A number is an abstract idea used in counting and measuring. A symbol which represents a number is called a numeral, but in common usage the word number is used for both the idea and the symbol. In addition to
their use in counting and measuring, numerals are often used for labels (telephone
numbers), for ordering (serial numbers), and for codes (ISBNs). In mathematics, the definition of number
has been extended over the years to include such numbers as zero, negative numbers, rational numbers, irrational numbers, and
complex numbers.
Certain procedures which input one or more numbers and output a number are called numerical operations. Unary operations input a single number and
output a single number. For example, the successor operation adds one to an integer: the successor of 4 is 5. More common are
binary operations which input two numbers and output a single number. Examples of
binary operations include addition, subtraction,
multiplication, division, and
exponentiation. The study of numerical operations is called arithmetic.
The branch of mathematics that studies abstract number systems such as groups, rings and fields is called abstract algebra.
Types of numbers
Numbers can be classified into sets, called number
systems. (For different methods of expressing numbers with symbols, such as the Roman
numerals, see numeral systems.)
Natural numbers
The most familiar numbers are the natural numbers or counting numbers: one,
two, three, ... . Some people also include zero in the natural numbers; however, others do not.
In the base ten number system, in almost universal use today, the symbols for natural numbers
are written using ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. In this base ten system,
the rightmost digit of a natural number has a place value of one, and every other digit has a place value ten times that of the
place value of the digit to its right. The symbol for the set of all natural numbers is N, also written UNIQ4a6bbbd91f2115d3-math-7e7ff903648dd1b800000001.
Integers
Negative numbers are numbers that are less than zero. They
are the opposite of positive numbers. For example, if a positive number indicates a bank deposit, then a negative number
indicates a withdrawal of the same amount. Negative numbers are usually written by writing a negative sign in front of the number
they are the opposite of. Thus the opposite of 7 is written −7. When the set of negative
whole numbers are combined with the positive whole numbers and zero, one obtains the
integers Z (German Zahl, plural Zahlen), also written
UNIQ4a6bbbd91f2115d3-math-7e7ff903648dd1b800000002.
Rational numbers
A rational number is a number that can be expressed as a fraction with an integer numerator and a non-zero natural
number denominator. The fraction m/n or

represents m equal parts, where n equal parts of that size make up one whole. Two different fractions may
correspond to the same rational number; for example 1/2 and 2/4 are equal, that is:
.
If the absolute value of m is greater than n, 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
zero. The set of all fractions includes the integers, since every integer can be written as a fraction with denominator 1. For
example −7 can be written −7/1. The symbol for the rational numbers is Q (for quotient), also written UNIQ4a6bbbd91f2115d3-math-7e7ff903648dd1b800000005.
Real numbers
The real numbers include all of the measuring numbers. Real numbers are usually
written using decimal numerals, in which a decimal point is placed to the right of the digit
with place value one. Following the decimal point, each digit has a place value one-tenth the place value of the digit to its
left. Thus

represents 1 hundred, 2 tens, 3 ones, 4 tenths, 5 hundredths, and 6 thousandths. In saying the number, the decimal is read
"point", thus: "one two three point four five six". In, for example, the US and UK, the decimal is represented by a
period, in continental Europe by a comma. Zero is often written as
0.0 and negative real numbers are written with a preceding minus sign:
.
Every rational number is also a real number. To write a fraction as a decimal, divide the numerator by the denominator. It is
not the case, however, that every real number is rational. If a real number cannot be written as a fraction of two integers, it
is called irrational. A decimal that can be written as a fraction either ends
(terminates) or forever repeats, because it is the answer to a problem in division. Thus the real number 0.5 can be written as
1/2 and the real number 0.333... (forever repeating threes) can be written as 1/3. On the other hand, the real number π
(pi), the ratio of the circumference of any circle to its
diameter, is
.
Since the decimal neither ends nor forever repeats, it cannot be written as a fraction, and is an example of an irrational
number. Other irrational numbers include

(the square root of 2, that is, the positive number whose square is 2).
Just as fractions can be written in more than one way, so too can decimals. For example, if we multiply both sides of the
equation

by three, we discover that
.
Thus 1.0 and 0.999... are two different decimal numerals representing the natural number 1.
There are infinitely many other ways of representing the number 1, for example 2/2, 3/3, 1.00, 1.000, and so on.
Every real number is either rational or irrational. Every real number corresponds to a point on the number line. The real numbers also have an important but highly technical property called the
least upper bound property. The symbol for the real numbers is R or
.
When a real number represents a measurement, there is always a margin of error. This is often indicated by rounding or
truncating a decimal, so that digits that suggest a greater accuracy than the measurement
itself are removed. The remaining digits are called significant digits. For example,
measurements with a ruler can seldom be made without a margin of error of at least 0.01 meters. If the sides of a
rectangle are measured as 1.23 meters and 4.56 meters, then multiplication gives an area for
the rectangle of 5.6088 square meters. Since only the first two digits after the decimal place are significant, this is usually
rounded to 5.61.
In abstract algebra, the real numbers are uniquely characterized by being the only
completely ordered field. They are not, however, an algebraically closed
field.
Complex 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 the question of whether a negative number can have a
square root. This led to the invention of a new number: the square root of negative one,
denoted by i, a symbol assigned by Leonhard Euler, and called the imaginary unit. The complex numbers consist of all numbers of the form

