, too,1 and 3 do divide 3 without remainder, however,4 is composite, since 2 is another number dividing 4 without remainder,4 =2 ·2. 5 is again prime, none of the numbers 2,3, next,6 is divisible by 2 or 3, since 6 =2 ·3. The image at the right illustrates that 12 is not prime,12 =3 ·4, no even number greater than 2 is prime because by definition, any such number n has at least three distinct divisors, namely 1,2, and n
Prime number
–
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.
15.
Practical number
–
In number theory, a practical number or panarithmic number is a positive integer n such that all smaller positive integers can be represented as sums of distinct divisors of n. Practical numbers were used by Fibonacci in his Liber Abaci in connection with the problem of representing rational numbers as Egyptian fractions, Fibonacci does not formally define practical numbers, but he gives a table of Egyptian fraction expansions for fractions with practical denominators. The name practical number is due to Srinivasan and he noted that the subdivision of money, weights and measures involved numbers like 4,12,16,20 and 28 which are usually supposed to be so inconvenient as to deserve replacement by powers of 10. He rediscovered the number theoretical property of such numbers and was the first to attempt a classification of numbers that was completed by Stewart. This characterization makes it possible to determine whether a number is practical by examining its prime factorization, every even perfect number and every power of two is also a practical number. Practical numbers have also shown to be analogous with prime numbers in many of their properties. If the ordered set of all divisors of the number n is d 1, d 2. D j with d 1 =1 and d j = n, in other words the ordered sequence of all divisors d 1 < d 2 <. < d j of a number has to be a complete sub-sequence. This partial characterization was extended and completed by Stewart and Sierpiński who showed that it is straightforward to determine whether a number is practical from its prime factorization, a positive integer greater than one with prime factorization n = p 1 α1. P k α k is if and only if each of its prime factors p i is small enough for p i −1 to have a representation as a sum of smaller divisors. The condition stated above is necessary and sufficient for a number to be practical, in the other direction, the condition is sufficient, as can be shown by induction. Since q ≤ σ and n / p k α k can be shown by induction to be practical, we can find a representation of q as a sum of divisors of n / p k α k. The divisors representing r, together with p k α k times each of the divisors representing q, the only odd practical number is 1, because if n >2 is an odd number, then 2 cannot be expressed as the sum of distinct divisors of n. More strongly, Srinivasan observes that other than 1 and 2, the product of two practical numbers is also a practical number. More strongly the least common multiple of any two numbers is also a practical number. Equivalently, the set of all numbers is closed under multiplication. From the above characterization by Stewart and Sierpiński it can be seen that if n is a practical number, in the set of all practical numbers there is a primitive set of practical numbers
Practical number
–
Overview
16.
Square number
–
In mathematics, a square number or perfect square is an integer that is the square of an integer, in other words, it is the product of some integer with itself. For example,9 is a number, since it can be written as 3 × 3. The usual notation for the square of a n is not the product n × n. The name square number comes from the name of the shape, another way of saying that a integer is a square number, is that its square root is again an integer. For example, √9 =3, so 9 is a square number, a positive integer that has no perfect square divisors except 1 is called square-free. For a non-negative integer n, the nth square number is n2, the concept of square can be extended to some other number systems. If rational numbers are included, then a square is the ratio of two integers, and, conversely, the ratio of two square integers is a square, e. g.49 =2. Starting with 1, there are ⌊√m⌋ square numbers up to and including m, the squares smaller than 602 =3600 are, The difference between any perfect square and its predecessor is given by the identity n2 −2 = 2n −1. Equivalently, it is possible to count up square numbers by adding together the last square, the last squares root, and the current root, that is, n2 =2 + + n. The number m is a number if and only if one can compose a square of m equal squares. Hence, a square with side length n has area n2, the expression for the nth square number is n2. This is also equal to the sum of the first n odd numbers as can be seen in the above pictures, 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. For example, the nth square number can be computed from the square by n2 =2 + + n =2 +. Alternatively, the nth square number can be calculated from the two by doubling the th square, subtracting the th square number, and adding 2. For example, 2 × 52 −42 +2 = 2 × 25 −16 +2 =50 −16 +2 =36 =62, a square number is also the sum of two consecutive triangular numbers. The sum of two square numbers is a centered square number. Every odd square is also an octagonal number
Square number
–
m = 1 2 = 1
17.
Liber Abaci
–
Liber Abaci is a historic book on arithmetic by Leonardo of Pisa, known later by his nickname Fibonacci. Liber Abaci was among the first Western books to describe Hindu–Arabic numbers traditionally described as Arabic Numerals, by addressing the applications of both commercial tradesmen and mathematicians, it contributed to convincing the public of the superiority of the Hindu–Arabic numeral system. The title of Liber Abaci means The Book of Calculation, the second version of Liber Abaci was dedicated to Michael Scot in 1227 CE. No versions of the original 1202 CE book have been found, the first section introduces the Hindu–Arabic numeral system, including methods for converting between different representation systems. The second section presents examples from commerce, such as conversions of currency and measurements, another example in this chapter, describing the growth of a population of rabbits, was the origin of the Fibonacci sequence for which the author is most famous today. The fourth section derives approximations, both numerical and geometrical, of irrational numbers such as square roots, the book also includes proofs in Euclidean geometry. Fibonaccis method of solving algebraic equations shows the influence of the early 10th-century Egyptian mathematician Abū Kāmil Shujāʿ ibn Aslam, there are three key differences between Fibonaccis notation and modern fraction notation. We generally write a fraction to the right of the number to which it is added. Fibonacci instead would write the same fraction to the left, i. e.132. That is, b a d c = a c + b c d, the notation was read from right to left. For example, 29/30 could be written as 124235 and this can be viewed as a form of mixed radix notation, and was very convenient for dealing with traditional systems of weights, measures, and currency. Sigler also points out an instance where Fibonacci uses composite fractions in which all denominators are 10, Fibonacci sometimes wrote several fractions next to each other, representing a sum of the given fractions. For instance, 1/3+1/4 = 7/12, so a notation like 14132 would represent the number that would now more commonly be written as the mixed number 2712, or simply the improper fraction 3112. Notation of this form can be distinguished from sequences of numerators and denominators sharing a fraction bar by the break in the bar. If all numerators are 1 in a written in this form, and all denominators are different from each other. This notation was also combined with the composite fraction notation. The complexity of this notation allows numbers to be written in different ways. In the Liber Abaci, Fibonacci says the following introducing the Modus Indorum or the method of the Indians, today known as Hindu–Arabic numerals or traditionally, just Arabic numerals
Liber Abaci
–
A page of the Liber Abaci from the Biblioteca Nazionale di Firenze showing (on right) the numbers of the Fibonacci sequence.
18.
