Number A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers c ...

s can be classified according to how they are represented or according to the properties that they have.

Main types

Natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''cardinal ...

s (

\mathbb

): The counting numbers are commonly called natural numbers; however, other definitions include 0, so that the non-negative integers are also called natural numbers. Natural numbers including 0 are also called ''whole numbers''. *

Integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...

s (

\mathbb

): Positive and negative counting numbers, as well as zero: . *

Rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all rat ...

s (

\mathbb

): Numbers that can be expressed as a

ratio In mathematics, a ratio shows how many times one number contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio of oranges to lemons is eight to six (that is, 8:6, which is equivalent to the ...

of an integer to a non-zero integer. All integers are rational, but there are rational numbers that are not integers, such as . *

Real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...

s (

\mathbb

): Numbers that correspond to points along a line. They can be positive, negative, or zero. All rational numbers are real, but the converse is not true. *

Irrational number In mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two inte ...

s: Real numbers that are not rational. * Imaginary numbers: Numbers that equal the product of a real number and the square root of −1. The number 0 is both real and purely imaginary. *

Complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...

s (

\mathbb

): Includes real numbers, imaginary numbers, and sums and differences of real and imaginary numbers. *

Hypercomplex number In mathematics, hypercomplex number is a traditional term for an element of a finite-dimensional unital algebra over the field of real numbers. The study of hypercomplex numbers in the late 19th century forms the basis of modern group represen ...

s include various number-system extensions: quaternions (

\mathbb

octonion In mathematics, the octonions are a normed division algebra over the real numbers, a kind of hypercomplex number system. The octonions are usually represented by the capital letter O, using boldface or blackboard bold \mathbb O. Octonions hav ...

s (

\mathbb

), and other less common variants.

Sedenion In abstract algebra, the sedenions form a 16-dimensional noncommutative and nonassociative algebra over the real numbers; they are obtained by applying the Cayley–Dickson construction to the octonions, and as such the octonions are isomorphic to ...

s (

\mathbb

), trigintaduonions (

\mathbb

tessarine In abstract algebra, a bicomplex number is a pair of complex numbers constructed by the Cayley–Dickson process that defines the bicomplex conjugate (w,z)^* = (w, -z), and the product of two bicomplex numbers as :(u,v)(w,z) = (u w - v z, u z ...

coquaternion In abstract algebra, the split-quaternions or coquaternions form an algebraic structure introduced by James Cockle in 1849 under the latter name. They form an associative algebra of dimension four over the real numbers. After introduction in ...

s, and

biquaternion In abstract algebra, the biquaternions are the numbers , where , and are complex numbers, or variants thereof, and the elements of multiply as in the quaternion group and commute with their coefficients. There are three types of biquaternions co ...

s. * -adic numbers: Various number systems constructed using limits of rational numbers, according to notions of "limit" different from the one used to construct the real numbers.

Number representations

* Decimal: The standard

Hindu–Arabic numeral system The Hindu–Arabic numeral system or Indo-Arabic numeral system Audun HolmeGeometry: Our Cultural Heritage 2000 (also called the Hindu numeral system or Arabic numeral system) is a positional decimal numeral system, and is the most common syste ...

using base ten. *

Binary Binary may refer to: Science and technology Mathematics * Binary number, a representation of numbers using only two digits (0 and 1) * Binary function, a function that takes two arguments * Binary operation, a mathematical operation that ta ...

: The base-two

numeral system A numeral system (or system of numeration) is a writing system for expressing numbers; that is, a mathematical notation for representing numbers of a given set, using digits or other symbols in a consistent manner. The same sequence of symbo ...

used by computers, with digits 0 and 1. * Ternary: The base-three numeral system with 0, 1, and 2 as digits. * Quaternary: The base-four numeral system with 0, 1, 2, and 3 as digits. * Hexadecimal: Base 16, widely used by computer system designers and programmers, as it provides a more human-friendly representation of binary-coded values. *

Octal The octal numeral system, or oct for short, is the base-8 number system, and uses the digits 0 to 7. This is to say that 10octal represents eight and 100octal represents sixty-four. However, English, like most languages, uses a base-10 number ...

