Portal:Arithmetic
Portal maintenance status: (August 2018)

The Arithmetic Portal
Arithmetic (from the Greek ἀριθμός arithmos, ' number' and τική [τέχνη], tiké [téchne], ' art' or ' craft') is a branch of mathematics that consists of the study of numbers, especially the properties of the traditional operations on them— addition, subtraction, multiplication, division, exponentiation and extraction of roots. Arithmetic is an elementary part of number theory, and number theory is considered to be one of the toplevel divisions of modern mathematics, along with algebra, geometry, and analysis. The terms arithmetic and higher arithmetic were used until the beginning of the 20th century as synonyms for number theory, and are sometimes still used to refer to a wider part of number theory. ( Full article...)
The Egyptians and Babylonians used all the elementary arithmetic operations as early as 2000 BC. Later Roman numerals, descended from tally marks used for counting. The continuous development of modern arithmetic starts with ancient Greece, although it originated much later than the Babylonian and Egyptian examples. Euclid is often credited as the first mathematician to separate study of arithmetic from philosophical and mystical beliefs. Greek numerals were used by Archimedes, Diophantus and others in a positional notation not very different from ours. The ancient Chinese had advanced arithmetic studies dating from the Shang Dynasty and continuing through the Tang Dynasty, from basic numbers to advanced algebra. The ancient Chinese used a positional notation similar to that of the Greeks. The gradual development of the Hindu–Arabic numeral system independently devised the placevalue concept and positional notation, which combined the simpler methods for computations with a decimal base and the use of a digit representing zero (0). This allowed the system to consistently represent both large and small integers. This approach eventually replaced all other systems. In the Middle Ages, arithmetic was one of the seven liberal arts taught in universities. The flourishing of algebra in the medieval Islamic world and in Renaissance Europe was an outgrowth of the enormous simplification of computation through decimal notation.
Selected general articles
 Image 1
In mathematics, a divisor of an integer $n$, also called a factor of $n$, is an integer $m$ that may be multiplied by some integer to produce $n$. In this case, one also says that $n$ is a multiple of $m.$ An integer $n$ is divisible by another integer $m$ if $m$ is a divisor of $n$; this implies dividing $n$ by $m$ leaves no remainder. ( Full article...)  Image 2
Subtraction is an arithmetic operation that represents the operation of removing objects from a collection. Subtraction is signified by the minus sign, −. For example, in the adjacent picture, there are 5 − 2 apples—meaning 5 apples with 2 taken away, resulting in a total of 3 apples. Therefore, the difference of 5 and 2 is 3; that is, 5 − 2 = 3. While primarily associated with natural numbers in arithmetic, subtraction can also represent removing or decreasing physical and abstract quantities using different kinds of objects including negative numbers, fractions, irrational numbers, vectors, decimals, functions, and matrices.
Subtraction follows several important patterns. It is anticommutative, meaning that changing the order changes the sign of the answer. It is also not associative, meaning that when one subtracts more than two numbers, the order in which subtraction is performed matters. Because 0 is the additive identity, subtraction of it does not change a number. Subtraction also obeys predictable rules concerning related operations, such as addition and multiplication. All of these rules can be proven, starting with the subtraction of integers and generalizing up through the real numbers and beyond. General binary operations that follow these patterns are studied in abstract algebra.
Performing subtraction on natural numbers is one of the simplest numerical tasks. Subtraction of very small numbers is accessible to young children. In primary education for instance, students are taught to subtract numbers in the decimal system, starting with single digits and progressively tackling more difficult problems. ( Full article...)  Image 3In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement for expressions in logical proofs.
Within an expression containing two or more occurrences in a row of the same associative operator, the order in which the operations are performed does not matter as long as the sequence of the operands is not changed. That is, (after rewriting the expression with parentheses and in infix notation if necessary) rearranging the parentheses in such an expression will not change its value. Consider the following equations:
:$(2+3)+4=2+(3+4)=9\,$ ( Full article...)  Image 4
A fraction (from Latin fractus, "broken") 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, onehalf, eightfifths, threequarters. A common, vulgar, or simple fraction (examples: ${\tfrac {1}{2}}$ and ${\tfrac {17}{3}}$) consists of a numerator displayed above a line (or before a slash like 1⁄2), and a nonzero denominator, displayed below (or after) that line. Numerators and denominators are also used in fractions that are not common, including compound fractions, complex fractions, and mixed numerals.
In positive common fractions, the numerator and denominator are natural numbers. The numerator represents a number of equal parts, and the denominator indicates how many of those parts make up a unit or a whole. The denominator cannot be zero, because zero parts can never make up a whole. For example, in the fraction 3/4, the numerator 3 indicates that the fraction represents 3 equal parts, and the denominator 4 indicates that 4 parts make up a whole. The picture to the right illustrates 3/4 of a cake.