Ptolemy
–
Claudius Ptolemy was a Greek writer, known as a mathematician, astronomer, geographer, astrologer, and poet of a single epigram in the Greek Anthology. He lived in the city of Alexandria in the Roman province of Egypt, wrote in Koine Greek, beyond that, few reliable details of his life are known. His birthplace has been given as Ptolemais Hermiou in the Thebaid in a statement by the 14th-century astronomer Theodore Meliteniotes. This is a very late attestation, however, and there is no reason to suppose that he ever lived elsewhere than Alexandria. Ptolemy wrote several 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 entitled the Mathematical Treatise. The second is the Geography, which is a discussion of the geographic knowledge of the Greco-Roman world. The third is the treatise in which he attempted to adapt horoscopic astrology to the Aristotelian natural philosophy of his day. This is sometimes known as the Apotelesmatika but more known as the Tetrabiblos from the Greek meaning Four Books or by the Latin Quadripartitum. The name Claudius is a Roman nomen, the fact that Ptolemy bore it indicates he lived under the Roman rule of Egypt with the privileges and political rights of Roman citizenship. It would have suited custom if the first of Ptolemys family to become a citizen took the nomen from a Roman called Claudius who was responsible for granting citizenship, if, as was common, this was the emperor, citizenship would have been granted between AD41 and 68. The astronomer would also have had a praenomen, which remains unknown and it occurs once in Greek mythology, and is of Homeric form. All the kings after him, until Egypt became a Roman province in 30 BC, were also Ptolemies, abu Mashar recorded a belief that a different member of this royal line composed the book on astrology and attributed it to Ptolemy. The correct answer is not known”, Ptolemy wrote in Greek and can be shown to have utilized Babylonian astronomical data. He was a Roman citizen, but most scholars conclude that Ptolemy was ethnically Greek and he was often known in later Arabic sources as the Upper Egyptian, suggesting he may have had origins in southern Egypt. Later Arabic astronomers, geographers and physicists referred to him by his name in Arabic, Ptolemys Almagest is the only surviving comprehensive ancient treatise on astronomy. Ptolemy presented his models in convenient tables, which could be used to compute the future or past position of the planets. The Almagest also contains a catalogue, which is a version of a catalogue created by Hipparchus
Ptolemy
–
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
–
Early Baroque artist's rendition
Ptolemy
–
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
–
Prima Europe tabula. A C15th copy of Ptolemy's map of Britain
19.
Almagest
–
The Almagest is the critical source of information on ancient Greek astronomy. It has also been valuable to students of mathematics because it documents the ancient Greek mathematician Hipparchuss work, Hipparchus wrote about trigonometry, but because his works appear to have been lost, mathematicians use Ptolemys book as their source for Hipparchuss work and ancient Greek trigonometry in general. The treatise was later titled Hē Megalē Syntaxis, and the form of this lies behind the Arabic name al-majisṭī. Ptolemy set up a public inscription at Canopus, Egypt, in 147 or 148, the late N. T. Hamilton found that the version of Ptolemys models set out in the Canopic Inscription was earlier than the version in the Almagest. Hence it cannot have been completed before about 150, a century after Ptolemy began observing. The Syntaxis Mathematica consists of thirteen sections, called books, an example illustrating how the Syntaxis was organized is given below. It is a 152-page 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 the vertical, Book III covers the length of the year, and the motion of the Sun. Ptolemy explains Hipparchus discovery of the precession of the equinoxes and begins explaining the theory of epicycles. Books IV and V cover the motion of the Moon, lunar parallax, the motion of the apogee. 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. The brightest stars were marked first magnitude, while the faintest visible to the eye were sixth magnitude. Each numerical magnitude was twice the brightness of the following one and this system is believed to have originated with Hipparchus. The stellar positions too are of Hipparchan origin, despite Ptolemys claim to the contrary, Book IX addresses general issues associated with creating models for the five naked eye planets, and the motion of Mercury. Book X covers the motions of Venus and Mars, Book XI covers the motions of Jupiter and Saturn. Book XII covers stations and retrograde motion, which occurs when planets appear to pause, Ptolemy understood these terms to apply to Mercury and Venus as well as the outer planets. Book XIII covers motion in latitude, that is, the deviation of planets from the ecliptic, the cosmology of the Syntaxis includes five main points, each of which is the subject of a chapter in Book I
Almagest
–
Ptolemy's Almagest became an authoritative work for many centuries.
Almagest
Almagest
–
Picture of George Trebizond's Latin translation of Almagest
20.
Mixed radix
–
Mixed radix numeral systems are non-standard positional numeral systems in which the numerical base varies from position to position. Such numerical representation applies when a quantity is expressed using a sequence of units that are each a multiple of the smaller one. 32,5,7,45,15,500. ∞,7,24,60,60,1000 or as 32∞577244560.15605001000 In the tabular format, the digits are written above their base, and a semicolon indicates the radix point. In numeral format, each digit has its base attached as a subscript. The base for each digit is the number of corresponding units that make up the larger unit. As a consequence there is no base for the first digit, the most familiar example of mixed radix systems is in timekeeping and calendars. Western time radices include decimal centuries, decades and years as well as duodecimal months, trigesimal days, overlapped with base 52 weeks, one variant uses tridecimal months, quaternary weeks, and septenary days. Time is further divided by quadrivigesimal hours, sexagesimal minutes and seconds, a mixed radix numeral system can often benefit from a tabular summary. m. On Wednesday, and 070201202602460 would be 12,02,24 a. m. on Sunday, ad hoc notations for mixed radix numeral systems are commonplace. The Maya calendar consists of several overlapping cycles of different radices, a short count tzolkin overlaps vigesimal named days with tridecimal numbered days. A haab consists of vigesimal days, octodecimal months, and base-52 years forming a round, in addition, a long count of vigesimal days, octodecimal winal, then vigesimal tun, katun, baktun, etc. tracks historical dates. So, for example, in the UK, banknotes are printed for £50, £20, £10 and £5, mixed-radix numbers of the same base can be manipulated using a generalization of manual arithmetic algorithms. APL and J include operators to convert to and from mixed-radix systems, another proposal is the so-called factorial number system, For example, the biggest number that could be represented with six digits would be 543210 which equals 719 in decimal, 5×5. It might not be clear at first sight but the factorial based numbering system is unambiguous and complete. Every number can be represented in one and only one way because the sum of respective factorials multiplied by the index is always the next factorial minus one, −1 There is a natural mapping between the integers 0. N. −1 and permutations of n elements in lexicographic order, the above equation is a particular case of the following general rule for any radix base representation which expresses the fact that any radix base representation is unambiguous and complete. The Art of Computer Programming, Volume 2, Seminumerical Algorithms, Über einfache Zahlensysteme, Zeitschrift für Math. Mixed Radix Calculator — Mixed Radix Calculator in C#
Mixed radix
–
Numeral systems
21.
Floor and ceiling functions
–
In mathematics and computer science, the floor and ceiling functions map a real number to the greatest preceding or the least succeeding integer, respectively. More precisely, floor = ⌊ x ⌋ is the greatest integer less than or equal to x, carl Friedrich Gauss introduced the square bracket notation for the floor function in his third proof of quadratic reciprocity. This remained the standard in mathematics until Kenneth E. Iverson introduced the names floor and ceiling, both notations are now used in mathematics, this article follows Iverson. e. The value of x rounded to an integer towards 0, 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 that is designed to use standard symbols, uses >. for ceiling. 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 ⌋. HTML4.0 uses the names, &lfloor, &rfloor, &lceil. 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, and n are integers, and Z is the set of integers. Floor and ceiling may be defined by the set equations ⌊ x ⌋ = max, ⌈ x ⌉ = min. Since there is exactly one integer in an interval of length one. Then ⌊ x ⌋ = m and ⌈ x ⌉ = n may also be taken as the definition of floor and these formulas can be used to simplify expressions involving floors and ceilings. In the language of order theory, the function is a residuated mapping. These formulas show how adding integers to the arguments affect the functions, negating the argument complements the fractional part, + = {0 if x ∈ Z1 if x ∉ Z. The floor, ceiling, and fractional part functions are idempotent, the result of nested floor or ceiling functions is the innermost function, ⌊ ⌈ x ⌉ ⌋ = ⌈ x ⌉, ⌈ ⌊ x ⌋ ⌉ = ⌊ x ⌋. If m and n are integers and n ≠0,0 ≤ ≤1 −1 | n |. If n is a positive integer ⌊ x + m n ⌋ = ⌊ ⌊ x ⌋ + m n ⌋, ⌈ x + m n ⌉ = ⌈ ⌈ x ⌉ + m n ⌉. For m =2 these imply n = ⌊ n 2 ⌋ + ⌈ n 2 ⌉
Floor and ceiling functions
–
Floor function
22.
