Portal:Arithmetic

The Arithmetic Portal

Arithmetic is an elementary branch of mathematics that deals with numerical operations like addition, subtraction, multiplication, and division. In a wider sense, it also includes exponentiation, extraction of roots, and taking logarithms. (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 place-value 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, the prime number theorem (PNT) describes the asymptotic distribution of the prime numbers among the positive integers. It formalizes the intuitive idea that primes become less common as they become larger by precisely quantifying the rate at which this occurs. The theorem was proved independently by Jacques Hadamard and Charles Jean de la Vallée Poussin in 1896 using ideas introduced by Bernhard Riemann (in particular, the Riemann zeta function). (Full article...)
Image 2
Cuisenaire rods: 5 (yellow) cannot be evenly divided in 2 (red) by any 2 rods of the same color/length, while 6 (dark green) can be evenly divided in 2 by 3 (lime green).

In mathematics, parity is the property of an integer of whether it is even or odd. An integer is even if it is divisible by 2, and odd if it is not. For example, −4, 0, and 82 are even numbers, while −3, 5, 23, and 69 are odd numbers. (Full article...)
Image 3
3 + 2 = 5 with apples, a popular choice in textbooks

Addition (usually signified by the plus symbol, +) is one of the four basic operations of arithmetic, the other three being subtraction, multiplication, and division. The addition of two whole numbers results in the total or sum of those values combined. For example, the adjacent image shows two columns of apples, one with three apples and the other with two apples, totaling to five apples. This observation is expressed as "3 + 2 = 5", which is read as "three plus two equals five". (Full article...)
Image 4
In mathematics, the greatest common divisor (GCD), also known as greatest common factor (GCF), of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For two integers $x$ , $y$ , the greatest common divisor of $x$ and $y$ is denoted $\gcd(x,y)$ . For example, the GCD of 8 and 12 is 4, that is, $gcd(8, 12) = 4$ . (Full article...)
Image 5
In statistics, the 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 $x$ at which the probability mass function takes its maximum value (i.e., $x = argmax x i P(X = x i)$ ). In other words, it is the value that is most likely to be sampled. (Full article...)
Image 6
Cuisenaire rods: 5 (yellow) cannot be evenly divided in 2 (red) by any 2 rods of the same color/length, while 6 (dark green) can be evenly divided in 2 by 3 (lime green).

In mathematics, parity is the property of an integer of whether it is even or odd. An integer is even if it is divisible by 2, and odd if it is not. For example, −4, 0, and 82 are even numbers, while −3, 5, 23, and 69 are odd numbers. (Full article...)
$Image 7 In arithmetic, quotition and partition are two ways of viewing fractions and division. In quotitive division one asks "how many parts are there?" while in partitive division one asks "what is the size of each part?" (Full article...)$

Image 7
In arithmetic, quotition and partition are two ways of viewing fractions and division. In quotitive division one asks "how many parts are there?" while in partitive division one asks "what is the size of each part?" (Full article...)
Image 8
In mathematics, summation is the addition of a sequence 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. (Full article...)
Image 9
The divisors of 10 illustrated with Cuisenaire rods: 1, 2, 5, and 10

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 or evenly divisible by another integer $m$ if $m$ is a divisor of $n$ ; this implies dividing $n$ by $m$ leaves no remainder. (Full article...)
Image 10
In mathematics, the prime number theorem (PNT) describes the asymptotic distribution of the prime numbers among the positive integers. It formalizes the intuitive idea that primes become less common as they become larger by precisely quantifying the rate at which this occurs. The theorem was proved independently by Jacques Hadamard and Charles Jean de la Vallée Poussin in 1896 using ideas introduced by Bernhard Riemann (in particular, the Riemann zeta function). (Full article...)
Image 11
In mathematics, a modular form is a holomorphic function on the complex upper half-plane, ${\mathcal {H}}$ , that roughly satisfies a functional equation with respect to the group action of the modular group and a growth condition. The theory of modular forms has origins in complex analysis, with important connections with number theory. Modular forms also appear in other areas, such as algebraic topology, sphere packing, and string theory. (Full article...)
$Image 12 A cake with one quarter (one fourth) removed. The remaining three fourths are shown by dotted lines and labeled by the fraction 1⁄4. A fraction (from Latin: fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight-fifths, three-quarters. A common, vulgar, or simple fraction (examples: ⁠1/2⁠ and ⁠17/3⁠) consists of an integer numerator, displayed above a line (or before a slash like 1⁄2), and a non-zero integer denominator, displayed below (or after) that line. If these integers are positive, then the numerator represents a number of equal parts, and the denominator indicates how many of those parts make up a unit or a whole. For example, in the fraction ⁠3/4⁠, the numerator 3 indicates that the fraction represents 3 equal parts, and the denominator 4 indicates that 4 parts make up a whole. The picture to the right illustrates ⁠3/4⁠ of a cake. (Full article...)$

