## Number |

A **number** is a ^{[1]} A written ^{[2]} A *number* may refer to a symbol, a word or phrase, or the mathematical object.

In ^{[3]} ^{[4]} ^{[5]} such as ^{[6]} which extend the real numbers with a ^{[4]}

Besides their practical uses, numbers have cultural significance throughout the world.^{[7]}^{[8]} For example, in Western society, the number ^{[7]} Though it is now regarded as ^{[9]} Numerology heavily influenced the development of ^{[9]}

During the 19th century, mathematicians began to develop many different abstractions which share certain properties of numbers and may be seen as extending the concept. Among the first were the ^{[10]}

- history
- main classification
- subclasses of the integers
- subclasses of the complex numbers
- extensions of the concept
- see also
- notes
- references
- external links

Numbers should be distinguished from **numerals**, the symbols used to represent numbers. The Egyptians invented the first ciphered numeral system, and the Greeks followed by mapping their counting numbers onto Ionian and Doric alphabets.^{[11]} Roman numerals, a system that used combinations of letters from the Roman alphabet, remained dominant in Europe until the spread of the superior ^{[12]} The key to the effectiveness of the system was the symbol for zero, which was developed by ancient ^{[12]}

Bones and other artifacts have been discovered with marks cut into them that many believe are ^{[13]} These tally marks may have been used for counting elapsed time, such as numbers of days, lunar cycles or keeping records of quantities, such as of animals.

A tallying system has no concept of place value (as in modern

The first known system with place value was the ^{[14]}

The first known documented use of zero dates to AD 628, and appeared in the * Brāhmasphuṭasiddhānta*, the main work of the

Brahmagupta's Brahmasphuṭasiddhanta is the first book that mentions zero as a number, hence Brahmagupta is usually considered the first to formulate the concept of zero. He gave rules of using zero with negative and positive numbers, such as 'Zero plus a positive number is a positive number, and a negative number plus zero is the negative number'. The Brahmasphutasiddhanta is the earliest known text to treat zero as a number in its own right, rather than as simply a placeholder digit in representing another number as was done by the Babylonians or as a symbol for a lack of quantity as was done by Ptolemy and the Romans.

The use of 0 as a number should be distinguished from its use as a placeholder numeral in *nfr* to denote zero balance in *Shunye* or *shunya* to refer to the concept of *void*. In mathematics texts this word often refers to the number zero.^{[15]} In a similar vein,

There are other uses of zero before Brahmagupta, though the documentation is not as complete as it is in the Brahmasphutasiddhanta.

Records show that the

The late

By 130 AD, *documented* use of a true zero in the Old World. In later *Syntaxis Mathematica* (*Almagest*), the Hellenistic zero had morphed into the

Another true zero was used in tables alongside *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

The abstract concept of negative numbers was recognized as early as 100–50 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

During the 600s, negative numbers were in use in * Brāhmasphuṭasiddhānta* 628, who used negative numbers to produce the general form

* Liber Abaci*, 1202) and later as losses (in

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, just as

It is likely that the concept of fractional numbers dates to *Elements*

The concept of

The earliest known use of irrational numbers was in the ^{[18]} The first existence proofs of irrational numbers is usually attributed to ^{[19]}

The 16th century brought final European acceptance of * Crelle,* 74),

The search for roots of

The existence of transcendental numbers^{[20]} was first established by *e* is transcendental and

The earliest known conception of mathematical

* Two New Sciences* discussed the idea of

In the 1960s,

A modern geometrical version of infinity is given by

The earliest fleeting reference to square roots of negative numbers occurred in the work of the mathematician and inventor

This was doubly unsettling since they did not even consider negative numbers to be on firm ground at the time. When

seemed capriciously inconsistent with the algebraic identity

which is valid for positive real numbers *a* and *b*, and was also used in complex number calculations with one of *a*, *b* positive and the other negative. The incorrect use of this identity, and the related identity

in the case when both *a* and *b* are negative even bedeviled *i* in place of to guard against this mistake.

The 18th century saw the work of

while

The existence of complex numbers was not completely accepted until *De Algebra tractatus*.

Also in 1799, Gauss provided the first generally accepted proof of the

*a* + *bi*, where *a* and *b* are integral, or rational (and *i* is one of the two roots of *x*^{2} + 1 = 0). His student, *a* + *bω*, where *ω* is a complex root of *x*^{3} − 1 = 0. Other such classes (called *x*^{k} − 1 = 0 for higher values of *k*. This generalization is largely due to

In 1850

*Elements* to the theory of primes; in it he proved the infinitude of the primes and the

In 240 BC,

In 1796,