Numbers (numerals)

A number originally was a count or a measurement. Mathematicians have extended this concept to include abstractions such as negative numbers and the square root of $$-1$$. In common usage, number symbols are often used as labels (highway numbers) or to indicate order (serial numbers).

Examples
The most familiar numbers are the counting numbers or natural numbers. Some writers include 0, thus: {0, 1, 2, ...}. Others do not: {1, 2, 3, ...}. In the base ten number system, now in almost universal use worldwide, the symbols for natural numbers are written using ten digits, 0 through 9. The symbol for the set of all natural numbers is $$\mathbb{N}.$$

If the negative whole numbers are combined with the positive whole numbers and zero, one obtains the integers $$\mathbb{Z}$$ (German Zahl, plural Zahlen).

Negative numbers are used to indicate an opposite. If a positive number is used to indicate distance to the right of some fixed point, a negative number indicates distance to the left. If a positive number indicates a bank deposit, a negative number indicates a withdrawal.

Rational numbers are made up of all numbers that can be expressed as a fraction, with integer numerator and non-zero natural number denominator. The fraction m/n represents the quantity arrived at when a whole is divided into n equal parts, and m of those equal parts are chosen. If m is greater than n, the fraction is greater than one. Fractions can be positive, negative, or zero. The set of all fractions includes the integers, since every integer can be written as a fraction with denominator 1. The symbol for the rational numbers is a bold face $$\mathbb{Q}$$ (for quotient).

The real numbers are made up of all numbers that can be expressed as a decimal. These are the measuring numbers, and in the base ten number system are written as a string of digits, with a period (US) or a comma (Europe) to the right of the ones place. The symbol for the real numbers is $$\mathbb{R}.$$

All measurements are necessarily approximations; the accuracy of the approximation depends on the accuracy of the measuring device. Therefore all measurements are properly represented by decimals that end, the last decimal place indicating the accuracy of the measurement. For example, 1.23 inches indicates a measurement accurate to the nearest hundredth of an inch. However, mathematically, when a rational number is expressed as a decimal, it may never end. Thus 1/3 becomes 0.3333... (unending threes). Mathematicians, therefore, consider both decimals that end and decimals that go on forever. The latter represent an infinite series. Some real numbers can be written as fractions, 0.3333... for example. Others cannot, 0.1010010001... for example. A decimal that can be written as a fraction is called rational, a decimal that cannot be written as a fraction is called irrational. A decimal is rational when it either ends or repeats forever.

There is a technical sense in which the real numbers are the ideal set of numbers. They are the only complete  ordered  field.

Moving to a greater level of abstraction, and away from counting and measuring, the real numbers can be extended to the complex numbers $$\mathbb{C}.$$ This set of numbers arose, historically, from consideration of the question of whether or not there was any sense in which negative numbers can have a square root. A new number was invented, the square root of negative one, denoted by i, a symbol assigned to this new number by Leonhard Euler. The complex numbers consist of all numbers of the form a + bi, where a and b are real numbers. If b is zero, then a + bi is real. If a is zero, then a + bi is called imaginary. The complex numbers are an algebraically closed field, meaning that every polynomial with complex coefficients can be factored into linear factors with complex coefficients.

The above symbols for frequently ooccurring sets of numbers may also be written in plain boldface characters, as $$\mathbf{N} \sub \mathbf{Z} \sub \mathbf{Q} \sub \mathbf{R} \sub \mathbf{C}.$$

While the natural numbers and the real numbers suffice for most everyday purposes, mathematicians have invented many other sets of numbers with specialized uses. Some are subsets of the complex numbers. For example the roots of polynomials with rational coefficients are called the algebraic numbers. Real numbers that are not algebraic are called transcendental numbers. The Gaussian integers are complex numbers a + bi where a and b are integers. Sets of numbers that are not subsets of the complex numbers include the quaternions $$\mathbb{H}$$, invented by Sir William Rowan Hamilton, in which multiplication is not commutative, and the octonions, in which multiplication is not associative.

Further generalizations
The arithmetical operations of numbers, such as addition, subtraction, multiplication and division, are generalized in the branch of mathematics called abstract algebra, the study of abstract number systems such as groups, rings and fields.

Elements of function fields of finite characteristic behave in some ways like numbers and are often regarded as a kind of number by number theorists.

