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The ancient Greeks even questioned whether 1 was a number. The late Olmec people of south-central Mexico began to use a symbol for zero, a shell glyph , in the New World, possibly by the 4th century BC but certainly by 40 BC, which became an integral part of Maya numerals and the Maya calendar. Maya arithmetic used base 4 and base 5 written as base George I.

By AD, Ptolemy , influenced by Hipparchus and the Babylonians, was using a symbol for 0 a small circle with a long overbar within a sexagesimal numeral system otherwise using alphabetic Greek numerals. Because it was used alone, not as just a placeholder, this Hellenistic zero was the first documented use of a true zero in the Old World.

In later Byzantine manuscripts of his Syntaxis Mathematica Almagest , the Hellenistic zero had morphed into the Greek letter Omicron otherwise meaning Another true zero was used in tables alongside Roman numerals by first known use by Dionysius Exiguus , but as a word, nulla meaning nothing , not as a symbol.

When division produced 0 as a remainder, nihil , also meaning nothing , was used. These medieval zeros were used by all future medieval computists calculators of Easter. An isolated use of their initial, N, was used in a table of Roman numerals by Bede or a colleague about , a true zero symbol.

The abstract concept of negative numbers was recognized as early as —50 BC in China. The Nine Chapters on the Mathematical Art contains methods for finding the areas of figures; red rods were used to denote positive coefficients , black for negative.

During the s, negative numbers were in use in India to represent debts. However, in the 12th century in India, Bhaskara gives negative roots for quadratic equations but says the negative value "is in this case not to be taken, for it is inadequate; people do not approve of negative roots".

European mathematicians, for the most part, resisted the concept of negative numbers until the 17th century, although Fibonacci allowed negative solutions in financial problems where they could be interpreted as debts chapter 13 of Liber Abaci , and later as losses in Flos.

At the same time, the Chinese were indicating negative numbers by drawing a diagonal stroke through the right-most non-zero digit of the corresponding positive number's numeral. He used them as exponents , but referred to them as "absurd numbers". As recently as the 18th century, it was common practice to ignore any negative results returned by equations on the assumption that they were meaningless. It is likely that the concept of fractional numbers dates to prehistoric times.

The Ancient Egyptians used their Egyptian fraction notation for rational numbers in mathematical texts such as the Rhind Mathematical Papyrus and the Kahun Papyrus. Classical Greek and Indian mathematicians made studies of the theory of rational numbers, as part of the general study of number theory. Of the Indian texts, the most relevant is the Sthananga Sutra , which also covers number theory as part of a general study of mathematics.

The concept of decimal fractions is closely linked with decimal place-value notation; the two seem to have developed in tandem. For example, it is common for the Jain math sutra to include calculations of decimal-fraction approximations to pi or the square root of 2.

The earliest known use of irrational numbers was in the Indian Sulba Sutras composed between and BC. The story goes that Hippasus discovered irrational numbers when trying to represent the square root of 2 as a fraction. However, Pythagoras believed in the absoluteness of numbers, and could not accept the existence of irrational numbers.

He could not disprove their existence through logic, but he could not accept irrational numbers, and so, allegedly and frequently reported, he sentenced Hippasus to death by drowning, to impede spreading of this disconcerting news. The 16th century brought final European acceptance of negative integral and fractional numbers. By the 17th century, mathematicians generally used decimal fractions with modern notation. It was not, however, until the 19th century that mathematicians separated irrationals into algebraic and transcendental parts, and once more undertook the scientific study of irrationals.

It had remained almost dormant since Euclid. In , the publication of the theories of Karl Weierstrass by his pupil E. Weierstrass's method was completely set forth by Salvatore Pincherle , and Dedekind's has received additional prominence through the author's later work and endorsement by Paul Tannery Weierstrass, Cantor, and Heine base their theories on infinite series, while Dedekind founds his on the idea of a cut Schnitt in the system of real numbers , separating all rational numbers into two groups having certain characteristic properties.

The search for roots of quintic and higher degree equations was an important development, the Abel—Ruffini theorem Ruffini , Abel showed that they could not be solved by radicals formulas involving only arithmetical operations and roots. Hence it was necessary to consider the wider set of algebraic numbers all solutions to polynomial equations. Galois linked polynomial equations to group theory giving rise to the field of Galois theory. Continued fractions , closely related to irrational numbers and due to Cataldi, , received attention at the hands of Euler , [26] and at the opening of the 19th century were brought into prominence through the writings of Joseph Louis Lagrange.