Image 12
A cake with one quarter (one fourth) removed. The remaining three fourths are shown by dotted lines and labeled by the fraction 1⁄4.

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, one-half, eight-fifths, three-quarters. A common, vulgar, or simple fraction (examples: ⁠1/2⁠ and ⁠17/3⁠) consists of an integer numerator, displayed above a line (or before a slash like 1⁄2), and a non-zero integer denominator, displayed below (or after) that line. If these integers are positive, then the numerator represents a number of equal parts, and the denominator indicates how many of those parts make up a unit or 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. (Full article...)
Image 13
The Riemann zeta function $ζ (z)$ plotted with domain coloring.

The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter $ζ$ (zeta), is a mathematical function of a complex variable defined as $\zeta (s)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}={\frac {1}{1^{s}}}+{\frac {1}{2^{s}}}+{\frac {1}{3^{s}}}+\cdots$ for $\operatorname {Re} (s)>1$ , and its analytic continuation elsewhere. (Full article...)
Image 14
20 / 4 = 5, illustrated here with apples. This is said verbally, "Twenty divided by four equals five."

Division is one of the four basic operations of arithmetic. The other operations are addition, subtraction, and multiplication. What is being divided is called the dividend, which is divided by the divisor, and the result is called the quotient. (Full article...)
$Image 15 The reciprocal function: y = 1/x. For every x except 0, y represents its multiplicative inverse. The graph forms a rectangular hyperbola. In mathematics, a multiplicative inverse or reciprocal for a number x, denoted by 1/x or x−1, is a number which when multiplied by x yields the multiplicative identity, 1. The multiplicative inverse of a fraction a/b is b/a. For the multiplicative inverse of a real number, divide 1 by the number. For example, the reciprocal of 5 is one fifth (1/5 or 0.2), and the reciprocal of 0.25 is 1 divided by 0.25, or 4. The reciprocal function, the function f(x) that maps x to 1/x, is one of the simplest examples of a function which is its own inverse (an involution). (Full article...)$

Image 15
The reciprocal function: y = 1/x. For every x except 0, y represents its multiplicative inverse. The graph forms a rectangular hyperbola.

In mathematics, a multiplicative inverse or reciprocal for a number x, denoted by 1/x or x⁻¹, is a number which when multiplied by x yields the multiplicative identity, 1. The multiplicative inverse of a fraction a/b is b/a. For the multiplicative inverse of a real number, divide 1 by the number. For example, the reciprocal of 5 is one fifth (1/5 or 0.2), and the reciprocal of 0.25 is 1 divided by 0.25, or 4. The reciprocal function, the function f(x) that maps x to 1/x, is one of the simplest examples of a function which is its own inverse (an involution). (Full article...)
Image 16
Order of operations

In mathematics and computer programming, the order of operations is a collection of rules that reflect conventions about which operations to perform first in order to evaluate a given mathematical expression. (Full article...)
$Image 17 The quotient of 12 apples by 3 apples is 4. In arithmetic, a quotient (from Latin: quotiens 'how many times', pronounced /ˈkwoʊʃənt/) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics. It has two definitions: either the integer part of a division (in the case of Euclidean division) or a fraction or ratio (in the case of a general division). For example, when dividing 20 (the dividend) by 3 (the divisor), the quotient is 6 (with a remainder of 2) in the first sense and '"`UNIQ--postMath-00000011-QINU`"' (a repeating decimal) in the second sense. (Full article...)$

Image 17
The quotient of 12 apples by 3 apples is 4.

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. It has two definitions: either the integer part of a division (in the case of Euclidean division) or a fraction or ratio (in the case of a general division). For example, when dividing 20 (the dividend) by 3 (the divisor), the quotient is 6 (with a remainder of 2) in the first sense and $6+{\tfrac {2}{3}}=6.66...$ (a repeating decimal) in the second sense. (Full article...)
Image 18
Four bags with three marbles per bag gives twelve marbles (4 × 3 = 12).