Extensions
Superreal, hyperreal and surreal numbers extend the real numbers by adding infinitesimal and infinitely large numbers. While real numbers may have infinitely long expansions to the right of the decimal point, one can also try to allow for infinitely long expansions to the left. The number system which results depends on what base is used for the digits: any base is possible, but a system with the best mathematical properties is obtained when the base is a prime number p. This leads to the p-adic numbers. For dealing with infinite collections, the natural numbers have been generalized to the ordinal numbers and to the cardinal numbers. The former give the ordering of the collection, the latter its size. (For the finite case, the ordinal and cardinal numbers are equivalent; but they differ in the infinite case.)

Numerals
Numbers should be distinguished from numerals, the symbols used to represent numbers. The number five can be represented by both the base ten numeral 5 and by the Roman numeral V. Notations used to represent numbers are discussed in the article numeral systems. An important development in the history of numerals was the development of a positional system, like modern decimals, which can represent very large numbers. The Roman numerals require extra symbols for larger numbers.

Numerals, natural numbers and integers

 * See also history of numeral systems, history of zero, first usage of negative numbers.


 * ca. 3400 BC - Mesopotamia, the Sumerians invent the first numeral system, a system of weights and measures, and are the first to construct cities
 * ca. 3100 BC - Egypt, earliest known decimal system allows indefinite counting by way of introducing new symbols,.
 * ca. 2700 BC - Indus Valley Civilization, the earliest use of negative numbers (see Negative Number: History)
 * 300s BC - Indian texts use the Sanskrit word "Shunya" to refer to the concept of 'void' (zero)
 * ca. 300 BC - Indian mathematician Pingala writes the "Chhandah-shastra", which contains the first use of zero (indicated by a dot) and also presents the first description of a binary numeral system,
 * ca. 250 BC - late Olmecs had already begun to use a true zero (a shell glyph) several centuries before Ptolemy in the New World. See 0 (number).
 * 50 BC - Indian numerals, the first positional notation base-10 numeral system, begins developing in India
 * 300 - the earliest known use of zero as a decimal digit is introduced by Indian mathematicians
 * 550 - Hindu mathematicians give zero a numeral representation in the positional notation Indian numeral system
 * 628 - Brahmagupta writes the Brahma-sphuta-siddhanta, where zero is clearly explained, and where the modern place-value Indian numeral system is fully developed with rules for manipulating both negative and positive numbers.
 * 773 - Kanka brings Brahmagupta's Brahma-sphuta-siddhanta to Baghdad to explain the Indian system of arithmetic astronomy and the Indian numeral system
 * 1100s - Indian numerals have been modified by Arab mathematicians to form the modern Hindu-Arabic numeral system (used universally in the modern world)
 * 1100s - the Hindu-Arabic numeral system reaches Europe through the Arabs
 * 1202 - Leonardo Fibonacci demonstrates the utility of Hindu-Arabic_numerals in his Book of the Abacus.

rational numbers, irrational numbers and reals

 * See also: Egyptian fractions, history of irrational numbers, History of Pi.


