Portal:Arithmetic
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The Arithmetic Portal
Arithmetic (from the Greek ἀριθμός arithmos, 'number' and τική [τέχνη], tiké [téchne], 'art') 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.
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 mathematics, summation is the addition of a sequence of any kind of numbers, called addends or summands; the result is their sum or total. Beside numbers, other types of values can be summed as well: functions, vectors, matrices, polynomials and, in general, elements of any type of mathematical objects on which an operation denoted "+" is defined.
Summations of infinite sequences are called series. They involve the concept of limit, and are not considered in this article.
The summation of an explicit sequence is denoted as a succession of additions. For example, summation of [1, 2, 4, 2] is denoted 1 + 2 + 4 + 2, and results in 9, that is, 1 + 2 + 4 + 2 = 9. Because addition is associative and commutative, there is no need of parentheses, and the result is the same irrespective of the order of the summands. Summation of a sequence of only one element results in this element itself. Summation of an empty sequence (a sequence with zero element), by convention, results in 0.
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 amount or sum of those values combined. The example in the adjacent picture shows a combination of three apples and two apples, making a total of five apples. This observation is equivalent to the mathematical expression "3 + 2 = 5" (i.e., "3 add 2 is equal to 5").
Besides counting items, addition can also be defined and executed without referring to concrete objects, using abstractions called numbers instead, such as integers, real numbers and complex numbers. Addition belongs to arithmetic, a branch of mathematics. In algebra, another area of mathematics, addition can also be performed on abstract objects such as vectors, matrices, subspaces and subgroups.
Addition has several important properties. It is commutative, meaning that order does not matter, and it is associative, meaning that when one adds more than two numbers, the order in which addition is performed does not matter (see Summation). Repeated addition of 1 is the same as counting; addition of 0 does not change a number. Addition also obeys predictable rules concerning related operations such as subtraction and multiplication.
In arithmetic, a quotient (from Latin: quotiens "how many times", pronounced /ˈkwoʊʃənt/) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as the integer part of a division (in the case of Euclidean division), or as a fraction or a ratio (in the case of proper division). For example, when dividing 20 (the dividend) by 3 (the divisor), the quotient is 6 in the Euclidean division sense, and $6{\tfrac {2}{3}}$ in the proper division sense. In the second sense, a quotient is simply the ratio of a dividend to its divisor. In science, a multiple is the product of any quantity and an integer. In other words, for the quantities a and b, we say that b is a multiple of a if b = na for some integer n, which is called the multiplier. If a is not zero, this is equivalent to saying that b/a is an integer.
In mathematics, when a and b are both integers, and b is a multiple of a, then a is called a divisor of b. One says also that a divides b. If a and b are not integers, mathematicians prefer generally to use integer multiple instead of multiple, for clarification. In fact, multiple is used for other kinds of product; for example, a polynomial p is a multiple of another polynomial q if there exists third polynomial r such that p = qr.
In some texts, "a is a submultiple of b" has the meaning of "b being an integer multiple of a". This terminology is also used with units of measurement (for example by the BIPM and NIST), where a submultiple of a main unit is a unit, named by prefixing the main unit, defined as the quotient of the main unit by an integer, mostly a power of 10^{3}. For example, a millimetre is the 1000fold submultiple of a metre. As another example, one inch may be considered as a 12fold submultiple of a foot, or a 36fold submultiple of a yard.
The Riemann zeta function or Euler–Riemann zeta function, ζ(s), is a function of a complex variable s that analytically continues the sum of the Dirichlet series
:$\zeta (s)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}},$
which converges when the real part of s is greater than 1. More general of ζ(s) for all s are given below. The Riemann zeta function plays a pivotal role in analytic number theory and has applications in physics, probability theory, and applied statistics.
As a function of a real variable, Leonhard Euler first introduced and studied it in the first half of the eighteenth century without using complex analysis, which was not available at the time. Bernhard Riemann's 1859 article "On the Number of Primes Less Than a Given Magnitude" extended the Euler definition to a complex variable, proved its meromorphic continuation and functional equation, and established a relation between its zeros and the distribution of prime numbers.
In mathematics, an Lfunction is a meromorphic function on the complex plane, associated to one out of several categories of mathematical objects. An Lseries is a Dirichlet series, usually convergent on a halfplane, that may give rise to an Lfunction via analytic continuation. The Riemann zeta function is an example of an Lfunction, and one important result involving Lfunctions is the Riemann hypothesis and its generalization.