: Base 8, occasionally used by computer system designers and programmers. * Duodecimal: Base 12, a numeral system that is convenient because of the many factors of 12. *

Sexagesimal Sexagesimal, also known as base 60 or sexagenary, is a numeral system with sixty as its base. It originated with the ancient Sumerians in the 3rd millennium BC, was passed down to the ancient Babylonians, and is still used—in a modified form ...

: Base 60, first used by the ancient Sumerians in the 3rd millennium BC, was passed down to the ancient Babylonians. * See positional notation for information on other bases. * Roman numerals: The numeral system of

ancient Rome In modern historiography, ancient Rome refers to Roman civilisation from the founding of the city of Rome in the 8th century BC to the collapse of the Western Roman Empire in the 5th century AD. It encompasses the Roman Kingdom (753–509 BC ...

, still occasionally used today, mostly in situations that do not require arithmetic operations. *

Tally marks Tally marks, also called hash marks, are a unary numeral system ( arguably). They are a form of numeral used for counting. They are most useful in counting or tallying ongoing results, such as the score in a game or sport, as no intermediate ...

: Usually used for counting things that increase by small amounts and do not change very quickly. * Fractions: A representation of a non-integer as a

of two integers. These include

improper fraction A fraction (from la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight ...

s as well as

mixed number A fraction (from la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight ...

s. *

Continued fraction In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer ...

: An expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on. *

Scientific notation Scientific notation is a way of expressing numbers that are too large or too small (usually would result in a long string of digits) to be conveniently written in decimal form. It may be referred to as scientific form or standard index form, o ...

: A method for writing very small and very

large numbers Large numbers are numbers significantly larger than those typically used in everyday life (for instance in simple counting or in monetary transactions), appearing frequently in fields such as mathematics, cosmology, cryptography, and statistical m ...

using powers of 10. When used in science, such a number also conveys the

precision Precision, precise or precisely may refer to: Science, and technology, and mathematics Mathematics and computing (general) * Accuracy and precision, measurement deviation from true value and its scatter * Significant figures, the number of digit ...

of measurement using

significant figures Significant figures (also known as the significant digits, ''precision'' or ''resolution'') of a number in positional notation are digits in the number that are reliable and necessary to indicate the quantity of something. If a number expres ...

. *

Knuth's up-arrow notation In mathematics, Knuth's up-arrow notation is a method of notation for very large integers, introduced by Donald Knuth in 1976. In his 1947 paper, R. L. Goodstein introduced the specific sequence of operations that are now called ''hyperoperati ...

and

Conway chained arrow notation Conway chained arrow notation, created by mathematician John Horton Conway, is a means of expressing certain extremely large numbers. It is simply a finite sequence of positive integers separated by rightward arrows, e.g. 2\to3\to4\to5\to6. As wit ...

: Notations that allow the concise representation of some extremely large integers such as Graham's number.

Signed numbers

* Positive numbers: Real numbers that are

greater than In mathematics, an inequality is a relation which makes a non-equal comparison between two numbers or other mathematical expressions. It is used most often to compare two numbers on the number line by their size. There are several different n ...

zero. * Negative numbers: Real numbers that are less than zero. Because zero itself has no sign, neither the positive numbers nor the negative numbers include zero. When zero is a possibility, the following terms are often used: * Non-negative numbers: Real numbers that are greater than or equal to zero. Thus a non-negative number is either zero or positive. * Non-positive numbers: Real numbers that are less than or equal to zero. Thus a non-positive number is either zero or negative.

Types of integer

* Even and odd numbers: An integer is even if it is a multiple of 2, and is odd otherwise. *

Prime number 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 ...

: A positive integer with exactly two positive

divisor 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 or evenly divisible by ...

s: itself and 1. The primes form an infinite sequence 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, ... *

Composite number A composite number is a positive integer that can be formed by multiplying two smaller positive integers. Equivalently, it is a positive integer that has at least one divisor other than 1 and itself. Every positive integer is composite, prime, ...

: A positive integer that can be factored into a product of smaller positive integers. Every integer greater than one is either prime or composite. *

Polygonal number In mathematics, a polygonal number is a number represented as dots or pebbles arranged in the shape of a regular polygon. The dots are thought of as alphas (units). These are one type of 2-dimensional figurate numbers. Definition and examples T ...

s: These are numbers that can be represented as dots that are arranged in the shape of a

regular polygon In Euclidean geometry, a regular polygon is a polygon that is direct equiangular (all angles are equal in measure) and equilateral (all sides have the same length). Regular polygons may be either convex, star or skew. In the limit, a sequence ...