A common fraction is a numeral which represents a rational number. That same number can also be represented as a decimal, a percent, or with a negative exponent. For example, 0.01, 1%, and 10^{−2} are all equal to the fraction 1/100. An integer can be thought of as having an implicit denominator of one (for example, 7 equals 7/1). ( Full article...)  Image 5
In arithmetic, a quotient (from Latin: quotiens "how many times", pronounced /ˈkwoʊʃənt/) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as the integer part of a division (in the case of Euclidean division), or as a fraction or a ratio (in the case of proper division). For example, when dividing 20 (the dividend) by 3 (the divisor), the quotient is 6 in the Euclidean division sense, and $6{\tfrac {2}{3}}$ in the proper division sense. In the second sense, a quotient is simply the ratio of a dividend to its divisor. ( Full article...)  Image 6In number theory, a multiplicative function is an arithmetic function f(n) of a positive integer n with the property that f(1) = 1 and whenever
a and b are coprime, then
:$f(ab)=f(a)f(b).$
An arithmetic function f(n) is said to be completely multiplicative (or totally multiplicative) if f(1) = 1 and f(ab) = f(a)f(b) holds for all positive integers a and b, even when they are not coprime. ( Full article...)  Image 7
A fraction (from Latin fractus, "broken") 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, onehalf, eightfifths, threequarters. A common, vulgar, or simple fraction (examples: ${\tfrac {1}{2}}$ and ${\tfrac {17}{3}}$) consists of a numerator displayed above a line (or before a slash like 1⁄2), and a nonzero denominator, displayed below (or after) that line. Numerators and denominators are also used in fractions that are not common, including compound fractions, complex fractions, and mixed numerals.
In positive common fractions, the numerator and denominator are natural numbers. The numerator represents a number of equal parts, and the denominator indicates how many of those parts make up a unit or a whole. The denominator cannot be zero, because zero parts can never make up a whole. For example, in the fraction 3/4, the numerator 3 indicates that the fraction represents 3 equal parts, and the denominator 4 indicates that 4 parts make up a whole. The picture to the right illustrates 3/4 of a cake.
A common fraction is a numeral which represents a rational number. That same number can also be represented as a decimal, a percent, or with a negative exponent. For example, 0.01, 1%, and 10^{−2} are all equal to the fraction 1/100. An integer can be thought of as having an implicit denominator of one (for example, 7 equals 7/1). ( Full article...)  Image 8
In mathematics, a cube root of a number x is a number y such that y^{3} = x. All nonzero real numbers, have exactly one real cube root and a pair of complex conjugate cube roots, and all nonzero complex numbers have three distinct complex cube roots. For example, the real cube root of 8, denoted ${\sqrt[{3}]{x}}$, is 2, because 2^{3} = 8, while the other cube roots of 8 are $1+i{\sqrt {3}}$ and $1i{\sqrt {3}}$. The three cube roots of −27i are
:$3i,\quad {\frac {3{\sqrt {3}}}{2}}{\frac {3}{2}}i,\quad {\text{and}}\quad {\frac {3{\sqrt {3}}}{2}}{\frac {3}{2}}i.$
In some contexts, particularly when the number whose cube root is to be taken is a real number, one of the cube roots (in this particular case the real one) is referred to as the principal cube root, denoted with the radical sign ${\sqrt[{3}]{~^{~}}}.$ The cube root is the inverse function of the cube function if considering only real numbers, but not if considering also complex numbers: although one has always $\left({\sqrt[{3}]{x}}\right)^{3}=x,$ the cube root of the cube of a number is not always this number. For example, –4 + 4i is a cube root of 8, and $(4+4i)^{3}=8$), but $4+4i\neq {\sqrt[{3}]{(4+4i)^{3}}}=2.$ ( Full article...)  Image 9
In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or process of changing the linear order of an ordered set.
Permutations differ from combinations, which are selections of some members of a set regardless of order. For example, written as tuples, there are six permutations of the set {1,2,3}, namely: (1,2,3), (1,3,2), (2,1,3), (2,3,1), (3,1,2), and (3,2,1). These are all the possible orderings of this threeelement set. Anagrams of words whose letters are different are also permutations: the letters are already ordered in the original word, and the anagram is a reordering of the letters. The study of permutations of finite sets is an important topic in the fields of combinatorics and group theory.
Permutations are used in almost every branch of mathematics, and in many other fields of science. In computer science, they are used for analyzing sorting algorithms; in quantum physics, for describing states of particles; and in biology, for describing RNA sequences. ( Full article...)  Image 10
Division is one of the four basic operations of arithmetic, the ways that numbers are combined to make new numbers. The other operations are addition, subtraction, and multiplication. The division sign ÷, a symbol consisting of a short horizontal line with a dot above and another dot below, is often used to indicate mathematical division. This usage, though widespread in anglophone countries, is neither universal nor recommended: the ISO 800002 standard for mathematical notation recommends only the solidus / or fraction bar for division, or the colon for ratios; it says that this symbol, ÷, "should not be used" for division.
At an elementary level the division of two natural numbers is, among other possible interpretations, the process of calculating the number of times one number is contained within another. This number of times is not always an integer (a number that can be obtained using the other arithmetic operations on the natural numbers).