The theory of Lfunctions has become a very substantial, and still largely conjectural, part of contemporary analytic number theory. In it, broad generalisations of the Riemann zeta function and the Lseries for a Dirichlet character are constructed, and their general properties, in most cases still out of reach of proof, are set out in a systematic way. Because of the Euler product formula there is a deep connection between Lfunctions and the theory of prime numbers. The unitary method is a technique for solving a problem by first finding the value of a single unit, and then finding the necessary value by multiplying the single unit value. In essence, this method is used to find the value of a unit from the value of a multiple, and hence the value of a multiple.
In statistics and probability theory, a median is a 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 advantage of the median in describing data compared to the mean (often simply described as the "average") is that it is not skewed so much by a small proportion of extremely large or small values, and so it may give a better idea of a "typical" value. For example, in understanding statistics like household income or assets, which vary greatly, the mean may be skewed by a small number of extremely high or low values. Median income, for example, may be a better way to suggest what a "typical" income is.
Because of this, 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 will not give an arbitrarily large or small result. 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 (i.e, X = x) at which the probability mass function takes its maximum value. In other words, it is the value that is most likely to be sampled.
Like the statistical mean and median, the mode is a way of expressing, in a (usually) single number, important information about a random variable or a population. The numerical value of the mode is the same as that of the mean and median in a normal distribution, and it may be very different in highly skewed distributions.
The mode is not necessarily unique to a given discrete distribution, since the probability mass function may take the same maximum value at several points x_{1}, x_{2}, etc. The most extreme case occurs in uniform distributions, where all values occur equally frequently.
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.
In mathematics, a percentage (from Latin per centum "by a hundred") is a number or ratio expressed as a fraction of 100. It is often denoted using the percent sign, "%", although the abbreviations "pct.", "pct" and sometimes "pc" are also used. A percentage is a dimensionless number (pure number); it has no unit of measurement. There are several kinds of mean in mathematics, especially in statistics.
For a data set, the arithmetic mean, also called the expected value or average, is the central value of a discrete 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 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 the 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.  The decimal numeral system (also called baseten positional numeral system, and occasionally called denary 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, or just decimal, or casually 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 . That is, fractions of the form a/10^{n}, where a is an integer, and n is a nonnegative integer.
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. 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.
In mathematics, parity is the property of an integer's inclusion in one of two categories: even or odd. An integer is even if it is divisible by two and odd if it is not even. For example, 6 is even because there is no remainder when dividing it by 2. By contrast, 3, 5, 7, 21 leave a remainder of 1 when divided by 2. Examples of even numbers include −4, 0, 82 and 178. In particular, zero is an even number. Some examples of odd numbers are −5, 3, 29, and 73.
A formal definition of an even number is that it is an integer of the form n = 2k, where k is an integer; it can then be shown that an odd number is an integer of the form n = 2k + 1 (or alternately, 2k  1). It is important to realize that the above definition of parity applies only to integer numbers, hence it cannot be applied to numbers like 1/2 or 4.201. See the section "Higher mathematics" below for some extensions of the notion of parity to a larger class of "numbers" or in other more general settings.
The sets of even and odd numbers can be defined as following:
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).
Multiplying a number is the same as dividing its reciprocal and vice versa. For example, multiplication by 4/5 (or 0.8) will give the same result as division by 5/4 (or 1.25). Therefore, multiplication by a number followed by multiplication of its reciprocal yields the original number (since their product is 1).
The term reciprocal was in common use at least as far back as the third edition of Encyclopædia Britannica (1797) to describe two numbers whose product is 1; geometrical quantities in inverse proportion are described as reciprocall in a 1570 translation of Euclid's Elements.
In mathematics, a square root of a number x is a number y such that y^{2} = x; in other words, a number y whose square (the result of multiplying the number by itself, or y ⋅ y) is x. For example, 4 and −4 are square roots of 16, because 4^{2} = (−4)^{2} = 16.
Every nonnegative real number x has a unique nonnegative square root, called the principal square root, which is denoted by √x, where the symbol √ is called the radical sign or radix. For example, the principal square root of 9 is 3, which is denoted by √9 = 3, because 3^{2} = 3 ⋅ 3 = 9 and 3 is nonnegative. The term (or number) whose square root is being considered is known as the radicand. The radicand is the number or expression underneath the radical sign, in this case 9.
Every positive number x has two square roots: √x, which is positive, and −√x, which is negative. Together, these two roots are denoted as ±√x (see ± shorthand). Although the principal square root of a positive number is only one of its two square roots, the designation "the square root" is often used to refer to the principal square root. For positive x, the principal square root can also be written in exponent notation, as x^{1/2}.
Square roots of negative numbers can be discussed within the framework of complex numbers. More generally, square roots can be considered in any context in which a notion of "squaring" of some mathematical objects is defined. These include algebras of matrices, endomorphism rings, among other mathematical structures. 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.  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 $\textstyle {\binom {S}{k}}$.
