Portal:Arithmetic
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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
 In number theory, 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).
The first such distribution found is π(N) ~ N/log(N), where π(N) is the primecounting function (the number of primes less than or equal to N) and log(N) is the natural logarithm of N. This means that for large enough N, the probability that a random integer not greater than N is prime is very close to 1 / log(N). Consequently, a random integer with at most 2n digits (for large enough n) is about half as likely to be prime as a random integer with at most n digits. For example, among the positive integers of at most 1000 digits, about one in 2300 is prime (log(10^{1000}) ≈ 2302.6), whereas among positive integers of at most 2000 digits, about one in 4600 is prime (log(10^{2000}) ≈ 4605.2). In other words, the average gap between consecutive prime numbers among the first N integers is roughly log(N). (Full article...)
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, 1 × 5 or 5 × 1, involve 5 itself.
However, 4 is composite because it is a product (2 × 2) in which both numbers are smaller than 4. Primes are central in number theory because of the fundamental theorem of arithmetic: every natural number greater than 1 is either a prime itself or can be factorized as a product of primes that is unique up to their order.
The property of being prime is called primality. A simple but slow method of checking the primality of a given number $n$, called trial division, tests whether $n$ is a multiple of any integer between 2 and ${\sqrt {n}}$. Faster algorithms include the Miller–Rabin primality test, which is fast but has a small chance of error, and the AKS primality test, which always produces the correct answer in polynomial time but is too slow to be practical. Particularly fast methods are available for numbers of special forms, such as Mersenne numbers. the largest known prime number is a Mersenne prime with 24,862,048 decimal digits.
There are infinitely many primes, as demonstrated by Euclid around 300 BC. No known simple formula separates prime numbers from composite numbers. However, the distribution of primes within the natural numbers in the large can be statistically modelled. The first result in that direction is the prime number theorem, proven at the end of the 19th century, which says that the probability of a randomly chosen number being prime is inversely proportional to its number of digits, that is, to its logarithm. (Full article...) In mathematics, a combination is a selection of items from a collection, 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 kcombination of a set S is a subset of k distinct elements of S. If the set has n elements, the number of kcombinations is equal to the binomial coefficient
:${\binom {n}{k}}={\frac {n(n1)\dotsb (nk+1)}{k(k1)\dotsb 1}},$
which can be written using factorials as $\textstyle {\frac {n!}{k!(nk)!}}$ whenever $k\leq n$, and which is zero when $k>n$. The set of all kcombinations of a set S is often denoted by $\textstyle {\binom {S}{k}}$. (Full article...)  The decimal numeral system (also called the baseten positional numeral system, and occasionally called denary /ˈdiːnəri/ or decanary) is the standard system for denoting integer and noninteger numbers. It is the extension to noninteger numbers of the Hindu–Arabic numeral system. The way of denoting numbers in the decimal system is often referred to as decimal notation.
A decimal numeral (also often just decimal or, less correctly, decimal number), refers generally to the notation of a number in the decimal numeral system. Decimals may sometimes be identified by a decimal separator (usually "." or "," as in 25.9703 or 3,1415). Decimal may also refer specifically to the digits after the decimal separator, such as in "3.14 is the approximation of π to two decimals".
The numbers that may be represented in the decimal system are the decimal fractions. That is, fractions of the form a/10^{n}, where a is an integer, and n is a nonnegative integer. (Full article...)  An integer (from the Latin integer meaning "whole") is colloquially defined as a number that can be written without a fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, 5+1/2, and √2 are not.
The set of integers consists of zero (0), the positive natural numbers (1, 2, 3, ...), also called whole numbers or counting numbers, and their additive inverses (the negative integers, i.e., −1, −2, −3, ...). The set of integers is often denoted by the boldface (Z) or blackboard bold $(\mathbb {Z} )$ letter "Z"—standing originally for the German word Zahlen ("numbers").
ℤ is a subset of the set of all rational numbers ℚ, which in turn is a subset of the real numbers ℝ. Like the natural numbers, ℤ is countably infinite. (Full article...)