The division with remainder or Euclidean division of two natural numbers provides an integer quotient, which is the number of times the second number is completely contained in the first number, and a remainder, which is the part of the first number that remains, when in the course of computing the quotient, no further full chunk of the size of the second number can be allocated. ( Full article...)  Image 11
The Riemann zeta function or Euler–Riemann zeta function, ζ(s), is a mathematical function of a complex variable s, and can be expressed as:
:$\zeta (s)=\sum _{n=1}^{\infty }n^{s}={\frac {1}{1^{s}}}+{\frac {1}{2^{s}}}+{\frac {1}{3^{s}}}+\cdots$
The Riemann zeta function plays a pivotal role in analytic number theory and has applications in physics, probability theory, and applied statistics. ( Full article...)  Image 12In mathematics and computer programming, the order of operations (or operator precedence) is a collection of rules that reflect conventions about which procedures to perform first in order to evaluate a given mathematical expression.
For example, in mathematics and most computer languages, multiplication is granted a higher precedence than addition, and it has been this way since the introduction of modern algebraic notation. Thus, the expression 1 + 2 × 3 is interpreted to have the value {{{1}}}, and not {{{1}}}. When exponents were introduced in the 16th and 17th centuries, they were given precedence over both addition and multiplication, and could be placed only as a superscript to the right of their base. Thus {{{1}}} and {{{1}}}.
These conventions exist to eliminate notational ambiguity, while allowing notation to be as brief as possible. Where it is desired to override the precedence conventions, or even simply to emphasize them, parentheses ( ) can be used to indicate an alternative order of operations (or to simply reinforce the default order of operations). For example, {{{1}}} forces addition to precede multiplication, while {{{1}}} forces addition to precede exponentiation. If multiple pairs of parentheses are required in a mathematical expression (such as in the case of nested parentheses), the parentheses may be replaced by brackets or braces to avoid confusion, as in {{{1}}}. ( Full article...)  Image 13In mathematics, summation is the addition of a sequence of any kind of numbers, called addends or summands; the result is their sum or total. Beside numbers, other types of values can be summed as well: functions, vectors, matrices, polynomials and, in general, elements of any type of mathematical objects on which an operation denoted "+" is defined.
Summations of infinite sequences are called series. They involve the concept of limit, and are not considered in this article.
The summation of an explicit sequence is denoted as a succession of additions. For example, summation of [1, 2, 4, 2] is denoted 1 + 2 + 4 + 2, and results in 9, that is, 1 + 2 + 4 + 2 = 9. Because addition is associative and commutative, there is no need of parentheses, and the result is the same irrespective of the order of the summands. Summation of a sequence of only one element results in this element itself. Summation of an empty sequence (a sequence with no elements), by convention, results in 0. ( Full article...)  Image 14
In mathematics, parity is the property of an integer of whether it is even or odd. An integer's parity is even if it is divisible by two with no remainders left and its parity is odd if it isn't; that is, its remainder is 1. For example, −4, 0, 82, and 178 are even because there is no remainder when dividing it by 2. By contrast, −3, 5, 7, 21 are odd numbers as they leave a remainder of 1 when divided by 2.
Even and odd numbers have opposite parities, e.g. 22 (even number) and 13 (odd number) have opposite parities. In particular, zero's parity is even. Any two consecutive integers have opposite parity.
A formal definition of an even number is that it is an integer of the form n = 2k, where k is an integer; it can then be shown that an odd number is an integer of the form n = 2k + 1 (or alternately, 2k − 1). It is important to realize that the above definition of parity applies only to integer numbers, hence it cannot be applied to numbers like 1/2 or 4.201. See the section "Higher mathematics" below for some extensions of the notion of parity to a larger class of "numbers" or in other more general settings. ( Full article...)  Image 15The mode is the value that appears most often in a set of data values. If X is a discrete random variable, the mode is the value x (i.e, X = x) at which the probability mass function takes its maximum value. In other words, it is the value that is most likely to be sampled.
Like the statistical mean and median, the mode is a way of expressing, in a (usually) single number, important information about a random variable or a population. The numerical value of the mode is the same as that of the mean and median in a normal distribution, and it may be very different in highly skewed distributions.
The mode is not necessarily unique to a given discrete distribution, since the probability mass function may take the same maximum value at several points x_{1}, x_{2}, etc. The most extreme case occurs in uniform distributions, where all values occur equally frequently. ( Full article...)  Image 16
In mathematics, a percentage (from Latin per centum "by a hundred") is a number or ratio expressed as a fraction of 100. It is often denoted using the percent sign, "%", although the abbreviations "pct.", "pct" and sometimes "pc" are also used. A percentage is a dimensionless number (pure number); it has no unit of measurement. ( Full article...)  Image 17
Multiplication (often denoted by the cross symbol ×, by the midline dot operator ⋅, by juxtaposition, or, on computers, by an asterisk *) is one of the four elementary mathematical operations of arithmetic, with the other ones being addition, subtraction and division. The result of a multiplication operation is called a product.
The multiplication of whole numbers may be thought of as a repeated addition; that is, the multiplication of two numbers is equivalent to adding as many copies of one of them, the multiplicand, as the quantity of the other one, the multiplier. Both numbers can be referred to as factors.