Multiplication 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. Multiplication is often denoted by the cross symbol, ×, by the mid-line dot operator, ·, by juxtaposition, or, on computers, by an asterisk, *. (Full article...)
$Image 19 In mathematics, the lowest common denominator or least common denominator (abbreviated LCD) is the lowest common multiple of the denominators of a set of fractions. It simplifies adding, subtracting, and comparing fractions. (Full article...)$

Image 19
In 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 20
In number theory, a multiplicative function is an arithmetic function $f$ of a positive integer $n$ with the property that $f(1)=1$ and
$f(ab)=f(a)f(b)$ whenever $a$ and $b$ are coprime. (Full article...)
Image 21
In mathematics, a combination is a selection of items from a set that has distinct members, such that the order of selection does not matter (unlike permutations). For example, given three fruits, say an apple, an orange and a pear, there are three combinations of two that can be drawn from this set: an apple and a pear; an apple and an orange; or a pear and an orange. More formally, a k-combination of a set S is a subset of k distinct elements of S. So, two combinations are identical if and only if each combination has the same members. (The arrangement of the members in each set does not matter.) If the set has n elements, the number of k-combinations, denoted by $C(n,k)$ or $C_{k}^{n}$ , is equal to the binomial coefficient (Full article...)
Image 22
In mathematics, the greatest common divisor (GCD), also known as greatest common factor (GCF), of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For two integers $x$ , $y$ , the greatest common divisor of $x$ and $y$ is denoted $\gcd(x,y)$ . For example, the GCD of 8 and 12 is 4, that is, $gcd(8, 12) = 4$ . (Full article...)
Image 23
The 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. (Full article...)
$Image 24 The rational numbers ⁠'"`UNIQ--postMath-0000001B-QINU`"'⁠ are included in the real numbers ⁠'"`UNIQ--postMath-0000001C-QINU`"'⁠, which are included in the complex numbers ⁠'"`UNIQ--postMath-0000001D-QINU`"'⁠, while rationals include the integers ⁠'"`UNIQ--postMath-0000001E-QINU`"'⁠, which in turn include the natural numbers ⁠'"`UNIQ--postMath-0000001F-QINU`"'⁠. In mathematics, a rational number is a number that can be expressed as the quotient or fraction ⁠'"`UNIQ--postMath-00000020-QINU`"'⁠ of two integers, a numerator p and a non-zero denominator q. For example, ⁠'"`UNIQ--postMath-00000021-QINU`"'⁠ is a rational number, as is every integer (for example, '"`UNIQ--postMath-00000022-QINU`"'). The set of all rational numbers, also referred to as "the rationals", the field of rationals or the field of rational numbers is usually denoted by boldface Q, or blackboard bold ⁠'"`UNIQ--postMath-00000023-QINU`"'⁠ (Full article...)$

Image 24
The rational numbers ⁠ $\mathbb {Q}$ ⁠ are included in the real numbers ⁠ $\mathbb {R}$ ⁠, which are included in the complex numbers ⁠ $\mathbb {C}$ ⁠, while rationals include the integers ⁠ $\mathbb {Z}$ ⁠, which in turn include the natural numbers ⁠ $\mathbb {N}$ ⁠.

In mathematics, a rational number is a number that can be expressed as the quotient or fraction ⁠ ${\tfrac {p}{q}}$ ⁠ of two integers, a numerator $p$ and a non-zero denominator $q$ . For example, ⁠ ${\tfrac {3}{7}}$ ⁠ is a rational number, as is every integer (for example, $-5={\tfrac {-5}{1}}$ ). The set of all rational numbers, also referred to as "the rationals", the field of rationals or the field of rational numbers is usually denoted by boldface $Q$ , or blackboard bold ⁠ $\mathbb {Q} .$ ⁠ (Full article...)
Image 25
This thermometer is indicating a negative Fahrenheit temperature (−4 °F).