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.
In mathematics, the irrational numbers are all the real numbers which are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two integers. When the ratio of lengths of two line segments is an irrational number, the line segments are also described as being incommensurable, meaning that they share no "measure" in common, that is, there is no length ("the measure"), no matter how short, that could be used to express the lengths of both of the two given segments as integer multiples of itself.
Among irrational numbers are the ratio π of a circle's circumference to its diameter, Euler's number e, the golden ratio φ, and the square root of two; in fact all square roots of natural numbers, other than of perfect squares, are irrational.
Like all real numbers, irrational numbers can be expressed in positional notation, notably as a decimal number. In the case of irrational numbers, the decimal expansion does not terminate, nor end with a repeating sequence. For example, the decimal representation of π starts with 3.14159, but no finite number of digits can represent π exactly, nor does it repeat. Conversely, a decimal expansion that terminates or repeats must be a rational number. These are provable properties of rational numbers and positional number systems, and are not used as definitions in mathematics. 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 a boldface letter 'Z' ("Z") or blackboard bold $\mathbb {Z}$ (Unicode U+2124 ℤ) standing for the German word Zahlen ([ˈtsaːlən], "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.
In mathematics, the distributive property of binary operations generalizes the distributive law from Boolean algebra and elementary algebra. In propositional logic, distribution refers to two valid rules of replacement. The rules allow one to reformulate conjunctions and disjunctions within logical proofs.
For example, in arithmetic:
: 2 ⋅ (1 + 3) = (2 ⋅ 1) + (2 ⋅ 3), but 2 / (1 + 3) ≠ (2 / 1) + (2 / 3).
On the lefthand side of the first equation, the 2 multiplies the sum of 1 and 3; on the righthand side, it multiplies the 1 and the 3 individually, with the products added afterward.
Because these give the same final answer (8), multiplication by 2 is said to distribute over the addition of 1 and 3.
Since one could have put any real numbers in place of 2, 1, and 3 above, and still have obtained a true equation, multiplication of real numbers distributes over addition of real numbers.
In mathematics, parity is the property of an integer's inclusion in one of two categories: even or odd. An integer is even if it is divisible by two and odd if it is not even. For example, 6 is even because there is no remainder when dividing it by 2. By contrast, 3, 5, 7, 21 leave a remainder of 1 when divided by 2. Examples of even numbers include −4, 0, 82 and 178. In particular, zero is an even number. Some examples of odd numbers are −5, 3, 29, and 73.
A formal definition of an even number is that it is an integer of the form n = 2k, where k is an integer; it can then be shown that an odd number is an integer of the form n = 2k + 1 (or alternately, 2k  1). It is important to realize that the above definition of parity applies only to integer numbers, hence it cannot be applied to numbers like 1/2 or 4.201. See the section "Higher mathematics" below for some extensions of the notion of parity to a larger class of "numbers" or in other more general settings.
The sets of even and odd numbers can be defined as following:
General images
The Tsinghua Bamboo Slips, Chinese Warring States era decimal multiplication table of 305 BC
The Ishango bone, found near Lake Edward, possibly displaying a numbering system from more than 20,000 years ago.
"Table of Pythagoras" on Napier's bones
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
Leibniz's Stepped Reckoner was the first calculator that could perform all four arithmetic operations.
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Selected biography
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 Ghaznavid dynasty, 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.
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 Greek 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 the 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—mostly in manuscripts from the Eastern Roman Empire, and a smaller part in Latin or Arabic translations.
Gerolamo (also Girolamo or Geronimo) Cardano (Italian: [dʒeˈrɔlamo karˈdano]; French: Jérôme Cardan; Latin: Hieronymus Cardanus; 24 September 1501 – 21 September 1576) was an Italian polymath, whose interests and proficiencies ranged from being a 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.
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, once again with remarkable accuracy. Additionally, he may have accurately calculated the distance from the Earth to the Sun and invented the leap day. 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 chief literary and political events from the conquest of Troy. Eratosthenes dated 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.
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".
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.
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.
Muḥammad ibn Mūsā alKhwārizmī (Persian: Muḥammad Khwārizmī محمد بن موسی خوارزمی; 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 (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.
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.
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 an 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.
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).
Pāṇini (Sanskrit: पाणिनि) (pronounced [paːɳɪnɪ], variously dated between fl. 4th century BCE and "6th to 5th century BCE") was an ancient Sanskrit philologist, grammarian, and a revered scholar in ancient India.
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. Brahmagupta (c. 598 CE – 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.
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