:$a\times b=\underbrace {b+\cdots +b} _{a{\text{ times}}}$ ( Full article...)  Image 18
In mathematics, two varying quantities are said to be in a relation of proportionality,
multiplicatively connected to a constant; that is, when either their ratio or their product yields a constant. The value of this constant is called the coefficient of proportionality or proportionality constant. If the ratio (y/x) of two variables (x and y) is equal to a constant {{{1}}}, then the variable in the numerator of the ratio (y) can be product of the other variable and the constant {{{1}}}. In this case y is said to be directly proportional to x with proportionality constant k. Equivalently one may write {{{1}}}; that is, x is directly proportional to y with proportionality constant {{{1}}}. If the term proportional is connected to two variables without further qualification, generally direct proportionality can be assumed.
 If the product of two variables (x ⋅ y) is equal to a constant {{{1}}}, then the two are said to be inversely proportional to each other with the proportionality constant k. Equivalently, both variables are directly proportional to the reciprocal of the respective other with proportionality constant k {{{1}}} and {{{1}}}.
If several pairs of variables share the same direct proportionality constant, the equation expressing the equality of these ratios is called a proportion, e.g., {{{1}}} (for details see Ratio). ( Full article...)  Image 19
In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line (or alternatively, a quantity that can be represented as an infinite decimal expansion). The adjective real in this context was introduced in the 17th century by René Descartes, who distinguished between real and imaginary roots of polynomials. The real numbers include all the rational numbers, such as the integer −5 and the fraction 4/3, and all the irrational numbers, such as √2 (1.41421356..., the square root of 2, an irrational algebraic number). Included within the irrationals are the real transcendental numbers, such as π (3.14159265...). In addition to measuring distance, real numbers can be used to measure quantities such as time, mass, energy, velocity, and many more. The set of real numbers is denoted using the symbol R or $\mathbb {R}$ and is sometimes called "the reals".
Real numbers can be thought of as points on an infinitely long line called the number line or real line, where the points corresponding to integers are equally spaced. Any real number can be determined by a possibly infinite decimal representation, such as that of 8.632, where each consecutive digit is measured in units onetenth the size of the previous one. The real line can be thought of as a part of the complex plane, and the real numbers can be thought of as a part of the complex numbers.
These descriptions of the real numbers are not sufficiently rigorous by the modern standards of pure mathematics. The discovery of a suitably rigorous definition of the real numbers—indeed, the realization that a better definition was needed—was one of the most important developments of 19thcentury mathematics. The current standard axiomatic definition is that real numbers form the unique Dedekindcomplete ordered field ($\mathbb {R}$ ; + ; · ; <), up to an isomorphism, whereas popular constructive definitions of real numbers include declaring them as equivalence classes of Cauchy sequences (of rational numbers), Dedekind cuts, or infinite decimal representations, together with precise interpretations for the arithmetic operations and the order relation. All these definitions satisfy the axiomatic definition and are thus equivalent. ( Full article...)  Image 20
A fraction (from Latin fractus, "broken") 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, onehalf, eightfifths, threequarters. A common, vulgar, or simple fraction (examples: ${\tfrac {1}{2}}$ and ${\tfrac {17}{3}}$) consists of a numerator displayed above a line (or before a slash like 1⁄2), and a nonzero denominator, displayed below (or after) that line. Numerators and denominators are also used in fractions that are not common, including compound fractions, complex fractions, and mixed numerals.
In positive common fractions, the numerator and denominator are natural numbers. The numerator represents a number of equal parts, and the denominator indicates how many of those parts make up a unit or a whole. The denominator cannot be zero, because zero parts can never make up a whole. For example, in the fraction 3/4, the numerator 3 indicates that the fraction represents 3 equal parts, and the denominator 4 indicates that 4 parts make up a whole. The picture to the right illustrates 3/4 of a cake.
A common fraction is a numeral which represents a rational number. That same number can also be represented as a decimal, a percent, or with a negative exponent. For example, 0.01, 1%, and 10^{−2} are all equal to the fraction 1/100. An integer can be thought of as having an implicit denominator of one (for example, 7 equals 7/1). ( Full article...)  Image 21
In mathematics, a ratio indicates how many times one number contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio of oranges to lemons is eight to six (that is, 8∶6, which is equivalent to the ratio 4∶3). Similarly, the ratio of lemons to oranges is 6∶8 (or 3∶4) and the ratio of oranges to the total amount of fruit is 8∶14 (or 4∶7).
The numbers in a ratio may be quantities of any kind, such as counts of people or objects, or such as measurements of lengths, weights, time, etc. In most contexts, both numbers are restricted to be positive.
A ratio may be specified either by giving both constituting numbers, written as "a to b" or "a∶b", or by giving just the value of their quotient a/b. Equal quotients correspond to equal ratios. ( Full article...)  Image 22In mathematics, the lowest common denominator or least common denominator (abbreviated LCD) is the lowest common multiple of the denominators of a set of fractions. It simplifies adding, subtracting, and comparing fractions. ( Full article...)