 * ca. 2800 BC - Ancient Indus Valley units of measurement earliest use of decimal fractions.
 * ca. 1000 BC - Vulgar fractions used by the Egyptians.
 * 800 BC - Baudhayana, author of the Baudhayana Sulba Sutra, a Vedic Sanskrit geometric text, contains the first use of the Pythagorean theorem, quadratic equations, and calculates the square root of 2 correct to five decimal places
 * ca. 600 BC - the other Vedic "Sulba Sutras" ("rule of chords" in Sanskrit) contain the first use of irrational numbers, the use of Pythagorean triples, evidence of a number of geometrical proofs, and approximation of &pi; at 3.16
 * 600 BC - Apastamba, author of the Apastamba Sulba Sutra, another Vedic Sanskrit geometric text, makes an attempt at squaring the circle and also calculates the square root of 2 correct to five decimal places
 * 530 BC - Pythagoras' group also discover the irrationality of the square root of two.
 * 150 BC - Jain mathematicians in India write the "Sthananga Sutra", which contains work on the theory of numbers, arithmetical operations, geometry, operations with fractions, simple equations, cubic equations, quartic equations, and permutations and combinations
 * 628 - Brahmagupta writes the Brahma-sphuta-siddhanta, where zero is clearly explained, and where the modern place-value Indian numeral system is fully developed. It also gives rules for manipulating both negative and positive numbers, methods for computing square roots, methods of solving linear and quadratic equations,
 * 1618 - John Napier publishes the first references to e in a work on logarithms.
 * 1761 -Lambert proved that &pi; cannot be rational, and that en is irrational if n is rational (unless n = 0), a proof, however, which left much to be desired.
 * 1794 - Legendre completed Lambert's proof, and showed that &pi; is not the square root of a rational number.
 * 1799 - Paolo Ruffini first proof, (largly ignored) of Abel–Ruffini theorem that the general quintic or higher equations cannot be solved by a general formula involving only arithmetical operations and roots,
 * 1824 - Niels Henrik Abel partially proves the Abel–Ruffini theorem.
 * 1832 - Évariste Galois developed techniques for determining whether a given equation could be solved by radicals which gave rise to the field of Galois theory.
 * 1840 - Joseph Liouville showed that neither e nor e2 can be a root of an integral quadratic equation.
 * 1844-1851 the existence of transcendental numbers was first established by Liouville (1844, 1851), [[ the proof being subsequently displaced by Georg Cantor (1873)
 * 1873 - Charles Hermite first proved e is transcendental,
 * 1882 - Ferdinand von Lindemann (1882), starting from Hermite's conclusions, showed the &pi; is transcendtal. Lindemann's proof was much simplified by Weierstrass (1885), still further by David Hilbert (1893), and has finally been made elementary by Hurwitz and Paul Albert Gordan.
 * 1874 - Georg Cantor shows that the set of all real numbers is uncountably infinite but the set of all algebraic numbers is countably infinite.  Contrary to widely held beliefs, his method was not his famous diagonal argument, which he published three years later.

Infinity

 * See also: history of infinity.


 * ca. 1800 BC - the Yajur Veda, one of the four Hindu Vedas, contains the earliest concept of infinity, and states that "if you remove a part from infinity or add a part to infinity, still what remains is infinity"
 * ca. 400 BC - Jain mathematicians in India write the "Surya Prajinapti", a mathematical text which classifies all numbers into three sets: enumerable, innumerable and infinite. It also recognises five different types of infinity: infinite in one and two directions, infinite in area, infinite everywhere, and infinite perpetually.
 * 400 - the "Bakhshali manuscript" is written by Jaina mathematicians, which describes a theory of the infinite containing different levels of infinity, shows an understanding of indices, as well as logarithms to base 2, and computes square roots of numbers as large as a million correct to at least 11 decimal places
 * 1895 -  Georg Cantor publishes a book about set theory containing the arithmetic of infinite cardinal numbers and the continuum hypothesis,

Complex numbers

 * See also: history of complex numbers.


 * 1st century - the earliest fleeting reference to square roots of negative numbers occurred in the work of the Greek mathematician and inventor Heron of Alexandria
 * 16th century - Complex numbers become more prominent when closed formulas for the roots of third and fourth degree polynomials were discovered by Italian mathematicians (see Niccolo Fontana Tartaglia, Gerolamo Cardano).
 * 1637 - First use of the term imaginary number by René Descartes, it was meant to be derogatory.
 * 1722 - Abraham De Moivre states De Moivre's theorem connecting trigonometric functions and complex numbers,
 * 1797 - Caspar Wessel associates vectors with complex numbers and studies complex number operations in geometrical terms,
 * 1799 - Carl Friedrich Gauss proves the fundamental theorem of algebra (every polynomial equation has a solution among the complex numbers),
 * 1850 - Victor Alexandre Puiseux distinguishes between poles and branch points and introduces the concept of essential singular points,

Prime numbers

 * See also: Prime numbers.


 * 300 BC - Euclid in his Elements studies geometry as an axiomatic system, proves the infinitude of prime numbers and presents the Euclidean algorithm; he states the law of reflection in  Catoptrics, and he proves the fundamental theorem of arithmetic
 * 240 BC - Eratosthenes uses his sieve algorithm to quickly isolate prime numbers,
 * 1796 - Adrien-Marie Legendre conjectures the prime number theorem on their asymptotic distribution.
 * 1859 - Bernhard Riemann formulates the Riemann hypothesis which has strong implications about the distribution of prime numbers,
 * 1896 - Jacques Hadamard and Charles de La Vallée-Poussin independently prove the prime number theorem,

Quaternions

 * See also: History of Quaternionians.


 * 1843 - William Hamilton discovers the calculus of quaternions and deduces that they are non-commutative,
 * 1854 - Arthur Cayley shows that quaternions can be used to represent rotations in four-dimensional space,