, including

Triangular number A triangular number or triangle number counts objects arranged in an equilateral triangle. Triangular numbers are a type of figurate number, other examples being square numbers and cube numbers. The th triangular number is the number of dots i ...

Square number In mathematics, a square number or perfect square is an integer that is the square of an integer; in other words, it is the product of some integer with itself. For example, 9 is a square number, since it equals and can be written as . The usu ...

Pentagonal number A pentagonal number is a figurate number that extends the concept of triangular and square numbers to the pentagon, but, unlike the first two, the patterns involved in the construction of pentagonal numbers are not rotationally symmetrical. The ...

Hexagonal number A hexagonal number is a figurate number. The ''n''th hexagonal number ''h'n'' is the number of ''distinct'' dots in a pattern of dots consisting of the ''outlines'' of regular hexagons with sides up to n dots, when the hexagons are overlaid so ...

Heptagonal number A heptagonal number is a figurate number that is constructed by combining heptagons with ascending size. The ''n''-th heptagonal number is given by the formula :H_n=\frac. The first few heptagonal numbers are: : 0, 1, 7, 18, 34, 55, 81, 112 ...

Octagonal number An octagonal number is a figurate number that represents an octagon. The octagonal number for ''n'' is given by the formula 3''n''2 - 2''n'', with ''n'' > 0. The first few octagonal numbers are : 1, 8, 21, 40, 65, 96, 133, 176, 225, 280, 34 ...

Nonagonal number A nonagonal number (or an enneagonal number) is a figurate number that extends the concept of triangular and square numbers to the nonagon (a nine-sided polygon). However, unlike the triangular and square numbers, the patterns involved in the constr ...

Decagonal number A decagonal number is a figurate number that extends the concept of triangular and square numbers to the decagon (a ten-sided polygon). However, unlike the triangular and square numbers, the patterns involved in the construction of decagonal number ...

s, Hendecagonal numbers, and

Dodecagonal number A dodecagonal number is a figurate number that represents a dodecagon. The dodecagonal number for ''n'' is given by the formula :D_=5n^2 - 4n The first few dodecagonal numbers are: : 0, 1, 12, 33, 64, 105, 156, 217, 288, 369, 460, 561, 672, 7 ...

s. * There are many other famous

integer sequence In mathematics, an integer sequence is a sequence (i.e., an ordered list) of integers. An integer sequence may be specified ''explicitly'' by giving a formula for its ''n''th term, or ''implicitly'' by giving a relationship between its terms. For ...

s, such as the sequence of

Fibonacci number In mathematics, the Fibonacci numbers, commonly denoted , form a sequence, the Fibonacci sequence, in which each number is the sum of the two preceding ones. The sequence commonly starts from 0 and 1, although some authors start the sequence from ...

s, the sequence of factorials, the sequence of

perfect number In number theory, a perfect number is a positive integer that is equal to the sum of its positive divisors, excluding the number itself. For instance, 6 has divisors 1, 2 and 3 (excluding itself), and 1 + 2 + 3 = 6, so 6 is a perfect number. ...

s, and so forth, many of which are enumerated in the

On-Line Encyclopedia of Integer Sequences The On-Line Encyclopedia of Integer Sequences (OEIS) is an online database of integer sequences. It was created and maintained by Neil Sloane while researching at AT&T Labs. He transferred the intellectual property and hosting of the OEIS to t ...

Algebraic numbers

* Algebraic number: Any number that is the

root In vascular plants, the roots are the organs of a plant that are modified to provide anchorage for the plant and take in water and nutrients into the plant body, which allows plants to grow taller and faster. They are most often below the su ...

of a non-zero

polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example ...

with rational coefficients. *

Transcendental number In mathematics, a transcendental number is a number that is not algebraic—that is, not the root of a non-zero polynomial of finite degree with rational coefficients. The best known transcendental numbers are and . Though only a few classes ...

: Any real or complex number that is not algebraic. Examples include and . *

Trigonometric number In mathematics, the values of the trigonometric functions can be expressed approximately, as in \cos (\pi/4) \approx 0.707, or exactly, as in \cos (\pi/ 4)= \sqrt 2 /2. While trigonometric tables contain many approximate values, the exact values ...