In mathematics, a negative number is the opposite of a positive real number. Equivalently, a negative number is a real number that is less than zero. Negative numbers are often used to represent the magnitude of a loss or deficiency. A debt that is owed may be thought of as a negative asset. If a quantity, such as the charge on an electron, may have either of two opposite senses, then one may choose to distinguish between those senses—perhaps arbitrarily—as positive and negative. Negative numbers are used to describe values on a scale that goes below zero, such as the Celsius and Fahrenheit scales for temperature. The laws of arithmetic for negative numbers ensure that the common-sense idea of an opposite is reflected in arithmetic. For example, −‍(−3) = 3 because the opposite of an opposite is the original value. (Full article...)

General images

The following are images from various arithmetic-related articles on Wikipedia.

Image 1Example of modular arithmetic using a clock: after adding 4 hours to 9 o'clock, the hand starts at the beginning again and points at 1 o'clock. (from Arithmetic)
Image 2Irrational numbers are sometimes required to describe magnitudes in geometry. For example, the length of the hypotenuse of a right triangle is irrational if its legs have a length of 1. (from Arithmetic)
Image 3Some historians interpret the Ishango bone as one of the earliest arithmetic artifacts. (from Arithmetic)
Image 4Some historians interpret the Ishango bone as one of the earliest arithmetic artifacts. (from Arithmetic)
Image 5Cycles of the unit digit of multiples of integers ending in 1, 3, 7 and 9 (upper row), and 2, 4, 6 and 8 (lower row) on a telephone keypad (from Multiplication table)
Image 6Different types of numbers on a number line. Integers are black, rational numbers are blue, and irrational numbers are green. (from Arithmetic)
Image 7Leibniz's stepped reckoner was the first calculator that could perform all four arithmetic operations. (from Arithmetic)
Image 8The Tsinghua Bamboo Slips, Chinese Warring States era decimal multiplication table of 305 BC (from Multiplication table)
Image 9Abacuses are tools to perform arithmetic operations by moving beads. (from Arithmetic)
Image 10Hieroglyphic numerals from 1 to 10,000 (from Arithmetic)
$Image 11Visualising adding fractions with different denominators by dividing a square in different directions (1), subdividing into common cells (2), adding the cells (3) and merging cells to simplify (4) (from Fraction)$

Image 11Visualising adding fractions with different denominators by dividing a square in different directions (1), subdividing into common cells (2), adding the cells (3) and merging cells to simplify (4) (from Fraction)
Image 12Using the number line method, calculating $5+2$ is performed by starting at the origin of the number line then moving five units to right for the first addend. The result is reached by moving another two units to the right for the second addend. (from Arithmetic)
Image 13Calculations in mental arithmetic are done exclusively in the mind without relying on external aids. (from Arithmetic)
Image 14Example of addition with carry. The black numbers are the addends, the green number is the carry, and the blue number is the sum. In the rightmost digit, the addition of 9 and 7 is 16, carrying 1 into the next pair of the digit to the left, making its addition 1 + 5 + 2 = 8. Therefore, 59 + 27 = 86. (from Elementary arithmetic)
Image 15Different types of numbers on a number line. Integers are black, rational numbers are blue, and irrational numbers are green. (from Arithmetic)
Image 16"Table of Pythagoras" on Napier's bones (from Multiplication table)
Image 17Using the number line method, calculating $5+2$ is performed by starting at the origin of the number line then moving five units to right for the first addend. The result is reached by moving another two units to the right for the second addend. (from Arithmetic)
Image 18Abacuses are tools to perform arithmetic operations by moving beads. (from Arithmetic)
Image 19Example of modular arithmetic using a clock: after adding 4 hours to 9 o'clock, the hand starts at the beginning again and points at 1 o'clock. (from Arithmetic)
Image 20Irrational numbers are sometimes required to describe magnitudes in geometry. For example, the length of the hypotenuse of a right triangle is irrational if its legs have a length of 1. (from Arithmetic)
Image 21If ${\tfrac {1}{2}}$ of a cake is to be added to ${\tfrac {1}{4}}$ of a cake, the pieces need to be converted into comparable quantities, such as cake-eighths or cake-quarters. (from Fraction)
Image 22Leibniz's stepped reckoner was the first calculator that could perform all four arithmetic operations. (from Arithmetic)
Image 23Calculations in mental arithmetic are done exclusively in the mind without relying on external aids. (from Arithmetic)
Image 24The main arithmetic operations are addition, subtraction, multiplication, and division. (from Arithmetic)
$Image 25A cake with one quarter (one fourth) removed. The remaining three fourths are shown by dotted lines and labeled by the fraction 1⁄4. (from Fraction)$