Sylvester's sequence
–
In number theory, Sylvesters sequence is an integer sequence in which each member of the sequence is the product of the previous members, plus one. The first few terms of the sequence are,2,3,7,43,1807,3263443,10650056950807,113423713055421844361000443, Sylvesters sequence is named after James Joseph Sylvester, who first investigated it in 1880. Values derived from this sequence have also used to construct finite Egyptian fraction representations of 1, Sasakian Einstein manifolds. Formally, Sylvesters 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 s i = s i −1 +1. 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. Specifically, it can be shown that s n = ⌊ E2 n +1 +12 ⌋, for a number E that is approximately 1.264084735305302. This would only be an algorithm if we had a better way of calculating E to the requisite number of places than calculating sn. The partial sums of this series have a form, ∑ i =0 j −11 s i =1 −1 s j −1 = s j −2 s j −1. Since this sequence of partial sums / converges to one, the series forms an infinite Egyptian fraction representation of the number one,1 =12 +13 +17 +143 +11807 + ⋯. The sum of the first k terms of the series provides the closest possible underestimate of 1 by any k-term Egyptian fraction. For example, the first four terms add to 1805/1806, alternatively, the terms of the sequence after the first can be viewed as the denominators of the odd greedy expansion of 1/2. As Sylvester himself observed, Sylvesters sequence seems to be unique in having such quickly growing values and this sequence provides an example showing that double-exponential growth is not enough to cause an integer sequence to be an irrationality sequence. Erdős & Graham conjectured that, in results of this type, badea surveys progress related to this conjecture, see also Brown. If i < j, it follows from the definition that sj ≡1, therefore, every two numbers in Sylvesters sequence are relatively prime. The sequence can be used to prove there are infinitely many prime numbers. More strongly, no prime factor of a number in the sequence can be congruent to 5, much remains unknown about the factorization of the numbers in the Sylvesters sequence. For instance, it is not known if all numbers in the sequence are squarefree, via this technique he found that 1166 out of the first three million primes are divisors of Sylvester numbers, and that none of these primes has a square that divides a Sylvester number
Sylvester's sequence
–
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.
23.
James Joseph Sylvester
–
James Joseph Sylvester FRS was an English mathematician. He made fundamental contributions to theory, invariant theory, number theory, partition theory. He played a role in American mathematics in the later half of the 19th century as a professor at the Johns Hopkins University. At his death, he was professor at Oxford, Sylvester was born James Joseph in London, England. His father, 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 St Johns College, Cambridge in 1831, for the same reason, he was unable to compete for a Fellowship or obtain a Smiths prize. In 1838 Sylvester became professor of philosophy at University College London. In 1841, he was awarded a BA and an MA by Trinity College, following his early retirement, Sylvester published a book entitled The Laws of Verse in which he attempted to codify a set of laws for prosody in poetry. In 1872, he received his B. A. and M. A. from Cambridge. In 1876 Sylvester again crossed the Atlantic Ocean to become the professor of mathematics at the new Johns Hopkins University in Baltimore. His salary was $5,000, which he demanded be paid in gold, after negotiation, agreement was reached on a salary that was not paid in gold. In 1878 he founded the American Journal of Mathematics, the only other mathematical journal in the US at that time was the Analyst, which eventually became the Annals of Mathematics. In 1883, he returned to England to take up the Savilian Professor of Geometry at Oxford University and he held this chair until his death, although in 1892 the University appointed a deputy professor to the same chair. Sylvester invented a number of mathematical terms such as matrix, graph. He coined the term totient for Eulers totient function φ and his collected scientific work fills four volumes. In Discrete geometry he is remembered for Sylvesters Problem and a result on the orchard problem, Sylvester House, a portion of an undergraduate dormitory at Johns Hopkins University, is named in his honor. Several professorships there are named in his honor also, the collected mathematical papers of James Joseph Sylvester, I, New York, AMS Chelsea Publishing, ISBN 978-0-8218-3654-5 Sylvester, James Joseph, Baker, Henry Frederick, ed
James Joseph Sylvester
–
James Joseph Sylvester
24.
Smooth number
–
In number theory, a smooth number is an integer which factors completely into small prime numbers. The term seems to have coined by Leonard Adleman. Smooth numbers are 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 and this definition includes numbers that lack some of the smaller prime factors, for example, both 10 and 12 are 5-smooth, despite the fact that they miss out prime factors 3 and 5 respectively. 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 cases of this recursion are small primes. 5-smooth or regular numbers play a role in Babylonian mathematics. They are also important in theory, and 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 hash function is one example of a constructive use of smoothness to obtain a provably secure design. Let Ψ denote the number of y-smooth integers less than or equal to x, if the smoothness bound B is fixed and small, there is a good estimate for Ψ, Ψ ∼1 π. ∏ p ≤ B log x log p. where π denotes the number of less than or equal to B. Otherwise, define the parameter u as u = log x / log y, then, Ψ = x ⋅ ρ + O where ρ is the Dickman function. The average size of the part of a number of given size is known as ζ. Further, m is called B-powersmooth if all prime powers p ν dividing m satisfy, for example,720 is 5-smooth but is not 5-powersmooth. It is 16-powersmooth since its greatest prime factor power is 24 =16, the number is also 17-powersmooth, 18-powersmooth, etc. B-smooth and B-powersmooth numbers have applications in number theory, such as in Pollards p −1 algorithm, for example, the Pohlig–Hellman algorithm for computing discrete logarithms has a running time of O for groups of B-smooth order
Smooth number
–
Overview
25.
Primary pseudoperfect number
–
The eight known primary pseudoperfect numbers are 2,6,42,1806,47058,2214502422,52495396602,8490421583559688410706771261086. The first four of these numbers are one less than the numbers in Sylvesters sequence. It is unknown whether there are many primary pseudoperfect numbers. The prime factors of primary pseudoperfect numbers sometimes may provide solutions to Známs problem, for instance, the prime factors of the primary pseudoperfect number 47058 form the solution set to Známs problem. Anne observes that there is one solution set of this type that has k primes in it, for each k ≤8. If a primary pseudoperfect number N is one less than a prime number, for instance,47058 is primary pseudoperfect, and 47059 is prime, so 47058 ×47059 =2214502422 is also primary pseudoperfect. Primary pseudoperfect numbers were first investigated and named by Butske, Jaje and those with 2 ≤ r ≤8, when reduced modulo 288, form the arithmetic progression 6,42,78,114,150,186,222, as was observed by Sondow and MacMillan. Giuga number Anne, Premchand, Egyptian fractions and the problem, The College Mathematics Journal, Mathematical Association of America,29, 296–300, doi,10. 2307/2687685. Butske, William, Jaje, Lynda M. Mayernik, Daniel R, on the equation ∑ p | N1 p +1 N =1, pseudoperfect numbers, and perfectly weighted graphs, Mathematics of Computation,69, 407–420, doi,10. 1090/S0025-5718-99-01088-1. Weisstein, Eric W. Primary Pseudoperfect Number
Primary pseudoperfect number
–
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.
26.