The existence of transcendental numbers [30] was first established by Liouville , Finally, Cantor showed that the set of all real numbers is uncountably infinite but the set of all algebraic numbers is countably infinite , so there is an uncountably infinite number of transcendental numbers.

The earliest known conception of mathematical infinity appears in the Yajur Veda , an ancient Indian script, which at one point states, "If you remove a part from infinity or add a part to infinity, still what remains is infinity. They distinguished between five types of infinity: infinite in one and two directions, infinite in area, infinite everywhere, and infinite perpetually.

Aristotle defined the traditional Western notion of mathematical infinity. He distinguished between actual infinity and potential infinity —the general consensus being that only the latter had true value. Galileo Galilei 's Two New Sciences discussed the idea of one-to-one correspondences between infinite sets. But the next major advance in the theory was made by Georg Cantor ; in he published a book about his new set theory , introducing, among other things, transfinite numbers and formulating the continuum hypothesis.

In the s, Abraham Robinson showed how infinitely large and infinitesimal numbers can be rigorously defined and used to develop the field of nonstandard analysis. The system of hyperreal numbers represents a rigorous method of treating the ideas about infinite and infinitesimal numbers that had been used casually by mathematicians, scientists, and engineers ever since the invention of infinitesimal calculus by Newton and Leibniz.

A modern geometrical version of infinity is given by projective geometry , which introduces "ideal points at infinity", one for each spatial direction. Each family of parallel lines in a given direction is postulated to converge to the corresponding ideal point.

This is closely related to the idea of vanishing points in perspective drawing. The earliest fleeting reference to square roots of negative numbers occurred in the work of the mathematician and inventor Heron of Alexandria in the 1st century AD , when he considered the volume of an impossible frustum of a pyramid. It was soon realized that these formulas, even if one was only interested in real solutions, sometimes required the manipulation of square roots of negative numbers.

This was doubly unsettling since they did not even consider negative numbers to be on firm ground at the time. See imaginary number for a discussion of the "reality" of complex numbers. A further source of confusion was that the equation. The incorrect use of this identity, and the related identity. The 18th century saw the work of Abraham de Moivre and Leonhard Euler. De Moivre's formula states:. The existence of complex numbers was not completely accepted until Caspar Wessel described the geometrical interpretation in Carl Friedrich Gauss rediscovered and popularized it several years later, and as a result the theory of complex numbers received a notable expansion.

The idea of the graphic representation of complex numbers had appeared, however, as early as , in Wallis 's De algebra tractatus. In the same year, Gauss provided the first generally accepted proof of the fundamental theorem of algebra , showing that every polynomial over the complex numbers has a full set of solutions in that realm. This generalization is largely due to Ernst Kummer , who also invented ideal numbers , which were expressed as geometrical entities by Felix Klein in In Victor Alexandre Puiseux took the key step of distinguishing between poles and branch points, and introduced the concept of essential singular points.

Prime numbers have been studied throughout recorded history. But most further development of the theory of primes in Europe dates to the Renaissance and later eras. In , Adrien-Marie Legendre conjectured the prime number theorem , describing the asymptotic distribution of primes. Other results concerning the distribution of the primes include Euler's proof that the sum of the reciprocals of the primes diverges, and the Goldbach conjecture , which claims that any sufficiently large even number is the sum of two primes.

Yet another conjecture related to the distribution of prime numbers is the Riemann hypothesis , formulated by Bernhard Riemann in Goldbach and Riemann's conjectures remain unproven and unrefuted. Numbers can be classified into sets , called number sets or number systems , such as the natural numbers and the real numbers. The main number systems are as follows:. Each of these number system is a subset of the next one.

So, for example, a rational number is also a real number, and every real number is also a complex number. This can be expressed symbolically as. The most familiar numbers are the natural numbers sometimes called whole numbers or counting numbers : 1, 2, 3, and so on. Traditionally, the sequence of natural numbers started with 1 0 was not even considered a number for the Ancient Greeks. However, in the 19th century, set theorists and other mathematicians started including 0 cardinality of the empty set , i.

In the base 10 numeral system, in almost universal use today for mathematical operations, the symbols for natural numbers are written using ten digits : 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. The radix or base is the number of unique numerical digits, including zero, that a numeral system uses to represent numbers for the decimal system, the radix is In this base 10 system, the rightmost digit of a natural number has a place value of 1, and every other digit has a place value ten times that of the place value of the digit to its right.

In set theory , which is capable of acting as an axiomatic foundation for modern mathematics, [34] natural numbers can be represented by classes of equivalent sets. For instance, the number 3 can be represented as the class of all sets that have exactly three elements. Alternatively, in Peano Arithmetic , the number 3 is represented as sss0, where s is the "successor" function i.