In mathematics, the additive inverse of a number a is the number that, when added to a, yields zero. This number is also known as the opposite (number), sign change, and negation. For a real number, it reverses its sign: the additive inverse (opposite number) of a positive number is negative, and the additive inverse of a negative number is positive. Zero is the additive inverse of itself.
The additive inverse of a is denoted by unary minus: −a (see also § Relation to subtraction below). For example, the additive inverse of 7 is −7, because 7 + (−7) = 0, and the additive inverse of −0.3 is 0.3, because −0.3 + 0.3 = 0.
Similarly, the additive inverse of a − b is −(a − b) which can be simplified to b − a. The additive inverse of 2x − 3 is 3 − 2x, because 2x − 3 + 3 − 2x = 0. (Full article...)
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), 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 tells us that the fraction represents 3 equal parts, and the denominator 4 tells us that 4 parts make up a whole. The picture to the right illustrates ${\tfrac {3}{4}}$ or ^{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...)
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), 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 tells us that the fraction represents 3 equal parts, and the denominator 4 tells us that 4 parts make up a whole. The picture to the right illustrates ${\tfrac {3}{4}}$ or ^{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...) In mathematics, the greatest common divisor (GCD) 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, $\gcd(8,12)=4$.
In the name "greatest common divisor", the adjective "greatest" may be replaced by "highest", and the word "divisor" may be replaced by "factor", so that other names include greatest common factor (gcf), etc. Historically, other names for the same concept have included greatest common measure.
This notion can be extended to polynomials (see Polynomial greatest common divisor) and other commutative rings (see § In commutative rings below). (Full article...)  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...)
In number theory, the fundamental theorem of arithmetic, also called the unique factorization theorem or the uniqueprimefactorization theorem, states that every integer greater than 1 either is a prime number itself or can be represented as the product of prime numbers and that, moreover, this representation is unique, up to (except for) the order of the factors. For example,
:$<br>1200=2^{4}\cdot 3\cdot 5^{2}=(2\cdot 2\cdot 2\cdot 2)\cdot 3\cdot (5\cdot 5)=5\cdot 2\cdot 5\cdot 2\cdot 3\cdot 2\cdot 2=\ldots <br>$
The theorem says two things for this example: first, that 1200 can be represented as a product of primes, and second, that no matter how this is done, there will always be exactly four 2s, one 3, two 5s, and no other primes in the product. (Full article...)
In mathematics, a negative number is a real number that is less than zero. Negative numbers represent opposites. If positive represents a movement to the right, negative represents a movement to the left. If positive represents above sea level, then negative represents below sea level. If positive represents a deposit, negative represents a withdrawal. They 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, a decrease in some quantity may be thought of as a negative increase. If a quantity 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 commonsense idea of an opposite is reflected in arithmetic. For example, −(−3) = 3 because the opposite of an opposite is the original value.
Negative numbers are usually written with a minus sign in front. For example, −3 represents a negative quantity with a magnitude of three, and is pronounced "minus three" or "negative three". To help tell the difference between a subtraction operation and a negative number, occasionally the negative sign is placed slightly higher than the minus sign (as a superscript). Conversely, a number that is greater than zero is called positive; zero is usually (but not always) thought of as neither positive nor negative. The positivity of a number may be emphasized by placing a plus sign before it, e.g. +3. In general, the negativity or positivity of a number is referred to as its sign.
Every real number other than zero is either positive or negative. The nonnegative whole numbers are referred to as natural numbers (i.e., 0, 1, 2, 3...), while the positive and negative whole numbers (together with zero) are referred to as integers. (Some definitions of the natural numbers exclude zero.) (Full article...)
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...)