Image 25A cake with one quarter (one fourth) removed. The remaining three fourths are shown by dotted lines and labeled by the fraction 1⁄4. (from Fraction)
Image 26Multiplication table from 1 to 10 drawn to scale with the upper-right half labeled with prime factorisations (from Multiplication table)
Image 27Hieroglyphic numerals from 1 to 10,000 (from Arithmetic)
Image 28The symbols for elementary-level math operations. From top-left going clockwise: addition, division, multiplication, and subtraction. (from Elementary arithmetic)
Image 29The main arithmetic operations are addition, subtraction, multiplication, and division. (from Arithmetic)

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Selected biography

$Image 1 Ḥasan Ibn al-Haytham (Latinized as Alhazen; /ælˈhæzən/; full name Abū ʿAlī al-Ḥasan ibn al-Ḥasan ibn al-Haytham أبو علي، الحسن بن الحسن بن الهيثم; c. 965 – c. 1040) was a medieval mathematician, astronomer, and physicist of the Islamic Golden Age from present-day Iraq. 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 al-Manāẓir (Arabic: كتاب المناظر, "Book of Optics"), written during 1011–1021, which survived in a Latin edition. The works of Alhazen were frequently cited during the scientific revolution by Isaac Newton, Johannes Kepler, Christiaan Huygens, and Galileo Galilei. Ibn al-Haytham was the first to correctly explain the theory of vision, and to argue that vision occurs in the brain, pointing to observations that it is subjective and affected by personal experience. He also stated the principle of least time for refraction which would later become Fermat's principle. He made major contributions to catoptrics and dioptrics by studying reflection, refraction and nature of images formed by light rays. Ibn al-Haytham was an early proponent of the concept that a hypothesis must be supported by experiments based on confirmable procedures or mathematical reasoning – an early pioneer in the scientific method five centuries before Renaissance scientists, he is sometimes described as the world's "first true scientist". He was also a polymath, writing on philosophy, theology and medicine. 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 al-Haytham is sometimes given the byname al-Baṣrī after his birthplace, or al-Miṣrī ("the Egyptian"). Al-Haytham was dubbed the "Second Ptolemy" by Abu'l-Hasan Bayhaqi and "The Physicist" by John Peckham. Ibn al-Haytham paved the way for the modern science of physical optics. (Full article...)$

Image 1

Ḥasan Ibn al-Haytham (Latinized as Alhazen; /ælˈhæzən/; full name Abū ʿAlī al-Ḥasan ibn al-Ḥasan ibn al-Haytham أبو علي، الحسن بن الحسن بن الهيثم; c. 965 – c. 1040) was a medieval mathematician, astronomer, and physicist of the Islamic Golden Age from present-day Iraq. 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 al-Manāẓir (Arabic: كتاب المناظر, "Book of Optics"), written during 1011–1021, which survived in a Latin edition. The works of Alhazen were frequently cited during the scientific revolution by Isaac Newton, Johannes Kepler, Christiaan Huygens, and Galileo Galilei.

Ibn al-Haytham was the first to correctly explain the theory of vision, and to argue that vision occurs in the brain, pointing to observations that it is subjective and affected by personal experience. He also stated the principle of least time for refraction which would later become Fermat's principle. He made major contributions to catoptrics and dioptrics by studying reflection, refraction and nature of images formed by light rays. Ibn al-Haytham was an early proponent of the concept that a hypothesis must be supported by experiments based on confirmable procedures or mathematical reasoning – an early pioneer in the scientific method five centuries before Renaissance scientists, he is sometimes described as the world's "first true scientist". He was also a polymath, writing on philosophy, theology and medicine.

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 al-Haytham is sometimes given the byname al-Baṣrī after his birthplace, or al-Miṣrī ("the Egyptian"). Al-Haytham was dubbed the "Second Ptolemy" by Abu'l-Hasan Bayhaqi and "The Physicist" by John Peckham. Ibn al-Haytham paved the way for the modern science of physical optics. (Full article...)
Image 2
Pāṇini (/ˈpɑːnɪni/; Sanskrit: पाणिनि, pāṇini [páːɳin̪i]) was a Sanskrit grammarian, logician, philologist, and revered scholar in ancient India during the mid-1st millennium BCE, dated variously by most scholars between the 6th–5th and 4th century BCE.