where a and b are real numbers. In the expression a + bi, the real number a is called the
real part and b is called the imaginary part. If the real part of a complex number is zero, then the number
is called an imaginary number or is referred to as purely imaginary; if the
imaginary part is zero, 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 symbol for the complex numbers is C or
.
In abstract algebra, the complex numbers are an example of an algebraically closed field, meaning that every polynomial
with complex coefficients can be factored into
linear factors. Like the real number system, the complex number system is a field
and is complete, but unlike the real numbers it is not ordered. That is, there is no meaning in saying that i is greater than 1, nor is there any meaning in
saying that that i is less than 1. In technical terms, the complex numbers lack the trichotomy property.
Complex numbers correspond to points on the complex plane, sometimes called the Argand
plane.
Each of the number systems mentioned above is a proper subset of the next number system.
Symbolically, N ⊂ Z ⊂ Q ⊂ R ⊂ C.
Other types
Superreal, hyperreal and surreal numbers extend the real numbers by adding infinitesimally small numbers and infinitely large
numbers, but still form fields.
The idea behind p-adic numbers is this: While real numbers may have infinitely long
expansions to the right of the decimal point, these numbers 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.
For dealing with infinite collections, the natural numbers have been generalized to the ordinal numbers and to the cardinal numbers. The former gives
the ordering of the collection, while the latter gives its size. For the finite set, the ordinal and cardinal numbers are
equivalent, but they differ in the infinite case.
There are also other sets of numbers with specialized uses. Some are subsets of the complex numbers. For example,
algebraic numbers are the roots of polynomials with
rational coefficients. Complex numbers that are not algebraic are called transcendental numbers.
Sets of numbers that are not subsets of the complex numbers include the quaternions
H, invented by Sir William Rowan Hamilton, in which multiplication is not
commutative, and the octonions, in which multiplication
is not associative. Elements of function fields of
finite characteristic behave in some ways like numbers and are often regarded
as numbers by number theorists.
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.
History
History of integers
The first numbers
- Further information: History of numeral systems
It is speculated that the first known use of numbers dates back to around 30000 BC, bones or other artifacts have been
discovered with marks cut into them which are often considered tally marks. The use of these
tally marks have been suggested to be anything from counting elapsed time, such as numbers of days, or keeping records of
amounts.
Tallying systems have no concept of place-value (such as in the currently used decimal notation), which limit its
representation of large numbers and as such is often considered that this is the first kind of abstract system that would be
used, and could be considered a Numeral System.
The first known system with place-value was the Mesopotamian base 60 system (ca. 3400
BC) and the earliest known base 10 system dates to 3100 BC in Egypt. [1]
History of zero
- Further information: History of zero
The use of zero as a number should be distinguished from its use as a placeholder numeral in place-value systems. Many ancient Indian texts use a Sanskrit word
Shunya to refer to the concept of void; in mathematics texts this word would often be used to refer to the number
zero. [2]. In a similar vein, Pāṇini (5th century
BC) used the null (zero) operator (ie a lambda production) in the Ashtadhyayi, his algebraic grammar for the Sanskrit language. (also see Pingala)
Records show that the Ancient Greeks seemed unsure about the status of zero as a
number: they asked themselves "how can 'nothing' be something?", leading to interesting philosophical and, by the Medieval period, religious arguments about the nature and existence of zero and the
vacuum. The paradoxes of Zeno of Elea depend in large part on the uncertain interpretation of zero. (The ancient Greeks even
questioned if 1 was a number.)
The late Olmec people of south-central Mexico began to use a
true 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, but did not influence Old World numeral systems.
By 130, Ptolemy, influenced by Hipparchus and the Babylonians, was using a symbol for zero (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 70).
Another true zero was used in tables alongside Roman numerals by 525 (first known use by Dionysius Exiguus), but as a word, nulla
meaning nothing, not as a symbol. When division produced zero 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 725, a true zero symbol.
An early documented use of the zero by Brahmagupta (in the Brahmasphutasiddhanta) dates to 628. He treated zero as a number and
discussed operations involving it, including division. By this time (7th century) the
concept had clearly reached Cambodia, and documentation shows the idea later spreading to
China and the Islamic world.
History of negative numbers
- Further information: First usage of negative numbers
The abstract concept of negative numbers was recognised as early as 100 BC - 50 BC. The Chinese ”Nine Chapters on the Mathematical Art” (Jiu-zhang Suanshu) contains