 Image 23In arithmetic, quotition and partition are two ways of viewing fractions and division.
In quotition division one asks, "how many parts are there?"; While in partition division one asks, "what is the size of each part?".
For example, the expression is ( Full article...)  Image 24The unitary method is a technique for solving a problem by first finding the value of a single unit, and then finding the necessary value by multiplying the single unit value. In essence, this method is used to find the value of a unit from the value of a multiple, and hence the value of a multiple. ( Full article...)
 Image 25The weighted arithmetic mean is similar to an ordinary arithmetic mean (the most common type of average), except that instead of each of the data points contributing equally to the final average, some data points contribute more than others. The notion of weighted mean plays a role in descriptive statistics and also occurs in a more general form in several other areas of mathematics.
If all the weights are equal, then the weighted mean is the same as the arithmetic mean. While weighted means generally behave in a similar fashion to arithmetic means, they do have a few counterintuitive properties, as captured for instance in Simpson's paradox. ( Full article...)
General images
Image 1Leibniz's Stepped Reckoner was the first calculator that could perform all four arithmetic operations.
Image 2"Table of Pythagoras" on Napier's bones
Image 7Cycles of the unit digit of multiples of integers ending in 1, 3, 5 and 7 (upper row), and 2, 4, 6 and 8 (lower row) on a telephone keypad
Image 8The Ishango bone, found near Lake Edward, possibly displaying a numbering system from more than 20,000 years ago.
Image 9The Tsinghua Bamboo Slips, Chinese Warring States era decimal multiplication table of 305 BC
Need help?
Do you have a question about Arithmetic that you can't find the answer to?
Consider asking it at the Wikipedia reference desk.
Selected biography
 Image 1
Pāṇini (Devanagari: पाणिनि, pronounced [paːɳɪnɪ]) was a Sanskrit philologist, grammarian, and revered scholar in ancient India, variously dated between the 6th and 4th century BCE.
Since the discovery and publication of his work by European scholars in the nineteenth century, Pāṇini has been considered the "first descriptive linguist", and even labelled as “the father of linguistics”.
Pāṇini's grammar was influential on such foundational linguists as Ferdinand de Saussure and Leonard Bloomfield. ( Full article...)  Image 2
Gerolamo (also Girolamo or Geronimo) Cardano (Italian: [dʒeˈrɔlamo karˈdano]; French: Jérôme Cardan; Latin: Hieronymus Cardanus; 24 September 1501 ( O. S.)– 21 September 1576 ( O. S.)) was an Italian polymath, whose interests and proficiencies ranged through those of mathematician, physician, biologist, physicist, chemist, astrologer, astronomer, philosopher, writer, and gambler. He was one of the most influential mathematicians of the Renaissance, and was one of the key figures in the foundation of probability and the earliest introducer of the binomial coefficients and the binomial theorem in the Western world. He wrote more than 200 works on science.
Cardano partially invented and described several mechanical devices including the combination lock, the gimbal consisting of three concentric rings allowing a supported compass or gyroscope to rotate freely, and the Cardan shaft with universal joints, which allows the transmission of rotary motion at various angles and is used in vehicles to this day. He made significant contributions to hypocycloids, published in De proportionibus, in 1570. The generating circles of these hypocycloids were later named Cardano circles or cardanic circles and were used for the construction of the first highspeed printing presses.
Today, he is well known for his achievements in algebra. In his 1545 book Ars Magna, he made the first systematic use of negative numbers in Europe, published with attribution the solutions of other mathematicians for the cubic and quartic equations, and acknowledged the existence of imaginary numbers. ( Full article...)  Image 3Brahmagupta ( c. 598 – c. 668 CE) was an Indian mathematician and astronomer. He is the author of two early works on mathematics and astronomy: the Brāhmasphuṭasiddhānta (BSS, "correctly established doctrine of Brahma", dated 628), a theoretical treatise, and the Khaṇḍakhādyaka ("edible bite", dated 665), a more practical text.
Brahmagupta was the first to give rules to compute with zero. The texts composed by Brahmagupta were in elliptic verse in Sanskrit, as was common practice in Indian mathematics. As no proofs are given, it is not known how Brahmagupta's results were derived. ( Full article...)  Image 4
Euclid ( /ˈjuːklɪd/; Ancient Greek: Εὐκλείδης – Eukleídēs, pronounced [eu̯.kleː.dɛːs]; fl. 300 BC), sometimes called Euclid of Alexandria to distinguish him from Euclid of Megara, was a Greek mathematician, often referred to as the "founder of geometry" or the "father of geometry". He was active in Alexandria during the reign of Ptolemy I (323–283 BC). His Elements is one of the most influential works in the history of mathematics, serving as the main textbook for teaching mathematics (especially geometry) from the time of its publication until the late 19th or early 20th century. In the Elements, Euclid deduced the theorems of what is now called Euclidean geometry from a small set of axioms. Euclid also wrote works on perspective, conic sections, spherical geometry, number theory, and mathematical rigour. ( Full article...)  Image 5
Abu Rayhan alBiruni /ælbɪˈruːni/ (973 – after 1050) was an Iranian scholar and polymath during the Islamic Golden Age. He has been variously called as the "founder of Indology", "Father of Comparative Religion", "Father of modern geodesy", and the first anthropologist.