The historical facts of his life are unknown, except only what can be inferred from his works, and legends recorded long after. His most notable work, the Aṣṭādhyāyī, is conventionally taken to mark the start of Classical Sanskrit. His work formally codified Classical Sanskrit as a refined and standardized language, making use of a technical metalanguage consisting of a syntax, morphology, and lexicon, organised according to a series of meta-rules.

Since the exposure of European scholars to his Aṣṭādhyāyī in the nineteenth century, Pāṇini has been considered the "first descriptive linguist", and even labelled as "the father of linguistics". His approach to grammar influenced such foundational linguists as Ferdinand de Saussure and Leonard Bloomfield. (Full article...)
Image 3

Etching of an ancient seal identified as Eratosthenes. Philipp Daniel Lippert [de], Dactyliothec, 1767.

Eratosthenes of Cyrene (/ɛrəˈtɒsθəniːz/; Ancient Greek: Ἐρατοσθένης [eratostʰénɛːs]; c. 276 BC – c. 195/194 BC) was an Ancient 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, even coining the terms geography and geographer.

He is best known for being the first person known to calculate the Earth's circumference, which he did by using the extensive survey results he could access in his role at the Library. His calculation was remarkably accurate (his error margin turned out to be less than 1%). He was the first to calculate Earth's axial tilt, which similarly 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 used Egyptian and Persian records to estimate the dates of the main events of the 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 and composite numbers.

He was a figure of influence in many fields who yearned to understand the complexities of the entire world. 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. Yet, according to an entry in the Suda (a 10th-century encyclopedia), some critics scorned him, calling him Number 2 because he always came in second in all his endeavours. (Full article...)
Image 4

A stamp of Zhang Heng issued by China Post in 1955

Zhang Heng (Chinese: 張衡; AD 78–139), formerly romanized Chang Heng, was a Chinese polymathic scientist and statesman who lived during the Eastern 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 present-day 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 water-powered 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 Greco-Roman Ptolemy (AD 86–161). (Full article...)
Image 5

Portrait of Luca Pacioli, traditionally attributed to Jacopo de' Barbari, 1495

Luca Bartolomeo de Pacioli, O.F.M. (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 and he was the first person to publish a work on the double-entry system of book-keeping on the continent. He was also called Luca di Borgo after his birthplace, Borgo Sansepolcro, Tuscany. (Full article...)
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Posthumous portrait of Thales by Wilhelm Meyer, based on a bust from the 4th century

Thales of Miletus (/ˈθeɪliːz/ THAY-leez; Ancient Greek: Θαλῆς; c. 626/623 – c. 548/545 BC) was an Ancient Greek pre-Socratic philosopher from Miletus in Ionia, Asia Minor. Thales was one of the Seven Sages, founding figures of Ancient Greece.

Beginning in eighteenth-century historiography, many came to regard him as the first philosopher in the Greek tradition, breaking from the prior use of mythology to explain the world and instead using natural philosophy. He is thus otherwise referred to as the first to have engaged in mathematics, science, and deductive reasoning.

Thales's view that all of nature is based on the existence of a single ultimate substance, which he theorized to be water, was widely influential among the philosophers of his time. Thales thought the Earth floated on water.

In mathematics, Thales is the namesake of Thales's theorem, and the intercept theorem can also be referred to as Thales's theorem. Thales was said to have calculated the heights of the pyramids and the distance of ships from the shore. In science, Thales was an astronomer who reportedly predicted the weather and a solar eclipse. The discovery of the position of the constellation Ursa Major is also attributed to Thales, as well as the timings of the solstices and equinoxes. He was also an engineer, known for having diverted the Halys River. Plutarch wrote that "at that time, Thales alone had raised philosophy from mere speculation to practice." (Full article...)
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An imaginary rendition of Al Biruni on a 1973 Soviet postage stamp

Abu Rayhan Muhammad ibn Ahmad al-Biruni /ælbɪˈruːni/ (Persian: ابوریحان بیرونی; Arabic: أبو الريحان البيروني; 973 – after 1050), known as al-Biruni, was a Khwarazmian Iranian scholar and polymath during the Islamic Golden Age. He has been called variously "Father of Comparative Religion", "Father of modern geodesy", Founder of Indology and the first anthropologist.