Irrational number
–
In mathematics, the irrational numbers are all the real numbers, which are not rational numbers, the latter being the numbers constructed from ratios of integers. Irrational numbers may also be dealt with via non-terminating continued fractions, for example, the decimal representation of the number π starts with 3.14159265358979, but no finite number of digits can represent π exactly, nor does it repeat. Mathematicians do not generally take terminating or repeating to be the definition of the concept of rational number, as a consequence of Cantors proof that the real numbers are uncountable and the rationals countable, it follows that almost all real numbers are irrational. The first proof of the existence of numbers is usually attributed to a Pythagorean. The then-current Pythagorean method would have claimed that there must be sufficiently small. However, Hippasus, in the 5th century BC, was able to deduce that there was in no common unit of measure. His reasoning is as follows, Start with a right triangle with side lengths of integers a, b. The ratio of the hypotenuse to a leg is represented by c, b, assume a, b, and 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, and therefore even. Since c2 is even, c must be even, since c is even, dividing c by 2 yields an integer. Squaring both sides of c = 2y yields c2 =2, or c2 = 4y2, substituting 4y2 for c2 in the first equation gives us 4y2= 2b2. Dividing by 2 yields 2y2 = b2, since y is an integer, and 2y2 = b2, b2 is divisible by 2, and therefore even. Since b2 is even, b must be even and we have just show that both b and c must be even. Hence they have a factor of 2. However this contradicts the assumption that they have no common factors and this contradiction proves that c and b cannot both be integers, and thus the existence of a number that cannot be expressed as a ratio of two integers. Greek mathematicians termed this ratio of incommensurable magnitudes alogos, or inexpressible. ”Another legend states that Hippasus was merely exiled for this revelation, the discovery of incommensurable ratios was indicative of another problem facing the Greeks, the relation of the discrete to the continuous. Brought into light by Zeno of Elea, who questioned the conception that quantities are discrete and composed of a number of units of a given size. ”However Zeno found that in fact “ in general are not discrete collections of units. That in fact, these divisions of quantity must necessarily be infinite, for example, consider a line segment, this segment can be split in half, that half split in half, the half of the half in half, and so on
Irrational number
–
The number is irrational.
27.
Double exponential function
–
A double exponential function is a constant raised to the power of an exponential function. The general formula is f = a b x = a, for example, if a = b =10, f =10 f =1010 f =10100 = googol f =101000 f =1010100 = googolplex. Factorials grow more quickly than exponential functions, but much more slowly 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 double exponential function is the double logarithm ln. Aho and Sloane observed that in several important integer sequences, each term is a constant plus the square of the previous term. They show that such sequences can be formed by rounding to the nearest integer the values of an exponential function in which the middle exponent is two. Integer sequences with this behavior include The Fermat numbers F =22 m +1 The harmonic primes, The primes p. The first few numbers, starting with 0, are 2,5,277,5195977. The Double Mersenne numbers M M =22 p −1 −1 The elements of Sylvesters sequence s n = ⌊ E2 n +1 +12 ⌋ where E ≈1.264084735305302 is Vardis constant. Additional sequences of this type include The prime numbers 2,11,1361, a = ⌊ A3 n ⌋ where A ≈1.306377883863 is Mills constant. In the worst case, a Gröbner basis may have a number of elements which is exponential in the number of variables. On the other hand, the complexity of Gröbner basis algorithms is doubly exponential in the number of variables as well as in the entry size. Finding a complete set of associative-commutative unifiers Satisfying CTL+ Quantifier elimination on real closed fields takes doubly exponential time. An example is Chans algorithm for computing convex hulls, which performs a sequence of computations using test values hi = 22i, thus, the overall time for the algorithm is O where h is the actual output size. Some number theoretical bounds are double exponential, odd perfect numbers with n distinct prime factors are known to be at most 24 n a result of Nielsen. The maximal volume of a polytope with k ≥1 interior lattice points is at most d ⋅15 d ⋅22 d +1 a result of Pikhurko. The largest known prime number in the era has grown roughly as a double exponential function of the year since Miller and Wheeler found a 79-digit prime on EDSAC1 in 1951. In population dynamics the growth of population is sometimes supposed to be double exponential
Double exponential function
–
A double exponential function (red curve) compared to a single exponential function (blue curve).
28.
Dorian M. Goldfeld
–
Dorian Morris Goldfeld is an American mathematician. He received his B. S. degree in 1967 from Columbia University and his doctoral dissertation, entitled Some Methods of Averaging in the Analytical Theory of Numbers, was completed under the supervision of Patrick X. Gallagher in 1969, also at Columbia. He has held positions at the University of California at Berkeley, Hebrew University, Tel Aviv University, Institute for Advanced Study, in Italy, at MIT, University of Texas at Austin, since 1985, he has been a professor at Columbia University. He is a member of the board of Acta Arithmetica. He is a co-founder and board member of SecureRF, a corporation that has developed the world’s first linear-based security solutions, Dorian Goldfelds research interests include various topics in number theory. In his thesis, he proved a version of Artins conjecture on primitive roots on the average 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 class number after a finite number of computations. He has also made contributions to the understanding of Siegel zeroes, to the ABC conjecture, to modular forms on GL, and to cryptography. Together with his wife, Dr. Iris Anshel, and father-in-law, Dr. Michael Anshel, in 1987 he received the Frank Nelson Cole Prize in Number Theory, one of the prizes in Number Theory, for his solution of Gauss class number problem for imaginary quadratic fields. He has also held the Sloan Fellowship and 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, in 2012 he became a fellow of the American Mathematical Society. Automorphic Representations and L-Functions for the General Linear Group, Volume 1, automorphic Representations and L-Functions for the General Linear Group, Volume 2. Editors, Gerritzen, Goldfeld, Kreuzer, Rosenberger and Shpilrain, cS1 maint, Multiple names, authors list Goldfeld, Dorian. Automorphic Forms and L-Functions for the Group GL, Dorian M. Goldfeld at the Mathematics Genealogy Project Dorian Goldfelds Home Page at Columbia University
Dorian M. Goldfeld
–
Dorian Goldfeld at The Analytic Theory of Automorphic Forms workshop, Oberwolfach, Germany (2011)
29.
Journal of Number Theory
–
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 by R. P. Bambah, P. Roquette, A. Ross, A. Woods and it is currently published monthly by Elsevier, with 6 volumes per year
Journal of Number Theory
–
Journal of Number Theory
30.
JSTOR
–
JSTOR is a digital library founded in 1995. Originally containing digitized back issues of journals, it now also includes books and primary sources. It provides full-text searches of almost 2,000 journals, more than 8,000 institutions in more than 160 countries have access to JSTOR, most access is by subscription, but some older public domain content is freely available to anyone. William G. Bowen, president of Princeton University from 1972 to 1988, JSTOR originally was conceived as a solution to one of the problems faced by libraries, especially research and university libraries, due to the increasing number of academic journals in existence. Most libraries found it prohibitively expensive in terms of cost and space to maintain a 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, online access and full-text search ability improved access dramatically. Bowen initially considered using CD-ROMs for distribution, JSTOR was initiated in 1995 at seven different library sites, and originally encompassed ten economics and history journals. JSTOR access improved based on feedback from its sites. Special software was put in place to make pictures and graphs clear, with the success of this limited project, Bowen and Kevin Guthrie, then-president of JSTOR, wanted to expand the number of participating journals. They met with representatives of the Royal Society of London and an agreement was made to digitize the Philosophical Transactions of the Royal Society dating from its beginning in 1665, 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 as an independent, self-sustaining nonprofit organization with offices in New York City and in Ann Arbor, Michigan. JSTOR content is provided by more than 900 publishers, the database contains more than 1,900 journal titles, in more than 50 disciplines. Each object is identified by an integer value, starting at 1. In addition to the site, the JSTOR labs group operates an open service that allows access to the contents of the archives for the purposes of corpus analysis at its Data for Research service. This site offers a facility with graphical indication of the article coverage. Users may create focused sets of articles and then request a dataset containing word and n-gram frequencies and they are notified when the dataset is ready and may download it in either XML or CSV formats. The service does not offer full-text, although academics may request that from JSTOR, JSTOR Plant Science is available in addition to the main site. The materials on JSTOR Plant Science are contributed through the Global Plants Initiative and are only to JSTOR
JSTOR
–
The JSTOR front page
31.