: Any number that is the sine or cosine of a rational multiple of . * Quadratic surd: A root of a

quadratic equation In algebra, a quadratic equation () is any equation that can be rearranged in standard form as ax^2 + bx + c = 0\,, where represents an unknown value, and , , and represent known numbers, where . (If and then the equation is linear, not q ...

with rational coefficients. Such a number is algebraic and can be expressed as the sum of a rational number and the

square root In mathematics, a square root of a number is a number such that ; in other words, a number whose ''square'' (the result of multiplying the number by itself, or  ⋅ ) is . For example, 4 and −4 are square roots of 16, because . ...

of a rational number. *

Constructible number In geometry and algebra, a real number r is constructible if and only if, given a line segment of unit length, a line segment of length , r, can be constructed with compass and straightedge in a finite number of steps. Equivalently, r is cons ...

: A number representing a length that can be constructed using a compass and straightedge. Constructible numbers form a subfield of the field of algebraic numbers, and include the quadratic surds. *

Algebraic integer In algebraic number theory, an algebraic integer is a complex number which is integral over the integers. That is, an algebraic integer is a complex root of some monic polynomial (a polynomial whose leading coefficient is 1) whose coefficients ...

: A root of a

monic polynomial In algebra, a monic polynomial is a single-variable polynomial (that is, a univariate polynomial) in which the leading coefficient (the nonzero coefficient of highest degree) is equal to 1. Therefore, a monic polynomial has the form: :x^n+c_x^+\c ...

with integer coefficients.

Non-standard numbers

Transfinite number In mathematics, transfinite numbers are numbers that are " infinite" in the sense that they are larger than all finite numbers, yet not necessarily absolutely infinite. These include the transfinite cardinals, which are cardinal numbers used to q ...

s: Numbers that are greater than any natural number. * Ordinal numbers: Finite and infinite numbers used to describe the

order type In mathematics, especially in set theory, two ordered sets and are said to have the same order type if they are order isomorphic, that is, if there exists a bijection (each element pairs with exactly one in the other set) f\colon X \to Y such ...

well-ordered set In mathematics, a well-order (or well-ordering or well-order relation) on a set ''S'' is a total order on ''S'' with the property that every non-empty subset of ''S'' has a least element in this ordering. The set ''S'' together with the well-ord ...

s. *

Cardinal number In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. T ...

s: Finite and infinite numbers used to describe the

cardinalities In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...

set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...

s. * Infinitesimals: These are smaller than any positive real number, but are nonetheless greater than zero. These were used in the initial development of

calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...

, and are used in synthetic differential geometry. *

Hyperreal number In mathematics, the system of hyperreal numbers is a way of treating infinite and infinitesimal (infinitely small but non-zero) quantities. The hyperreals, or nonstandard reals, *R, are an extension of the real numbers R that contains numbers ...

s: The numbers used in

non-standard analysis The history of calculus is fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal numbers. The standard way to resolve these debates is to define the operations of calculus using epsilon–delta ...

. These include infinite and infinitesimal numbers which possess certain properties of the real numbers. *

Surreal number In mathematics, the surreal number system is a totally ordered proper class containing the real numbers as well as infinite and infinitesimal numbers, respectively larger or smaller in absolute value than any positive real number. The surreals ...

s: A number system that includes the hyperreal numbers as well as the ordinals.

Computability and definability

Computable number In mathematics, computable numbers are the real numbers that can be computed to within any desired precision by a finite, terminating algorithm. They are also known as the recursive numbers, effective numbers or the computable reals or recursive ...

: A real number whose digits can be computed by some

algorithm In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing ...

. *

Definable number Informally, a definable real number is a real number that can be uniquely specified by its description. The description may be expressed as a construction or as a formula of a formal language. For example, the positive square root of 2, \sqrt, ca ...

: A real number that can be defined uniquely using a first-order formula with one

free variable In mathematics, and in other disciplines involving formal languages, including mathematical logic and computer science, a free variable is a notation (symbol) that specifies places in an expression where substitution may take place and is not ...

in the language of

set theory Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly conce ...

References

{{DEFAULTSORT:Types of numbers Mathematics-related lists

Types Type may refer to: Science and technology Computing * Typing, producing text via a keyboard, typewriter, etc. * Data type In computer science and computer programming, a data type (or simply type) is a set of possible values and a set of allo ...