Al-Biruni was well versed in physics, mathematics, astronomy, and natural sciences, and also distinguished himself as a historian, chronologist, and linguist. He studied almost all the sciences of his day and was rewarded abundantly for his tireless research in many fields of knowledge. Royalty and other powerful elements in society funded al-Biruni's research and sought him out with specific projects in mind. Influential in his own right, al-Biruni was himself influenced by the scholars of other nations, such as the Greeks, from whom he took inspiration when he turned to the study of philosophy. A gifted linguist, he was conversant in Khwarezmian, Persian, Arabic, and Sanskrit, and also knew Greek, Hebrew, and Syriac. He spent much of his life in Ghazni, then capital of the Ghaznavids, in modern-day central-eastern Afghanistan. In 1017, he travelled to the Indian subcontinent and wrote a treatise on Indian culture entitled Tārīkh al-Hind ("The History of India"), after exploring the Hindu faith practiced in India. He was, for his time, an admirably impartial writer on the customs and creeds of various nations, his scholarly objectivity earning him the title al-Ustadh ("The Master") in recognition of his remarkable description of early 11th-century India. (Full article...)
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Statue in Madrid, erected 2020

Muhammad ibn Musa al-Khwarizmi (Arabic: محمد بن موسى الخوارزميّ; c. 780 – c. 850), or simply al-Khwarizmi, was a vastly influential mathematician from the Islamic Golden Age who produced Arabic-language works in mathematics, astronomy, and geography. Around 820, he worked at the House of Wisdom in Baghdad, the contemporary capital city of the Abbasid Caliphate.

His popularizing treatise on algebra, compiled between 813 and 833 as Al-Jabr (The Compendious Book on Calculation by Completion and Balancing), presented the first systematic solution of linear and quadratic equations. One of his achievements in algebra was his demonstration of how to solve quadratic equations by completing the square, for which he provided geometric justifications. Because al-Khwarizmi was the first person 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 English term algebra comes from the short-hand title of his aforementioned treatise (الجبر Al-Jabr, transl. "completion" or "rejoining"). His name gave rise to the English terms algorism and algorithm; the Spanish, Italian, and Portuguese terms algoritmo; and the Spanish term guarismo and Portuguese term algarismo, both meaning 'digit'.

In the 12th century, Latin translations of al-Khwarizmi's textbook on Indian arithmetic (Algorithmo de Numero Indorum), which codified the various Indian numerals, introduced the decimal-based positional number system to the Western world. Likewise, Al-Jabr, translated into Latin by the English scholar Robert of Chester in 1145, was used until the 16th century as the principal mathematical textbook of European universities.

Al-Khwarizmi revised Geography, the 2nd-century Greek-language treatise by Ptolemy, listing the longitudes and latitudes of cities and localities. He further produced a set of astronomical tables and wrote about calendric works, as well as the astrolabe and the sundial. Al-Khwarizmi made important contributions to trigonometry, producing accurate sine and cosine tables and the first table of tangents. (Full article...)
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Euclid by Jusepe de Ribera, c. 1630–1635

Euclid (/ˈjuːklɪd/; Ancient Greek: Εὐκλείδης; fl. 300 BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the Elements treatise, which established the foundations of geometry that largely dominated the field until the early 19th century. His system, now referred to as Euclidean geometry, involved innovations in combination with a synthesis of theories from earlier Greek mathematicians, including Eudoxus of Cnidus, Hippocrates of Chios, Thales and Theaetetus. With Archimedes and Apollonius of Perga, Euclid is generally considered among the greatest mathematicians of antiquity, and one of the most influential in the history of mathematics.

Very little is known of Euclid's life, and most information comes from the scholars Proclus and Pappus of Alexandria many centuries later. Medieval Islamic mathematicians invented a fanciful biography, and medieval Byzantine and early Renaissance scholars mistook him for the earlier philosopher Euclid of Megara. It is now generally accepted that he spent his career in Alexandria and lived around 300 BC, after Plato's students and before Archimedes. There is some speculation that Euclid studied at the Platonic Academy and later taught at the Musaeum; he is regarded as bridging the earlier Platonic tradition in Athens with the later tradition of Alexandria.