methods for finding the areas of figures; red rods were used to denote positive coefficients, black for negative. This is the earliest known mention of negative numbers in the East; the first reference in a western work was in the 3rd century in
Greece. Diophantus referred to the equation equivalent to
4x + 20 = 0 (the solution would be negative) in Arithmetica, saying that the equation gave an absurd result.
During the 600s, negative numbers were in use in India to
represent debts. Diophantus’ previous reference was discussed more explicitly by Indian
mathematician Brahmagupta, in Brahma-Sphuta-Siddhanta 628, who used negative numbers to produce the
general form quadratic formula that remains in use today. 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 debits (chapter 13 of Liber
Abaci, 1202) 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
nonzero digit of the corresponding positive number's numeral[citation needed]. The first use of negative numbers in a European work was by
Chuquet during the 15th century. He used them as
exponents, but referred to them as “absurd numbers”.
As recently as the 18th century, the Swiss
mathematician Leonhard Euler believed that negative numbers were greater than
infinity[citation needed], and it was common practice to ignore any negative results returned by
equations on the assumption that they were meaningless, just as René Descartes did with
negative solutions in a cartesian coordinate system.
History of rational, irrational, and real numbers
- Further information: History of irrational numbers and History of pi
History of rational numbers
It is likely that the concept of fractional numbers dates to prehistoric times. Even the
Ancient Egyptians wrote math texts describing how to convert general fractions into their special notation. Classical Greek
and Indian mathematicians made studies of the theory of rational numbers, as part of the general study of number theory. The best
known of these is Euclid's Elements, dating to roughly 300
BC. 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 sutras to include calculations of decimal-fraction
approximations to pi or the square root of two. Similarly,
Babylonian math texts had always used sexagesimal fractions with great frequency.
History of irrational numbers
The earliest known use of irrational numbers was in the Indian Sulba Sutras composed between 800-500
BC. [citation needed] The first existence proofs of
irrational numbers is usually attributed to Pythagoras, more specifically to the
Pythagorean Hippasus of Metapontum, who produced a (most
likely geometrical) proof of the irrationality of the square root of 2. 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 his beliefs would not accept the existence of irrational numbers and so
he sentenced Hippasus to death by drowning.
The sixteenth century saw the final acceptance by Europeans of negative, integral and fractional
numbers. The seventeenth century saw decimal fractions with the modern notation quite generally used by mathematicians. But it
was not until the nineteenth century that the irrationals were separated into algebraic and transcendental parts, and a
scientific study of theory of irrationals was taken once more. It had remained almost dormant since Euclid. The year 1872 saw the publication of the theories of Karl
Weierstrass (by his pupil Kossak), Heine
(Crelle, 74), Georg Cantor (Annalen, 5),
and Richard Dedekind. Méray had taken in 1869 the same
point of departure as Heine, but the theory is generally referred to the year 1872. Weierstrass's
method has been completely set forth by Salvatore Pincherle (1880), and Dedekind's
has received additional prominence through the author's later work (1888) and the recent endorsement by Paul Tannery (1894). 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 subject has received later contributions at the hands of Weierstrass,
Kronecker (Crelle, 101), and Méray.
Continued fractions, closely related to irrational numbers (and due to Cataldi,
1613), received attention at the hands of Euler, and at the opening of the nineteenth
century were brought into prominence through the writings of Joseph Louis
Lagrange. Other noteworthy contributions have been made by Druckenmüller (1837),
Kunze (1857), Lemke (1870), and Günther (1872). Ramus (1855) first connected the subject with determinants, resulting, with the subsequent contributions of Heine, Möbius, and Günther, in the theory of Kettenbruchdeterminanten.
Dirichlet also added to the general theory, as have numerous contributors to the applications of the subject.
Transcendental numbers and reals
The first results concerning transcendental numbers were Lambert's 1761 proof
that π cannot be rational, and also that en is irrational if n is rational (unless n = 0).
(The constant e was first referred to in Napier's 1618 work on logarithms.) Legendre extended this proof to showed that π is not the square root of a rational number. The
search for roots of quintic and higher degree equations was an important development,
the Abel–Ruffini theorem (Ruffini 1799,
Abel 1824) showed that they could not be solved by radicals (formula involving only arithmetical operations and roots). Hence it was necessary to consider the
wider set of algebraic numbers (all solutions to polynomial equations).
Galois (1832) linked polynomial equations to group
theory giving rise to the field of Galois theory.
Even the set of algebraic numbers was not sufficient and the full set of real number includes transcendental numbers. The existence of which was first established by Liouville (1844, 1851).