List of types of numbers

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An Euler diagram showing some types of real numbers

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

Main types[edit]

Natural numbers (\scriptstyle\mathbb{N}
The counting numbers {1, 2, 3, ...}, are called natural numbers. They include all the counting numbers i.e. from 1 to infinity.
Whole numbers 
They are the natural numbers including zero. Not all whole numbers are natural numbers, but all natural numbers are whole numbers.
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]
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.
Irrational numbers (\scriptstyle\mathbb{I})
A real number that is not rational is called irrational.
Complex numbers (\scriptstyle\mathbb{C})
Includes real numbers and imaginary numbers, such as the square root of negative one.
Hypercomplex numbers 
Includes various number-system extensions: quaternions (\scriptstyle\mathbb{H}), octonions (\scriptstyle\mathbb{O}), tessarines, coquaternions, and biquaternions.

Number representations[edit]

The standard Hindu–Arabic numeral system using base ten.
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.
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[edit]

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[edit]

Even and odd numbers 
A number is even if it is a multiple of two, and is odd otherwise.
Prime number 
A number with exactly two positive divisors.
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[edit]

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

Algebraic numbers[edit]

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.
Quadratic surd 
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[edit]

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.
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[edit]

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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