AlBiruni was well versed in physics, mathematics, astronomy, and natural sciences, and also distinguished himself as a historian, chronologist and linguist. He studied almost all fields of science and was compensated for his research and strenuous work. Royalty and powerful members of society sought out AlBiruni to conduct research and study to uncover certain findings. In addition to this type of influence, AlBiruni was also influenced by other nations, such as the Greeks, who he took inspiration from when he turned to studies of philosophy. He was conversant in Khwarezmian, Persian, Arabic, Sanskrit, and also knew Greek, Hebrew and Syriac. He spent much of his life in Ghazni, then capital of the Ghaznavids, in modernday centraleastern Afghanistan. In 1017 he travelled to the Indian subcontinent and authored a study of Indian culture Tārīkh alHind (History of India) after exploring the Hindu faith practiced in India. He was an impartial writer on customs and creeds of various nations, and was given the title alUstadh ("The Master") for his remarkable description of early 11thcentury India.
In Iran, Abu Rayhan Biruni's birthday is celebrated as the day of the surveying engineer. ( Full article...)  Image 6
Eratosthenes of Cyrene ( /ɛrəˈtɒsθəniːz/; Greek: Ἐρατοσθένης ὁ Κυρηναῖος, romanized: Eratosthénēs ho Kurēnaĩos, IPA: [eratostʰénɛːs]; {{{1}}} – c. 195/194 BC) was a Greek polymath: a mathematician, geographer, poet, astronomer, and music theorist. He was a man of learning, becoming the chief librarian at the Library of Alexandria. His work is comparable to what is now known as the study of geography, and he introduced some of the terminology still used today.
He is best known for being the first person known to calculate the circumference of the Earth, which he did by using the extensive survey results he could access in his role at the Library; his calculation was remarkably accurate. He was also the first to calculate Earth's axial tilt, which also proved to have remarkable accuracy. He created the first global projection of the world, incorporating parallels and meridians based on the available geographic knowledge of his era.
Eratosthenes was the founder of scientific chronology; he endeavoured to revise the dates of the main events of the semimythological Trojan War, dating the Sack of Troy to 1183 BC. In number theory, he introduced the sieve of Eratosthenes, an efficient method of identifying prime numbers.
He was a figure of influence in many fields. According to an entry in the Suda (a 10thcentury encyclopedia), his critics scorned him, calling him Beta (the second letter of the Greek alphabet) because he always came in second in all his endeavours. Nonetheless, his devotees nicknamed him Pentathlos after the Olympians who were well rounded competitors, for he had proven himself to be knowledgeable in every area of learning. Eratosthenes yearned to understand the complexities of the entire world. ( Full article...)  Image 7
Pythagoras of Samos ( c. 570 – c. 495 BC) was an ancient Ionian Greek philosopher and the eponymous founder of Pythagoreanism. His political and religious teachings were well known in Magna Graecia and influenced the philosophies of Plato, Aristotle, and, through them, Western philosophy. Knowledge of his life is clouded by legend, but he appears to have been the son of Mnesarchus, a gemengraver on the island of Samos. Modern scholars disagree regarding Pythagoras's education and influences, but they do agree that, around 530 BC, he travelled to Croton in southern Italy, where he founded a school in which initiates were sworn to secrecy and lived a communal, ascetic lifestyle. This lifestyle entailed a number of dietary prohibitions, traditionally said to have included vegetarianism, although modern scholars doubt that he ever advocated for complete vegetarianism.
The teaching most securely identified with Pythagoras is metempsychosis, or the "transmigration of souls", which holds that every soul is immortal and, upon death, enters into a new body. He may have also devised the doctrine of musica universalis, which holds that the planets move according to mathematical equations and thus resonate to produce an inaudible symphony of music. Scholars debate whether Pythagoras developed the numerological and musical teachings attributed to him, or if those teachings were developed by his later followers, particularly Philolaus of Croton. Following Croton's decisive victory over Sybaris in around 510 BC, Pythagoras's followers came into conflict with supporters of democracy and Pythagorean meeting houses were burned. Pythagoras may have been killed during this persecution, or escaped to Metapontum, where he eventually died.
In antiquity, Pythagoras was credited with many mathematical and scientific discoveries, including the Pythagorean theorem, Pythagorean tuning, the five regular solids, the Theory of Proportions, the sphericity of the Earth, and the identity of the morning and evening stars as the planet Venus. It was said that he was the first man to call himself a philosopher ("lover of wisdom") and that he was the first to divide the globe into five climatic zones. Classical historians debate whether Pythagoras made these discoveries, and many of the accomplishments credited to him likely originated earlier or were made by his colleagues or successors. Some accounts mention that the philosophy associated with Pythagoras was related to mathematics and that numbers were important, but it is debated to what extent, if at all, he actually contributed to mathematics or natural philosophy.