American Mathematical Monthly
–
The American Mathematical Monthly is a mathematical journal founded by Benjamin Finkel in 1894. It is published ten times each year by the Mathematical Association of America, the American Mathematical Monthly is an expository journal intended for a wide audience of mathematicians, from undergraduate students to research professionals. Articles are chosen on the basis of their broad interest and reviewed and edited for quality of exposition as well as content, in this the American Mathematical Monthly fulfills a different role from that of typical 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 Americas website, the MAA gives the Lester R. Ford Awards annually to authors of articles of expository excellence published in the American Mathematical Monthly
American Mathematical Monthly
–
American Mathematical Monthly
32.
International Standard Book Number
–
The International Standard Book Number is a unique numeric commercial book identifier. An ISBN is assigned to each edition and variation of a book, for example, an e-book, a paperback and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, the method of assigning an ISBN is nation-based and 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 in 1967 based upon the 9-digit Standard Book Numbering created in 1966, the 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108. Occasionally, a book may appear without a printed ISBN if it is printed privately or the author does not follow the usual ISBN procedure, however, this can be rectified later. Another identifier, the International Standard Serial Number, identifies periodical publications such as magazines, the ISBN configuration of recognition was generated in 1967 in the United Kingdom by David Whitaker and in 1968 in the US by Emery Koltay. The 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108, 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 to an ISBN by prefixing the digit 0. For example, the edition of Mr. J. G. Reeder Returns, published by Hodder in 1965, has SBN340013818 -340 indicating the publisher,01381 their serial number. This can be converted to ISBN 0-340-01381-8, the check digit does not need to be re-calculated, since 1 January 2007, ISBNs have contained 13 digits, a format that is compatible with Bookland European Article Number EAN-13s. An ISBN is assigned to each edition and variation of a book, for example, an ebook, a paperback, and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, a 13-digit ISBN can be separated into its parts, and when this is done it is customary to separate the parts with hyphens or spaces. Separating the parts of a 10-digit ISBN is also done with either hyphens or spaces, figuring out how to correctly separate a given ISBN number is complicated, because most of the parts do not use a fixed number of digits. ISBN issuance is country-specific, in that ISBNs are issued by the ISBN registration agency that is responsible for country or territory regardless of the publication language. Some ISBN registration agencies are based in national libraries or within ministries of culture, in other cases, the ISBN registration service is provided by organisations such as bibliographic data providers that are not government funded. In Canada, ISBNs are issued at no cost with the purpose of encouraging Canadian culture. In the United Kingdom, United States, and some countries, where the service is provided by non-government-funded organisations. Australia, ISBNs are issued by the library services agency Thorpe-Bowker
International Standard Book Number
–
A 13-digit ISBN, 978-3-16-148410-0, as represented by an EAN-13 bar code
33.
Ronald Graham
–
He has done important work in scheduling theory, computational geometry, Ramsey theory, and quasi-randomness. Graham was born in Taft, California, in 1962, he received his Ph. D. in mathematics from the University of California, Berkeley and began working at Bell Labs and later AT&T Labs. He was director of information sciences in AT&T Labs, but retired from AT&T in 1999 after 37 years and his 1977 paper considered a problem in Ramsey theory, and 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 scientists Erdős number is the minimum number of coauthored publications away from a publication with Erdős. He co-authored almost 30 papers with Erdős, and was also a good friend, Erdős often stayed with Graham, and 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 brain cancer, between 1993 and 1994 Graham served as the president of the American Mathematical Society. He has published about 320 papers and five books, including Concrete Mathematics with Donald Knuth and he is married to Fan Chung Graham, who is the Akamai Professor in Internet Mathematics at the University of California, San Diego. He has four children, daughters Ché, Laura and Christy, in 2003, Graham won the American Mathematical Societys annual Steele Prize for Lifetime Achievement. The prize was awarded on January 16 that year, at the Joint Mathematics Meetings in Baltimore, in 1999 he was inducted as a Fellow of the Association for Computing Machinery. Graham has won other prizes over the years, he was one of the laureates of the prestigious Pólya Prize the first year it was ever awarded. And the Carl Allendoerfer prize which was established in 1976 for the reasons, however for a different magazine. In 2012 he became a fellow of the American Mathematical Society, with Paul Erdős, Old and new results in combinatorial number theory. L’Enseignement Mathématique,1980 with Fan Chung, Erdős on Graphs, a. K. Peters,1998 with Jaroslav Nešetřil, The mathematics of Paul Erdős. Springer,1997 Rudiments of Ramsey Theory, American Mathematical Society,1981 with Donald E. Knuth & Oren Patashnik, Concrete Mathematics, a foundation for computer science. Addison-Wesley,1989,1994 with Joel H, spencer & Bruce L. Rothschild, Ramsey Theory. Wiley,1980,1990 with Martin Grötschel & László Lovász, Handbook of Combinatorics
Ronald Graham
–
Ronald Graham
Ronald Graham
–
Ronald Graham juggling a four ball fountain (1986)
Ronald Graham
–
Ronald Graham, his wife Fan Chung, and Paul Erdős, Japan 1986
34.
Richard K. Guy
–
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 theory, geometry, recreational mathematics, combinatorics. He is best known for co-authorship of Winning Ways for your Mathematical Plays and he has also published over 300 papers. For this paper he received the MAA Lester R. Ford Award, Guy was born 30 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 and he attended Warwick School for Boys, the third oldest school in Britain, but was not enthusiastic about most of the curriculum. He was good at sports, however, and excelled in mathematics, at the age of 17 he read Dicksons History of the Theory of Numbers. He said it was better than the works of Shakespeare. By then he had developed a passion for mountain climbing. In 1935 Guy entered Gonville and Caius College, at the University of Cambridge as a result of winning several scholarships, to win the most important of these he had to travel to Cambridge and write exams for two days. His interest in games began while at Cambridge where he became a composer of chess problems. In 1938, he graduated with an honours degree, he himself thinks that his failure to get a first may have been related to his obsession with chess. Although his parents advised against it, Guy decided to become a teacher. He met his future wife Nancy Louise Thirian through her brother Michael who was a fellow scholarship winner at Gonville and he and Louise shared loves of mountains and dancing. He wooed her through correspondence, and they married in December 1940, in November 1942, Guy received an emergency commission in the Meteorological Branch of the Royal Air Force, with the rank of flight lieutenant. He was posted to Reykjavik, and later to Bermuda, as a meteorologist and he tried to get permission for Louise to join him but was refused. While in Iceland, he did some glacier travel, skiing and mountain climbing, marking the beginning of another love affair. When Guy returned to England after the war, he went back to teaching, this time at Stockport Grammar School, in 1947 the family moved to London, where he got a job teaching math at Goldsmiths College. In 1951 he moved to Singapore, where he taught at the University of Malaya for the next decade and he then spent a few years at the Indian Institute of Technology in Delhi, India
Richard K. Guy
–
Richard K. Guy in June 2005
35.
American Journal of Mathematics
–
The American Journal of Mathematics is a bimonthly mathematics journal published by the Johns Hopkins University Press. Story was associate editor in charge, he was replaced by Thomas Craig in 1880, for volume 7 Simon Newcomb became chief editor with Craig managing until 1894. Then with volume 16 it was Edited by Thomas Craig with the Co-operation of Simon Newcomb until 1898, fields medalist Cédric Villani has speculated that the most famous article in its long history may be a 1958 paper by John Nash, Continuity of solutions of parabolic and elliptic equations. The American Journal of Mathematics is a general-interest mathematics journal covering all the areas of contemporary mathematics. According to the Journal Citation Reports, its 2009 impact factor is 1.337, as of June,2012, the editors are Christopher D. Sogge, editor-in-chief, William Minicozzi II, Freydoon Shahidi, and Vyacheslav Shokurov
American Journal of Mathematics
–
American Journal of Mathematics
36.