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...) There are several kinds of mean in mathematics, especially in statistics:
For a data set, the arithmetic mean, also known as average or arithmetic average, is a central value of a finite set of numbers: specifically, the sum of the values divided by the number of values. The arithmetic mean of a set of numbers x_{1}, x_{2}, ..., x_{n} is typically denoted by ${\bar {x}}$. If the data set were based on a series of observations obtained by sampling from a statistical population, the arithmetic mean is the sample mean (denoted ${\bar {x}}$) to distinguish it from the mean, or expected value, of the underlying distribution, the population mean (denoted $\mu$ or $\mu _{x}$).
In probability and statistics, the population mean, or expected value, is a measure of the central tendency either of a probability distribution or of a random variable characterized by that distribution. In a discrete probability distribution of a random variable X, the mean is equal to the sum over every possible value weighted by the probability of that value; that is, it is computed by taking the product of each possible value x of X and its probability p(x), and then adding all these products together, giving $\mu =\sum xp(x)....$. An analogous formula applies to the case of a continuous probability distribution. Not every probability distribution has a defined mean (see the Cauchy distribution for an example). Moreover, the mean can be infinite for some distributions. (Full article...)
In mathematics, the natural numbers are those used for counting (as in "there are six coins on the table") and ordering (as in "this is the third largest city in the country"). In common mathematical terminology, words colloquially used for counting are "cardinal numbers", and words used for ordering are "ordinal numbers". The natural numbers can, at times, appear as a convenient set of codes (labels or "names"), that is, as what linguists call nominal numbers, forgoing many or all of the properties of being a number in a mathematical sense. The set of natural numbers is often denoted by the symbol $\mathbb {N}$.
Some definitions, including the standard ISO 800002, begin the natural numbers with 0, corresponding to the nonnegative integers 0, 1, 2, 3, ... (often collectively denoted by the symbol $\mathbb {N} ,$ or $\mathbb {N} _{0}$ for emphasizing that zero is included), whereas others start with 1, corresponding to the positive integers 1, 2, 3, ... (sometimes collectively denoted by the symbol $\mathbb {N} ,$ $\mathbb {N} _{1}$, or $\mathbb {N} ^{*}$ for emphasizing that zero is excluded).
Texts that exclude zero from the natural numbers sometimes refer to the natural numbers together with zero as the whole numbers, while in other writings, that term is used instead for the integers (including negative integers).^{[dubious – discuss]} (Full article...) Scientific notation is a way of expressing numbers that are too large or too small (usually would result in a long string of digits) to be conveniently written in decimal form. It may be referred to as scientific form or standard index form, or standard form in the UK. This base ten notation is commonly used by scientists, mathematicians, and engineers, in part because it can simplify certain arithmetic operations. On scientific calculators it is usually known as "SCI" display mode.
In scientific notation, nonzero numbers are written in the form
:m × 10^{n}or m times ten raised to the power of n, where n is an integer, and the coefficient m is a nonzero real number (usually between 1 and 10 in absolute value, and nearly always written as a terminating decimal). The integer n is called the exponent and the real number m is called the significand or mantissa. The term "mantissa" can be ambiguous where logarithms are involved, because it is also the traditional name of the fractional part of the common logarithm. If the number is negative then a minus sign precedes m, as in ordinary decimal notation. In normalized notation, the exponent is chosen so that the absolute value (modulus) of the significand m is at least 1 but less than 10.
Decimal floating point is a computer arithmetic system closely related to scientific notation. (Full article...)
In arithmetic and number theory, the least common multiple, lowest common multiple, or smallest common multiple of two integers a and b, usually denoted by lcm(a, b), is the smallest positive integer that is divisible by both a and b. Since division of integers by zero is undefined, this definition has meaning only if a and b are both different from zero. However, some authors define lcm(a,0) as 0 for all a, which is the result of taking the lcm to be the least upper bound in the lattice of divisibility.
The lcm is the "lowest common denominator" (lcd) that can be used before fractions can be added, subtracted or compared. The lcm of more than two integers is also welldefined: it is the smallest positive integer that is divisible by each of them. (Full article...) In 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...)
