# List of types of numbers

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

## Main types

Natural numbers ($\scriptstyle\mathbb{N}$)
The counting numbers {1, 2, 3, ...}, are called natural numbers. Some authors include 0, so that the natural numbers are {0, 1, 2, 3, ...}.
Whole numbers
The numbers {0, 1, 2, 3, ...}.
Integers ($\scriptstyle\mathbb{Z}$)
Positive and negative counting numbers, as well as zero:{..., -2, -1, 0, 1, 2,...}.
Rational numbers ($\scriptstyle\mathbb{Q}$)
Numbers that can be expressed as a fraction of an integer and a non-zero integer.[1] All integers are rational, but the converse is not true.
Real numbers ($\scriptstyle\mathbb{R}$)
All numbers that can be expressed as the limit of a sequence of rational numbers. Every real number corresponds to a point on the number line. All rational numbers are real, but the converse is not true.
Irrational numbers ($\scriptstyle\mathbb{I}$)
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 imaginary.
Complex numbers ($\scriptstyle\mathbb{C}$)
Includes real numbers, imaginary numbers, and sums and differences of real and imaginary numbers.
Hypercomplex numbers include various number-system extensions
quaternions ($\scriptstyle\mathbb{H}$), octonions ($\scriptstyle\mathbb{O}$), tessarines, coquaternions, and biquaternions.
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 using base ten.
Binary
The base-two numeral system used by computers. See positional notation for information on other bases.
Roman numerals
The numeral system of ancient Rome, still occasionally used today.
Fractions
A representation of a non-integer as a ratio of two integers. These include improper fractions as well as mixed numbers.
Scientific notation
A method for writing very small and very large numbers using powers of 10. When used in science, such a number also conveys the precision of measurement using significant figures.
Knuth's up-arrow notation and Conway chained arrow notation
Notations that allow the concise representation of extremely large integers such as Graham's number.

## Signed numbers

Positive numbers
Real numbers that are greater than 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 integers

Even and odd numbers
An integer is even if it is a multiple of two, and is odd otherwise.
Prime number
An integer with exactly two positive divisors: itself and 1. The primes form an infinite sequence 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, ....
Composite number
A number that can be factored into a product of smaller integers. Every integer greater than one is either prime or composite.
Square number
A number that can be written as the square of an integer.

There are many other famous integer sequences, such as the sequence of Fibonacci numbers, the sequence of factorials, the sequence of perfect numbers, and so forth.

### Polygonal numbers

These are numbers that can be represented as dots that are arranged in the shape of a regular polygon.

## Algebraic numbers

Algebraic number
Any number that is the root of a non-zero polynomial with rational coefficients.
Transcendental number
Any real or complex number that is not algebraic. Examples include e and π.
An algebraic number that is the root of a quadratic equation. Such a number can be expressed as the sum of a rational number and the square root of a rational.
Constructible number
A number representing a length that can be constructed using a compass and straightedge. These are a subset of the algebraic numbers, and include the quadratic surds.
Algebraic integer
An algebraic number that is the root of a monic polynomial with integer coefficients.

## Non-standard numbers

Transfinite numbers
Numbers that are greater than any natural number.
Ordinal numbers
Infinite numbers used to describe the order types of well-ordered sets. These include the cardinal numbers, which are used to describe the cardinalities of sets.
Infinitesimals
Nilpotent numbers. These are smaller than any positive real number, but are nonetheless greater than zero. These were used in the initial development of calculus, and are used in synthetic differential geometry.
Hyperreal numbers
The numbers used in non-standard analysis. These include infinite and infinitesimal numbers which possess certain properties of the real numbers.
Surreal numbers
A number system that includes the hyperreal numbers as well as the ordinals. The surreal numbers are the largest possible ordered field.

## Computability and definability

Computable number
A real number whose digits can be computed using an algorithm.
Definable number
A real number that can be defined uniquely using a first-order formula with one free variable in the language of set theory.

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