Pythagoras influenced Plato, whose dialogues, especially his Timaeus, exhibit Pythagorean teachings. Pythagorean ideas on mathematical perfection also impacted ancient Greek art. His teachings underwent a major revival in the first century BC among Middle Platonists, coinciding with the rise of Neopythagoreanism. Pythagoras continued to be regarded as a great philosopher throughout the Middle Ages and his philosophy had a major impact on scientists such as Nicolaus Copernicus, Johannes Kepler, and Isaac Newton. Pythagorean symbolism was used throughout early modern European esotericism, and his teachings as portrayed in Ovid's Metamorphoses influenced the modern vegetarian movement. ( Full article...)  Image 8
Muḥammad ibn Mūsā alKhwārizmī ( Persian: محمد بن موسی خوارزمی, romanized: Moḥammad ben Musā Khwārazmi; c. 780 – c. 850), Arabized as alKhwarizmi and formerly Latinized as Algorithmi, was a Persian polymath who produced vastly influential works in mathematics, astronomy, and geography. Around 820 CE he was appointed as the astronomer and head of the library of the House of Wisdom in Baghdad.
AlKhwarizmi's popularizing treatise on algebra ( The Compendious Book on Calculation by Completion and Balancing, c. 813–833 CE) presented the first systematic solution of linear and quadratic equations. One of his principal achievements in algebra was his demonstration of how to solve quadratic equations by completing the square, for which he provided geometric justifications. Because he was the first to treat algebra as an independent discipline and introduced the methods of "reduction" and "balancing" (the transposition of subtracted terms to the other side of an equation, that is, the cancellation of like terms on opposite sides of the equation), he has been described as the father or founder of algebra. The term algebra itself comes from the title of his book (the word aljabr meaning "completion" or "rejoining"). His name gave rise to the terms algorism and algorithm, as well as Spanish and Portuguese terms algoritmo, and Spanish guarismo and Portuguese algarismo meaning " digit".
In the 12th century, Latin translations of his textbook on arithmetic (Algorithmo de Numero Indorum) which codified the various Indian numerals, introduced the decimal positional number system to the Western world. The Compendious Book on Calculation by Completion and Balancing, translated into Latin by Robert of Chester in 1145, was used until the sixteenth century as the principal mathematical textbook of European universities.
In addition to his bestknown works, he revised Ptolemy's Geography, listing the longitudes and latitudes of various cities and localities. He further produced a set of astronomical tables and wrote about calendaric works, as well as the astrolabe and the sundial. He also made important contributions to trigonometry, producing accurate sine and cosine tables, and the first table of tangents. ( Full article...)  Image 9
Thales of Miletus ( /ˈθeɪliːz/ THAYleez; Greek: Θαλῆς (ὁ Μιλήσιος), Thalēs; c. 624/623 – c. 548/545 BC) was a Greek mathematician, astronomer and preSocratic philosopher from Miletus in Ionia, Asia Minor. He was one of the Seven Sages of Greece. Many, most notably Aristotle, regarded him as the first philosopher in the Greek tradition, and he is otherwise historically recognized as the first individual in Western civilization known to have entertained and engaged in scientific philosophy.
Thales is recognized for breaking from the use of mythology to explain the world and the universe, and instead explaining natural objects and phenomena by naturalistic theories and hypotheses, in a precursor to modern science. Almost all the other preSocratic philosophers followed him in explaining nature as deriving from a unity of everything based on the existence of a single ultimate substance, instead of using mythological explanations. Aristotle regarded him as the founder of the Ionian School and reported Thales' hypothesis that the originating principle of nature and the nature of matter was a single material substance: water.
In mathematics, Thales used geometry to calculate the heights of pyramids and the distance of ships from the shore. He is the first known individual to use deductive reasoning applied to geometry, by deriving four corollaries to Thales' theorem. He is the first known individual to whom a mathematical discovery has been attributed. ( Full article...)  Image 10
Ḥasan Ibn alHaytham ( Latinized as Alhazen /ælˈhæzən/; full name Abū ʿAlī alḤasan ibn alḤasan ibn alHaytham أبو علي، الحسن بن الحسن بن الهيثم; c. 965 – c. 1040) was a Muslim Arab mathematician, astronomer, and physicist of the Islamic Golden Age. Referred to as "the father of modern optics", he made significant contributions to the principles of optics and visual perception in particular. His most influential work is titled Kitāb alManāẓir ( Arabic: كتاب المناظر, "Book of Optics"), written during 1011–1021, which survived in a Latin edition. A polymath, he also wrote on philosophy, theology and medicine.
Ibn alHaytham was the first to explain that vision occurs when light reflects from an object and then passes to one's eyes. He was also the first to demonstrate that vision occurs in the brain, rather than in the eyes. Building upon a naturalistic, empirical method pioneered by Aristotle in ancient Greece, Ibn alHaytham was an early proponent of the concept that a hypothesis must be supported by experiments based on confirmable procedures or mathematical evidence—an early pioneer in the scientific method five centuries before Renaissance scientists.