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), 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 tells us that the fraction represents 3 equal parts, and the denominator 4 tells us that 4 parts make up a whole. The picture to the right illustrates ${\tfrac {3}{4}}$ or ^{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...)
In arithmetic and number theory, the least common multiple, lowest common multiple, or smallest common multiple of two integers a and b, usually denoted by lcm(a, b), is the smallest positive integer that is divisible by both a and b. Since division of integers by zero is undefined, this definition has meaning only if a and b are both different from zero. However, some authors define lcm(a,0) as 0 for all a, which is the result of taking the lcm to be the least upper bound in the lattice of divisibility.
The lcm is the "lowest common denominator" (lcd) that can be used before fractions can be added, subtracted or compared. The lcm of more than two integers is also welldefined: it is the smallest positive integer that is divisible by each of them. (Full article...)
Rounding means replacing a number with an approximate value that has a shorter, simpler, or more explicit representation. For example, replacing $23.4476 with $23.45, the fraction 312/937 with 1/3, or the expression √2 with 1.414.
Rounding is often done to obtain a value that is easier to report and communicate than the original. Rounding can also be important to avoid misleadingly precise reporting of a computed number, measurement or estimate; for example, a quantity that was computed as 123,456 but is known to be accurate only to within a few hundred units is usually better stated as "about 123,500".
On the other hand, rounding of exact numbers will introduce some roundoff error in the reported result. Rounding is almost unavoidable when reporting many computations – especially when dividing two numbers in integer or fixedpoint arithmetic; when computing mathematical functions such as square roots, logarithms, and sines; or when using a floatingpoint representation with a fixed number of significant digits. In a sequence of calculations, these rounding errors generally accumulate, and in certain illconditioned cases they may make the result meaningless. (Full article...)
In mathematics, the distributive property of binary operations generalizes the distributive law from elementary algebra, which asserts that one has always
:$x\cdot (y+z)=x\cdot y+x\cdot z.$
For example, one has
: 2 ⋅ (1 + 3) = (2 ⋅ 1) + (2 ⋅ 3).
One says that multiplication distributes over addition.
This basic property of numbers is assumed in the definition of most algebraic structures that have two operations called addition and multiplication, such as complex numbers, polynomials, matrices, rings, and fields. It is also encountered in Boolean algebra and mathematical logic, where each of the logical and (denoted ∧) and the logical or (denoted ∨) distributes over the other. (Full article...)
In statistics and probability theory, the median is the value separating the higher half from the lower half of a data sample, a population, or a probability distribution. For a data set, it may be thought of as "the middle" value. The basic feature of the median in describing data compared to the mean (often simply described as the "average") is that it is not skewed by a small proportion of extremely large or small values, and therefore provides a better representation of a "typical" value. Median income, for example, may be a better way to suggest what a "typical" income is, because income distribution can be very skewed. The median is of central importance in robust statistics, as it is the most resistant statistic, having a breakdown point of 50%: so long as no more than half the data are contaminated, the median is not an arbitrarily large or small result. (Full article...)
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...)
General images
Leibniz's Stepped Reckoner was the first calculator that could perform all four arithmetic operations.
The Tsinghua Bamboo Slips, Chinese Warring States era decimal multiplication table of 305 BC
Cycles 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
"Table of Pythagoras" on Napier's bones
The Ishango bone, found near Lake Edward, possibly displaying a numbering system from more than 20,000 years ago.
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Selected biography
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...)
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.
The English name Euclid is the anglicized version of the Greek name Εὐκλείδης, which means "renowned, glorious". (Full article...)
Pāṇini (Devanagari: पाणिनि, pronounced [paːɳɪnɪ]) was a Sanskrit philologist, grammarian, and a 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...)
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...)
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 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 the tilt of the Earth's axis, 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...)
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...)
Zhang Heng (Chinese: 張衡; AD 78–139), formerly romanized as Chang Heng, was a Chinese polymathic scientist and statesman from Nanyang 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...)
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...)
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...)
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...)
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...)
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...) 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 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...)
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