Scientific American
–
Scientific American is an American popular science magazine. Many famous scientists, including Albert Einstein, have contributed articles in the past 170 years and it is the oldest continuously published monthly magazine in the United States. Scientific American was founded by inventor and publisher Rufus M. Porter in 1845 as a weekly newspaper. Throughout its early years, much emphasis was placed on reports of what was going on at the U. S, current issues include a this date in history section, featuring excerpts from articles originally published 50,100, and 150 years earlier. Topics include humorous incidents, wrong-headed theories, and noteworthy advances in the history of science, Porter sold the publication to Alfred Ely Beach and Orson Desaix Munn a mere ten months after founding it. Until 1948, it remained owned by Munn & Company, under Munns grandson, Orson Desaix Munn III, 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 fell into decline. Thus the partners—publisher Gerard Piel, editor Dennis Flanagan, and general manager Donald H. Miller, Miller retired in 1979, Flanagan and Piel in 1984, when Gerard Piels son Jonathan became president and editor, circulation had grown fifteen-fold since 1948. In 1986, it was sold to the Holtzbrinck group of Germany, in the fall of 2008, Scientific American was put under the control of Nature Publishing Group, a division of Holtzbrinck. Donald Miller died in December 1998, Gerard Piel in September 2004, Mariette DiChristina is the current editor-in-chief, after John Rennie stepped down in June 2009. Scientific American published its first foreign edition in 1890, the Spanish-language La America Cientifica, a Russian edition V Mire Nauki was launched in the Soviet Union in 1983, and continues in the present-day Russian Federation. Kexue, a simplified Chinese edition launched in 1979, was the first Western magazine published in the Peoples Republic of China, founded in Chongqing, the simplified Chinese magazine was transferred to Beijing in 2001. Later in 2005, an edition, Global Science, was published instead of Kexue. A traditional Chinese edition, known as 科學人, was introduced to Taiwan in 2002, the Hungarian edition Tudomány existed between 1984 and 1992. In 1986, an Arabic edition, Oloom magazine, was published, in 2002, a Portuguese edition was launched in Brazil. From 1902 to 1911, Scientific American supervised the publication of the Encyclopedia Americana and it originally styled itself The Advocate of Industry and Enterprise and Journal of Mechanical and other Improvements. On the front page of the first issue was the engraving of Improved Rail-Road Cars, the masthead had a commentary as follows, Scientific American published every Thursday morning at No.11 Spruce Street, New York, No.16 State Street, Boston, and No. 2l Arcade Philadelphia, by Rufus Porter, five copies will be sent to one address six months for four dollars in advance
Scientific American
–
Cover of the March 2005 issue
Scientific American
–
PDF of first issue: Scientific American Vol. 1, No. 01 published August 28, 1845
Scientific American
–
Special Navy Supplement, 1898
37.
David Eppstein
–
David Arthur Eppstein is an American computer scientist and mathematician. He is a Chancellors Professor of computer science at University of California and he is known for his work in computational geometry, graph algorithms, and recreational mathematics. In 2011, he was named an ACM Fellow and he joined the UC Irvine faculty in 1990, and was co-chair of the Computer Science Department there from 2002 to 2005. In 2014, he was named a Chancellors Professor, since 2007, Eppstein has been an administrator at the English Wikipedia. In 1984 Eppstein was awarded with a National Science Foundation Graduate Research Fellowship, in 1992, Eppstein received a National Science Foundation Young Investigator Award along with six other UC-Irvine academics. In 2011, he was named an ACM Fellow for his contributions to graph algorithms, Eppstein, D. Galil, Z. Italiano, G. F. Nissenzweig, A. Sparsification—a technique for speeding up dynamic graph algorithms. Amenta, N. Bern, M. Eppstein, D, the Crust and the β-Skeleton, Combinatorial Curve Reconstruction. Donald Bren School of Information and Computer Sciences, University of California, david Eppstein at the Mathematics Genealogy Project
David Eppstein
–
David Eppstein
38.
The Wolfram Demonstrations Project
–
It is hosted by Wolfram Research, whose stated goal is to bring computational exploration to the widest possible audience. At its launch, it contained 1300 demonstrations but has grown to over 10,000, the site won a Parents Choice Award in 2008. Each Demonstration also has a description of the concept. Demonstrations are now easily embeddable into any website or blog, each Demonstration page includes a snippet of JavaScript code in the Share section of the sidebar. The website is organized by topic, for example, science, mathematics, computer science, art, biology and they cover a variety of levels, from elementary school mathematics to much more advanced topics such as quantum mechanics and models of biological organisms. The site is aimed at educators and students, as well as researchers who wish to present their ideas to the broadest possible audience. Wolfram Researchs staff organizes and edits the Demonstrations, which may be created by any user of Mathematica, then freely published, the Demonstrations are open-source, which means that they not only demonstrate the concept itself but also show how to implement it. The use of the web to transmit small interactive programs is reminiscent of Suns Java applets, Adobes Flash, however, those creating Demonstrations have access to the algorithmic and visualization capabilities of Mathematica making it more suitable for technical demonstrations. The Demonstrations Project also has similarities to user-generated content websites like Wikipedia and its business model is similar to Adobes Acrobat and Flash strategy of charging for development tools but providing a free reader
The Wolfram Demonstrations Project
–
Surface tangents.
The Wolfram Demonstrations Project
–
Shell growth.
The Wolfram Demonstrations Project
–
Legal structures.
39.
Ratio
–
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 eight oranges and six lemons, thus, a ratio can be a fraction as opposed to a whole number. Also, in example the ratio of lemons to oranges is 6,8. The numbers compared in a ratio can be any quantities of a kind, such as objects, persons, lengths. A ratio is written a to b or a, b, 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 word ratio is often used instead for this more general notion as well. The numbers A and B are sometimes called terms with A being the antecedent, the proportion expressing the equality of the ratios A, B and C, D is written A, B = C, D or A, B, C, D. This latter form, when spoken or written in the English language, is expressed as A is to B as C is to D. A, B, C and D are called the terms of the proportion. A and D are called the extremes, and B and C are called the means, the equality of three or more proportions is called a continued proportion. Ratios are sometimes used three or more terms. The ratio of the dimensions of a two by four that is ten inches long is 2,4,10, a good concrete mix is sometimes quoted as 1,2,4 for the ratio of cement to sand to gravel. It is impossible to trace the origin of the concept of ratio because the ideas from which it developed would have been familiar to preliterate cultures. For example, the idea of one village being twice as large as another is so basic that it would have been understood in prehistoric society, however, it is possible to trace the origin of the word ratio to the Ancient Greek λόγος. Early translators rendered this into Latin as ratio, a more modern interpretation of Euclids meaning is more akin to computation or reckoning. Medieval writers used the word to indicate ratio and proportionalitas for the equality of ratios, Euclid collected the results appearing in the Elements from earlier sources. The Pythagoreans developed a theory of ratio and proportion as applied to numbers, the discovery of a theory of ratios that does not assume commensurability is probably due to Eudoxus of Cnidus. The exposition of the theory of proportions that appears in Book VII of The Elements reflects the earlier theory of ratios of commensurables, the existence of multiple theories seems unnecessarily complex to modern sensibility since ratios are, to a large extent, identified with quotients. This is a recent development however, as can be seen from the fact that modern geometry textbooks still use distinct terminology and notation for ratios
Ratio
–
The ratio of width to height of standard-definition television.
40.