In the Elements, Euclid deduced the theorems from a small set of axioms. He also wrote works on perspective, conic sections, spherical geometry, number theory, and mathematical rigour. In addition to the Elements, Euclid wrote a central early text in the optics field, Optics, and lesser-known works including Data and Phaenomena. Euclid's authorship of On Divisions of Figures and Catoptrics has been questioned. He is thought to have written many lost works. (Full article...)
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17th-century German depiction of Heron

Hero of Alexandria (/ˈhɪəroʊ/; Ancient Greek: Ἥρων ὁ Ἀλεξανδρεύς, Hērōn hò Alexandreús, also known as Heron of Alexandria /ˈhɛrən/; probably 1st or 2nd century AD) was a Greek mathematician and engineer who was active in Alexandria in Egypt during the Roman era. He has been described as the greatest experimentalist of antiquity and a representative of the Hellenistic scientific tradition.

Hero published a well-recognized description of a steam-powered device called an aeolipile, also known as "Hero's engine". Among his most famous inventions was a windwheel, constituting the earliest instance of wind harnessing on land. In his work Mechanics, he described pantographs. Some of his ideas were derived from the works of Ctesibius.

In mathematics, he wrote a commentary on Euclid's Elements and a work on applied geometry known as the Metrica. 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 in manuscripts from the Byzantine Empire and, to a lesser extent, in Latin or Arabic translations. (Full article...)
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Bust of Pythagoras of Samos in the
Capitoline Museums, Rome

Pythagoras of Samos (Ancient Greek: Πυθαγόρας; c. 570 – c. 495 BC) was an ancient Ionian Greek philosopher, polymath, 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. Modern scholars disagree regarding Pythagoras's education and influences, but most agree that he travelled to Croton in southern Italy around 530 BC, where he founded a school in which initiates were allegedly sworn to secrecy and lived a communal, ascetic lifestyle.

In antiquity, Pythagoras was credited with mathematical and scientific discoveries, such as the Pythagorean theorem, Pythagorean tuning, the five regular solids, the theory of proportions, the sphericity of the Earth, the identity of the morning and evening stars as the planet Venus, and the division of the globe into five climatic zones. He was reputedly the first man to call himself a philosopher ("lover of wisdom"). Historians debate whether Pythagoras made these discoveries and pronouncements, as some of the accomplishments credited to him likely originated earlier or were made by his colleagues or successors, such as Hippasus and Philolaus.

The teaching most securely identified with Pythagoras is the "transmigration of souls" or metempsychosis, 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 ratios and thus resonate to produce an inaudible symphony of music. Following Croton's decisive victory over Sybaris in around 510 BC, Pythagoras's followers came into conflict with supporters of democracy, and their meeting houses were burned. Pythagoras may have been killed during this persecution, or he may have escaped to Metapontum and died there.

Pythagoras influenced Plato whose dialogues (especially Timaeus) exhibit Pythagorean ideas. A major revival of his teachings occurred 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 Pythagoreanism had an influence on scientists such as Nicolaus Copernicus, Johannes Kepler, and Isaac Newton. Pythagorean symbolism was also used throughout early modern European esotericism, and his teachings as portrayed in Ovid's Metamorphoses would later influence the modern vegetarian movement. (Full article...)
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17th-century portrait engraving of Cardano

Gerolamo Cardano (Italian: [dʒeˈrɔːlamo karˈdaːno]; also Girolamo or Geronimo; French: Jérôme Cardan; Latin: Hieronymus Cardanus; 24 September 1501– 21 September 1576) was an Italian polymath whose interests and proficiencies ranged through those of mathematician, physician, biologist, physicist, chemist, astrologer, astronomer, philosopher, music theorist, writer, and gambler. He became one of the most influential mathematicians of the Renaissance and one of the key figures in the foundation of probability; he introduced 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, later named "Cardano circles" or "cardanic circles", were used for the construction of the first high-speed printing presses.

Today, Cardano 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 cubic and quartic equations, and acknowledged the existence of imaginary numbers. (Full article...)
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Brahmagupta (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 Khandakhadyaka ("edible bite", dated 665), a more practical text.

In 628 CE, Brahmagupta first described gravity as an attractive force, and used the term "gurutvākarṣaṇam (गुरुत्वाकर्षणम्)" in Sanskrit to describe it. He is also credited with the first clear description of the quadratic formula (the solution of the quadratic equation) in his main work, the Brāhma-sphuṭa-siddhānta. (Full article...)