Born in Basra, he spent most of his productive period in the Fatimid capital of Cairo and earned his living authoring various treatises and tutoring members of the nobilities. Ibn alHaytham is sometimes given the byname alBaṣrī after his birthplace, or alMiṣrī ("of Egypt"). AlHaytham was dubbed the "Second Ptolemy" by Abu'lHasan Bayhaqi and "The Physicist" by John Peckham. Ibn alHaytham paved the way for the modern science of physical optics. ( Full article...)  Image 11
Hero of Alexandria ( /ˈhɪəroʊ/; Greek: Ἥρων ὁ Ἀλεξανδρεύς, Heron ho Alexandreus; also known as Heron of Alexandria /ˈhɛrən/; c. 10 AD – c. 70 AD) was a GrecoEgyptian mathematician and engineer who was active in his native city of Alexandria, Roman Egypt. He is often considered the greatest experimenter of antiquity and his work is representative of the Hellenistic scientific tradition.
Hero published a wellrecognized description of a steampowered device called an aeolipile (sometimes called a "Hero engine"). Among his most famous inventions was a windwheel, constituting the earliest instance of wind harnessing on land. He is said to have been a follower of the atomists. In his work Mechanics, he described pantographs. Some of his ideas were derived from the works of Ctesibius.
In mathematics he is mostly remembered for Heron's formula, a way to calculate the area of a triangle using only the lengths of its sides.
Much of Hero's original writings and designs have been lost, but some of his works were preserved including in manuscripts from the Eastern Roman Empire and to a lesser extent, in Latin or Arabic translations. ( Full article...)  Image 12
Zhang Heng ( Chinese: 張 衡; AD 78–139), formerly romanized as Chang Heng, was a Chinese polymathic scientist and statesman who lived during the Han dynasty. Educated in the capital cities of Luoyang and Chang'an, he achieved success as an astronomer, mathematician, seismologist, hydraulic engineer, inventor, geographer, cartographer, ethnographer, artist, poet, philosopher, politician, and literary scholar.
Zhang Heng began his career as a minor civil servant in Nanyang. Eventually, he became Chief Astronomer, Prefect of the Majors for Official Carriages, and then Palace Attendant at the imperial court. His uncompromising stance on historical and calendrical issues led to his becoming a controversial figure, preventing him from rising to the status of Grand Historian. His political rivalry with the palace eunuchs during the reign of Emperor Shun (r. 125–144) led to his decision to retire from the central court to serve as an administrator of Hejian Kingdom in presentday Hebei. Zhang returned home to Nanyang for a short time, before being recalled to serve in the capital once more in 138. He died there a year later, in 139.
Zhang applied his extensive knowledge of mechanics and gears in several of his inventions. He invented the world's first waterpowered armillary sphere to assist astronomical observation; improved the inflow water clock by adding another tank; and invented the world's first seismoscope, which discerned the cardinal direction of an earthquake 500 km (310 mi) away. He improved previous Chinese calculations for pi. In addition to documenting about 2,500 stars in his extensive star catalog, Zhang also posited theories about the Moon and its relationship to the Sun: specifically, he discussed the Moon's sphericity, its illumination by reflected sunlight on one side and the hidden nature of the other, and the nature of solar and lunar eclipses. His fu (rhapsody) and shi poetry were renowned in his time and studied and analyzed by later Chinese writers. Zhang received many posthumous honors for his scholarship and ingenuity; some modern scholars have compared his work in astronomy to that of the GrecoRoman Ptolemy (AD 86–161). ( Full article...)  Image 13
Fra Luca Bartolomeo de Pacioli (sometimes Paccioli or Paciolo; c. 1447 – 19 June 1517) was an Italian mathematician, Franciscan friar, collaborator with Leonardo da Vinci, and an early contributor to the field now known as accounting. He is referred to as "The Father of Accounting and Bookkeeping" in Europe and he was the second person to publish a work on the doubleentry system of bookkeeping on the continent. He was also called Luca di Borgo after his birthplace, Borgo Sansepolcro, Tuscany. ( Full article...)
Subcategories
 Select [►] to view subcategories
Related portals
Related topics
 Addition of natural numbers
 Additive inverse
 Arithmetic coding
 Arithmetic mean
 Arithmetic progression
 Arithmetic properties
 Associativity
 Commutativity
 Distributivity
 Elementary arithmetic
 Finite field arithmetic
 Geometric progression
 Integer
 List of important publications in mathematics
 Mental calculation
 Number line
 Stepped reckoner
Associated Wikimedia
The following Wikimedia Foundation sister projects provide more on this subject:
Wikibooks
Books
Commons
Media
Wikinews
News
Wikiquote
Quotations
Wikisource
Texts
Wikiversity
Learning resources
Wiktionary
Definitions
Wikidata
Database
Portals
 Portals with short description
 Portals with triaged subpages from August 2018
 All portals with triaged subpages
 Portals with no named maintainer
 Automated articleslideshow portals with 51–100 articles in article list
 Automated articleslideshow portals with 11–15 articles in article list
 Unredirected portals with existing subpages