Binary number
–
The base-2 system is a positional notation with a radix of 2. Because of its implementation in digital electronic circuitry using logic gates. Each digit is referred to as a bit, the modern binary number system was devised by Gottfried Leibniz in 1679 and appears in his article Explication de lArithmétique Binaire. Systems related to binary numbers have appeared earlier in multiple cultures including ancient Egypt, China, Leibniz was specifically inspired by the Chinese I Ching. The scribes of ancient Egypt used two different systems for their fractions, Egyptian fractions and Horus-Eye fractions, the method used for ancient Egyptian multiplication is also closely related to binary numbers. This method can be seen in use, for instance, in the Rhind Mathematical Papyrus, 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 and it is based on taoistic duality of yin and yang. Eight trigrams and a set of 64 hexagrams, analogous to the three-bit and six-bit binary numerals, were in use at least as early as the Zhou Dynasty of ancient China. The Song Dynasty scholar Shao Yong rearranged the hexagrams in a format that resembles modern binary numbers, the Indian scholar Pingala developed a binary system for describing prosody. He used binary numbers in the form of short and long syllables, Pingalas Hindu classic titled Chandaḥśāstra describes the formation of a matrix in order to give a unique value to each meter. The binary representations in Pingalas system increases towards the right, the residents of the island of Mangareva in French Polynesia were using a hybrid binary-decimal system before 1450. Slit drums with binary tones are used to encode messages across Africa, sets of binary combinations similar to the I Ching have also been used in traditional African divination systems such as Ifá as well as in medieval Western geomancy. The base-2 system utilized in geomancy had long been applied in sub-Saharan Africa. Leibnizs system uses 0 and 1, like the modern binary numeral system, Leibniz was first introduced to the I Ching through his contact with the French Jesuit Joachim Bouvet, who visited China in 1685 as a missionary. Leibniz saw the I Ching hexagrams as an affirmation of the universality of his own beliefs as a Christian. Binary numerals were central to Leibnizs theology and he believed that binary numbers were symbolic of the Christian idea of creatio ex nihilo or creation out of nothing. Is not easy to impart to the pagans, is the ex nihilo through Gods almighty power. In 1854, British mathematician George Boole published a paper detailing an algebraic system of logic that would become known as Boolean algebra
Binary number
–
Numeral systems
Binary number
–
Arithmetic values represented by parts of the Eye of Horus
Binary number
–
Gottfried Leibniz
Binary number
–
George Boole
41.
Decimal fraction
–
This article aims to be an accessible introduction. For the mathematical definition, see Decimal representation, the decimal numeral system has ten as its base, which, in decimal, is written 10, as is the base in every positional numeral system. It is the base most widely used by modern civilizations. Decimal fractions have terminating decimal representations and other fractions have repeating decimal representations, Decimal notation is the writing of numbers in a base-ten numeral system. Examples are Brahmi numerals, Greek numerals, Hebrew numerals, Roman numerals, Roman numerals have symbols for the decimal powers and secondary symbols for half these values. Brahmi numerals have symbols for the nine numbers 1–9, the nine decades 10–90, plus a symbol for 100, Chinese numerals have symbols for 1–9, and additional symbols for powers of ten, which in modern usage reach 1072. Positional decimal systems include a zero and use symbols for the ten values to represent any number, positional notation uses positions for each power of ten, units, tens, hundreds, thousands, etc. The position of each digit within a number denotes the multiplier multiplied with that position has a value ten times that of the position to its right. There were at least two independent sources of positional decimal systems in ancient civilization, the Chinese counting rod system. Ten is the number which is the count of fingers and thumbs on both hands, the English word digit as well as its translation in many languages is also the anatomical term for fingers and toes. In English, decimal means tenth, decimate means reduce by a tenth, however, the symbols used in different areas are not identical, for instance, Western Arabic numerals differ from the forms used by other Arab cultures. A decimal fraction is a fraction the denominator of which is a power of ten. g, Decimal fractions 8/10, 1489/100, 24/100000, and 58900/10000 are expressed in decimal notation as 0.8,14.89,0.00024,5.8900 respectively. In English-speaking, some Latin American and many Asian countries, a period or raised period is used as the separator, in many other countries, particularly in Europe. The integer part, or integral part of a number is the part to the left of the decimal separator. The part from the separator to the right is the fractional part. It is usual for a number that consists only of a fractional part to have a leading zero in its notation. Any rational number with a denominator whose only prime factors are 2 and/or 5 may be expressed as a decimal fraction and has a finite decimal expansion. 1/2 =0.5 1/20 =0.05 1/5 =0.2 1/50 =0.02 1/4 =0.25 1/40 =0.025 1/25 =0.04 1/8 =0.125 1/125 =0.008 1/10 =0
Decimal fraction
–
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
–
Numeral systems
Decimal fraction
–
Ten fingers on two hands, the possible starting point of the decimal counting.
Decimal fraction
–
Diagram of the world's earliest decimal multiplication table (c. 305 BC) from the Warring States period
42.
Golden ratio
–
In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. The figure on the right illustrates the geometric relationship, expressed algebraically, for quantities a and b with a > b >0, a + b a = a b = def φ, where the Greek letter phi represents the golden ratio. Its value is, φ =1 +52 =1.6180339887 …, A001622 The golden ratio is also called the golden mean or golden section. Other names include extreme and mean ratio, medial section, divine proportion, divine section, golden proportion, golden cut, the golden ratio appears in some patterns in nature, including the spiral arrangement of leaves and other plant parts. The golden ratio has also used to analyze the proportions of natural objects as well as man-made systems such as financial markets. Two quantities a and b are said to be in the golden ratio φ if a + b a = a b = φ, one method for finding the value of φ is to start with the left fraction. Through simplifying the fraction and substituting in b/a = 1/φ, a + b a =1 + b a =1 +1 φ, multiplying by φ gives φ +1 = φ2 which can be rearranged to φ2 − φ −1 =0. First, the line segment A B ¯ is about doubled and then the semicircle with the radius A S ¯ around the point S is drawn, now the semicircle is drawn with the radius A B ¯ around the point B. The arising intersection point E corresponds 2 φ, next up, the perpendicular on the line segment A E ¯ from the point D will be establish. The subsequent parallel F S ¯ to the line segment C M ¯, produces, as it were and it is well recognizable, this triangle and the triangle M S C are similar to each other. The hypotenuse F S ¯ has due to the cathetuses S D ¯ =1 and D F ¯ =2 according the Pythagorean theorem, finally, the circle arc is drawn with the radius 5 around the point F. The golden ratio has been claimed to have held a fascination for at least 2,400 years. But the fascination with the Golden Ratio is not confined just to mathematicians, biologists, artists, musicians, historians, architects, psychologists, and even mystics have pondered and debated the basis of its ubiquity and appeal. In fact, it is fair to say that the Golden Ratio has inspired thinkers of all disciplines like no other number in the history of mathematics. Ancient Greek mathematicians first studied what we now call the golden ratio because of its frequent appearance in geometry, the division of a line into extreme and mean ratio is important in the geometry of regular pentagrams and pentagons. Euclid explains a construction for cutting a line in extreme and mean ratio, throughout the Elements, several propositions and their proofs employ the golden ratio. The golden ratio is explored in Luca Paciolis book De divina proportione, since the 20th century, the golden ratio has been represented by the Greek letter φ or less commonly by τ. Timeline according to Priya Hemenway, Phidias made the Parthenon statues that seem to embody the golden ratio, plato, in his Timaeus, describes five possible regular solids, some of which are related to the golden ratio
Golden ratio
–
Michael Maestlin, first to publish a decimal approximation of the golden ratio, in 1597
Golden ratio
–
Line segments in the golden ratio
Golden ratio
–
Many of the proportions of the Parthenon are alleged to exhibit the golden ratio.
Golden ratio
–
The drawing of a man's body in a pentagram suggests relationships to the golden ratio.