Many different representations are possible; all that is needed to formally represent 3 is to inscribe a certain symbol or pattern of symbols three times. The negative of a positive integer is defined as a number that produces 0 when it is added to the corresponding positive integer. Negative numbers are usually written with a negative sign a minus sign. Here the letter Z comes from German Zahl 'number'.

The set of integers forms a ring with the operations addition and multiplication. The natural numbers form a subset of the integers. As there is no common standard for the inclusion or not of zero in the natural numbers, the natural numbers without zero are commonly referred to as positive integers , and the natural numbers with zero are referred to as non-negative integers.

A rational number is a number that can be expressed as a fraction with an integer numerator and a positive integer denominator. Negative denominators are allowed, but are commonly avoided, as every rational number is equal to a fraction with positive denominator.

Fractions are written as two integers, the numerator and the denominator, with a dividing bar between them. If the absolute value of m is greater than n supposed to be positive , then the absolute value of the fraction is greater than 1. Fractions can be greater than, less than, or equal to 1 and can also be positive, negative, or 0. The set of all rational numbers includes the integers since every integer can be written as a fraction with denominator 1. The symbol for the real numbers is R , also written as R.

Every real number corresponds to a point on the number line. The following paragraph will focus primarily on positive real numbers. The treatment of negative real numbers is according to the general rules of arithmetic and their denotation is simply prefixing the corresponding positive numeral by a minus sign , e. Most real numbers can only be approximated by decimal numerals, in which a decimal point is placed to the right of the digit with place value 1.

Each digit to the right of the decimal point has a place value one-tenth of the place value of the digit to its left. For example, A real number can be expressed by a finite number of decimal digits only if it is rational and its fractional part has a denominator whose prime factors are 2 or 5 or both, because these are the prime factors of 10, the base of the decimal system.

Thus, for example, one half is 0. Representing other real numbers as decimals would require an infinite sequence of digits to the right of the decimal point. If this infinite sequence of digits follows a pattern, it can be written with an ellipsis or another notation that indicates the repeating pattern.

Such a decimal is called a repeating decimal. Forever repeating 3s are also written as 0. It turns out that these repeating decimals including the repetition of zeroes denote exactly the rational numbers, i. A real number that is not rational is called irrational. When pi is written as. Another well-known number, proven to be an irrational real number, is. Not only these prominent examples but almost all real numbers are irrational and therefore have no repeating patterns and hence no corresponding decimal numeral.

They can only be approximated by decimal numerals, denoting rounded or truncated real numbers. Any rounded or truncated number is necessarily a rational number, of which there are only countably many. All measurements are, by their nature, approximations, and always have a margin of error.

Thus Digits that suggest a greater accuracy than the measurement itself does, should be removed. The remaining digits are then called significant digits. For example, measurements with a ruler can seldom be made without a margin of error of at least 0. If the sides of a rectangle are measured as 1. Since not even the second digit after the decimal place is preserved, the following digits are not significant.

Therefore, the result is usually rounded to 5. Just as the same fraction can be written in more than one way, the same real number may have more than one decimal representation. For example, 0. A given real number has only the following decimal representations: an approximation to some finite number of decimal places, an approximation in which a pattern is established that continues for an unlimited number of decimal places or an exact value with only finitely many decimal places.

In this last case, the last non-zero digit may be replaced by the digit one smaller followed by an unlimited number of 9's, or the last non-zero digit may be followed by an unlimited number of zeros. Thus the exact real number 3. Similarly, a decimal numeral with an unlimited number of 0's can be rewritten by dropping the 0's to the right of the rightmost nonzero digit, and a decimal numeral with an unlimited number of 9's can be rewritten by increasing by one the rightmost digit less than 9, and changing all the 9's to the right of that digit to 0's.

Finally, an unlimited sequence of 0's to the right of a decimal place can be dropped. For example, 6. Finally, if all of the digits in a numeral are 0, the number is 0, and if all of the digits in a numeral are an unending string of 9's, you can drop the nines to the right of the decimal place, and add one to the string of 9s to the left of the decimal place.

The real numbers also have an important but highly technical property called the least upper bound property. It can be shown that any ordered field , which is also complete , is isomorphic to the real numbers. Moving to a greater level of abstraction, the real numbers can be extended to the complex numbers. This set of numbers arose historically from trying to find closed formulas for the roots of cubic and quadratic polynomials.

The complex numbers consist of all numbers of the form. Because of this, complex numbers correspond to points on the complex plane , a vector space of two real dimensions. If the real part of a complex number is 0, then the number is called an imaginary number or is referred to as purely imaginary ; if the imaginary part is 0, then the number is a real number. Thus the real numbers are a subset of the complex numbers.

If the real and imaginary parts of a complex number are both integers, then the number is called a Gaussian integer. The fundamental theorem of algebra asserts that the complex numbers form an algebraically closed field , meaning that every polynomial with complex coefficients has a root in the complex numbers. Like the reals, the complex numbers form a field , which is complete , but unlike the real numbers, it is not ordered.

That is, there is no consistent meaning assignable to saying that i is greater than 1, nor is there any meaning in saying that i is less than 1. In technical terms, the complex numbers lack a total order that is compatible with field operations. An even number is an integer that is "evenly divisible" by two, that is divisible by two without remainder ; an odd number is an integer that is not even.

The old-fashioned term "evenly divisible" is now almost always shortened to " divisible ". A prime number , often shortened to just prime , is an integer greater than 1 that is not the product of two smaller positive integers.

The first few prime numbers are 2, 3, 5, 7, and There is no such simple formula as for odd and even numbers to generate the prime numbers. The primes have been widely studied for more than years and have led to many questions, only some of which have been answered. The study of these questions belongs to number theory. Goldbach's conjecture is an example of a still unanswered question: "Is every even number the sum of two primes?

One answered question, as to whether every integer greater than one is a product of primes in only one way, except for a rearrangement of the primes, was confirmed; this proven claim is called the fundamental theorem of arithmetic.

A proof appears in Euclid's Elements. Many subsets of the natural numbers have been the subject of specific studies and have been named, often after the first mathematician that has studied them. Example of such sets of integers are Fibonacci numbers and perfect numbers.

For more examples, see Integer sequence. A total; a sum: the number of feet in a mile. An indefinite quantity of units or individuals: The crowd was small in number. A number of people complained. A large quantity; a multitude: Numbers of people visited the fair. Numerical superiority: The South had leaders, the North numbers. Grammar The indication, as by inflection, of the singularity, duality, or plurality of a linguistic form. Metrical feet or lines; verses: "These numbers will I tear, and write in prose" Shakespeare.

Numbers used with a sing. One of the separate offerings in a program of music or other entertainment: The band's second number was a march. Slang A frequently repeated, characteristic speech, argument, or performance: suspects doing their usual number—protesting innocence. Slang A person or thing singled out for a particular characteristic: a crafty number. To assign a number to or mark with a number: Did you number the pages of the report?

To determine the number or amount of; count: Tickets sold for the show were numbered at To total in number or amount; add up to: The ships in the harbor number around To include in a group or category: He was numbered among the lost.

To limit or restrict in number: Our days are numbered. To call off numbers; count: numbering to ten. To have as a total; amount to a number: The applicants numbered in the thousands. In unison as numbers are called out by a leader: performing calisthenics by the numbers. Usage Note: As a collective noun number may take either a singular or a plural verb. It takes a singular verb when it is preceded by the definite article the: The number of skilled workers is increasing. It takes a plural verb when preceded by the indefinite article a: A number of the workers have learned new skills.

All rights reserved. Mathematics a concept of quantity that is or can be derived from a single unit, the sum of a collection of units, or zero.

Every number occupies a unique position in a sequence, enabling it to be used in counting. It can be assigned to one or more sets that can be arranged in a hierarchical classification: every number is a complex number ; a complex number is either an imaginary number or a real number , and the latter can be a rational number or an irrational number ; a rational number is either an integer or a fraction , while an irrational number can be a transcendental number or an algebraic number.

See complex number , imaginary number , real number , rational number , irrational number , integer , fraction , transcendental number , algebraic number See also cardinal number , ordinal number.

Classical Music a self-contained part of an opera or other musical score, esp one for the stage. Recreational Drugs slang a cannabis cigarette: roll another number. Grammar a grammatical category for the variation in form of nouns, pronouns, and any words agreeing with them, depending on how many persons or things are referred to, esp as singular or plural in number and in some languages dual or trial. Military by numbers military of a drill procedure, etc performed step by step, each move being made on the call of a number.

Copyright , , by Random House, Inc. Switch to new thesaurus. Fibonacci number - a number in the Fibonacci sequence. Arabic numeral , Hindu numeral , Hindu-Arabic numeral - one of the symbols 1,2,3,4,5,6,7,8,9,0. ABA transit number , bank identification number , BIN - an identification number consisting of a two-part code assigned to banks and savings associations; the first part shows the location and the second identifies the bank itself.

Social Security number - the number of a particular individual's Social Security account. Based on WordNet 3.