43.
Silver ratio
–
This defines the silver ratio as an irrational mathematical constant, whose value of one plus the square root of 2 is approximately 2.4142135623. The silver ratio is denoted by δS and these fractions provide accurate rational approximations of the silver ratio, analogous to the approximation of the golden ratio by ratios of consecutive Fibonacci numbers. Multiplying by δS and rearranging gives δ S2 −2 δ S −1 =0, using the quadratic formula, two solutions can be obtained. In fact it is the second smallest quadratic PV number after the golden ratio and this means the distance from δ n S to the nearest integer is 1/δ n S ≈0. 41n. Thus, the sequence of parts of δ n S, n =1,2,3. In particular, this sequence is not equidistributed mod 1, the paper sizes under ISO216 are rectangles in the proportion 1, √2, sometimes called A4 rectangles. Removing a largest possible square from a sheet of such paper leaves a rectangle with proportions 1, √2 −1 which is the same as 1 + √2,1, removing a largest square from one of these sheets leaves one again with aspect ratio 1, √2. A rectangle whose aspect ratio is the ratio is sometimes called a silver rectangle by analogy with golden rectangles. Confusingly, silver rectangle can also refer to the paper sizes specified by ISO216, however, only the 1, √2 rectangles have the property that by cutting the rectangle in half across its long side produces two smaller rectangles of the same aspect ratio. The silver rectangle is connected to the regular octagon, if the edge length of a regular octagon is t, then the inradius of the octagon is δSt, and the area of the octagon is 2δSt2. Metallic means Ammann–Beenker tiling Buitrago, Antonia Redondo, polygons, Diagonals, and the Bronze Mean, Nexus Network Journal 9,2, Architecture and Mathematics, p. 321-2. An Introduction to Continued Fractions, The Silver Means, Fibonacci Numbers, Silver rectangle and its sequence at Tartapelago by Giorgio Pietrocola
Silver ratio
–
Silver ratio within the octagon
44.
Interval (music)
–
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 and they can be formed using the notes of various kinds of non-diatonic scales. Some of the very smallest ones are called commas, and describe small discrepancies, observed in some tuning systems, Intervals can be arbitrarily small, and 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. This means that successive increments of pitch by the same result in an exponential increase of frequency. For this reason, intervals are often measured in cents, a derived from the logarithm of the frequency ratio. In Western music theory, the most common naming 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, each appropriate to a different context, frequency ratios or cents, the size of an interval between two notes may be measured by the ratio of their frequencies. When a musical instrument is tuned using a just intonation tuning system, Intervals with small-integer ratios are often called just intervals, or pure intervals. Most commonly, however, musical instruments are tuned using a different tuning system. As a consequence, the size of most equal-tempered intervals cannot be expressed by small-integer ratios, for instance, an equal-tempered fifth has a frequency ratio of 2 7⁄12,1, approximately equal to 1.498,1, or 2.997,2. For a comparison between the size of intervals in different tuning systems, see section Size in different tuning systems, the standard system for comparing interval sizes is with cents. The cent is a unit of measurement. If frequency is expressed in a scale, and along that scale the distance between a given frequency and its double is divided into 1200 equal parts, each of these parts is one cent. In twelve-tone equal temperament, a system in which all semitones have the same size. Hence, in 12-TET the cent can be defined as one hundredth of a semitone
Interval (music)
–
Melodic and harmonic intervals. Play (help · info)
45.
Percentage
–
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, a percentage is a dimensionless number. For example, 45% is equal to 45⁄100,45,100, percentages are often used to express a proportionate part of a total. If 50% of the number of students in the class are male. If there are 1000 students, then 500 of them are male, 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 a 6% increase, while many percentage values are between 0 and 100, there is no mathematical restriction and percentages may take on other values. For example, it is common to refer to 111% or −35%, especially for percent changes, 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 centesima rerum venalium, computation with these fractions was equivalent to computing percentages. Many of these texts applied these methods to profit and loss, interest rates, by the 17th century it was standard to quote interest rates in hundredths. The term per cent is derived from the Latin per centum, the sign for per cent evolved by gradual contraction of the Italian term per cento, meaning for a hundred. The per was often abbreviated as p. and eventually disappeared entirely, the cento was contracted to two circles separated by a horizontal line, from which the modern % symbol is derived. The percent value is computed by multiplying the value of the ratio by 100. For example, to find 50 apples as a percentage of 1250 apples, first compute the ratio 50⁄1250 =0.04, and then multiply by 100 to obtain 4%. The percent value can also be found by multiplying first, so in this example the 50 would be multiplied by 100 to give 5,000, and this result would be divided by 1250 to give 4%. To calculate a percentage of a percentage, convert both percentages to fractions of 100, or to decimals, and multiply them, for example, 50% of 40% is, 50⁄100 × 40⁄100 =0.50 ×0.40 =0.20 = 20⁄100 = 20%. It is not correct to divide by 100 and use the percent sign at the same time, whenever we talk about a percentage, it is important to specify what it is relative to, i. e. what is the total that corresponds to 100%. The following problem illustrates this point, in a certain college 60% of all students are female, and 10% of all students are computer science majors. If 5% of female students are computer science majors, what percentage of computer science majors are female and we are asked to compute the ratio of female computer science majors to all computer science majors
Percentage
–
Contents
46.
Egyptian fraction
–
An Egyptian fraction is a finite sum of distinct unit fractions, such as 12 +13 +116. That is, each fraction in the expression has an equal to 1 and a denominator that is a positive integer. The value of an expression of type is a positive rational number a/b. Every positive rational number can be represented by an Egyptian fraction, in modern mathematical notation, Egyptian fractions have been superseded by vulgar fractions and decimal notation. However, Egyptian fractions continue to be an object of study in modern theory and recreational mathematics. Beyond their historical use, Egyptian fractions have some 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, Eye of Horus, and Egyptian mathematics. Egyptian fraction notation was developed in the Middle Kingdom of Egypt, five early texts in which Egyptian fractions appear were the Egyptian Mathematical Leather Roll, the Moscow Mathematical Papyrus, the Reisner Papyrus, the Kahun Papyrus and the Akhmim Wooden Tablet. A later text, the Rhind Mathematical Papyrus, introduced improved ways of writing Egyptian fractions, the Rhind papyrus was written by Ahmes and dates from the Second Intermediate Period, it includes a table of Egyptian fraction expansions for rational numbers 2/n, as well as 84 word problems. Solutions to each problem were written out in scribal shorthand, with the 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. To write the unit used in their Egyptian fraction notation, in hieroglyph script. Similarly in hieratic script they drew a line over the letter representing the number. For example, The Egyptians had special symbols for 1/2, 2/3, the remaining number after subtracting one of these special fractions was written using as a sum of distinct unit fractions according to the usual Egyptian fraction notation. These have been called Horus-Eye fractions after a theory that they were based on the parts of the Eye of Horus symbol, the unit fraction 1/n is expressed as n, and the fraction 2/n is expressed as n, and the plus sign “＋” is omitted. For example, 2/3 = 1/2 + 1/6 is expressed as 3 =26, modern historians of mathematics have studied the Rhind papyrus and other ancient sources in an attempt to discover the methods the Egyptians used in calculating with Egyptian fractions. In particular, study in this area has concentrated on understanding the tables of expansions for numbers of the form 2/n in the Rhind papyrus, although these expansions can generally be described as algebraic identities, the methods used by the Egyptians may not correspond directly to these identities. This method is available for not only odd prime denominators but also all odd denominators, for larger prime denominators, an expansion of the form 2/p = 1/A + 2A − p/Ap was used, where A is a number with many divisors between p/2 and p
Egyptian fraction
–
Eye of Horus