A number is a mathematical object used to count, measure and label. The original examples are the natural numbers 1, 2, 3, and so forth.

Numerals

Numbers should be distinguished from numerals, the symbols used to represent numbers.

Main classification

Main number systems

${\displaystyle \mathbb {N} }$	Natural	0, 1, 2, 3, 4, ... or 1, 2, 3, 4, ...
${\displaystyle \mathbb {Z} }$	Integer	..., −5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5, ...
${\displaystyle \mathbb {Q} }$	Rational	⁠a/b⁠ where a and b are integers and b is not 0
${\displaystyle \mathbb {R} }$	Real	The limit of a convergent sequence of rational numbers
${\displaystyle \mathbb {C} }$	Complex	a + bi where a and b are real numbers and i is the square root of −1

Rational numbers

A rational number is a number that can be expressed as a fraction with an integer numerator and a positive integer denominator. Negative denominators are allowed, but are commonly avoided, as every rational number is equal to a fraction with positive denominator. Fractions are written as two integers, the numerator and the denominator, with a dividing bar between them. The fraction ⁠m/n⁠ represents m parts of a whole divided into n equal parts. Two different fractions may correspond to the same rational number; for example ⁠1/2⁠ and ⁠2/4⁠ are equal, that is:

${\displaystyle {1 \over 2}={2 \over 4}.\,}$

If the absolute value of m is greater than n (supposed to be positive), 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 0. The set of all rational numbers 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 ${\displaystyle \mathbb {Q} }$.

Real numbers

The real numbers include all the measuring numbers. The symbol for the real numbers is R, also written as ${\displaystyle \mathbb {R} }$.

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 trying to find closed formulas for the roots of cubic and quartic polynomials. This led to expressions involving the square roots of negative numbers, and eventually to the definition of a new number: a square root of −1, denoted by i, a symbol assigned by Leonhard Euler, and called the imaginary unit. The complex numbers consist of all numbers of the form

${\displaystyle \,a+bi}$

where a and b are real numbers. Because of this, complex numbers correspond to points on the complex plane, a vector space of two real dimensions. 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 0, then the number is called an imaginary number or is referred to as purely imaginary; if the imaginary part is 0, 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 ${\displaystyle \mathbb {C} }$.

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 i is less than 1. In technical terms, the complex numbers lack the trichotomy property.

Each of the number systems mentioned above is a proper subset of the next number system. Symbolically, ${\displaystyle \mathbb {N} \subset \mathbb {Z} \subset \mathbb {Q} \subset \mathbb {R} \subset \mathbb {C} }$.

Prime numbers

A prime number is an integer greater than 1 that is not the product of two smaller positive integers. The prime numbers have been widely studied for more than 2000 years and have led to many questions, only some of which have been answered. The study of these questions is called number theory. An example of a question that is still unanswered is whether every even number is the sum of two primes. This is called Goldbach's conjecture.

A question that has been answered is whether every integer greater than one is a product of primes in only one way, except for a rearrangement of the primes. This is called fundamental theorem of arithmetic. A proof appears in Euclid's Elements.

Prime numbers

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.

Main types

Natural numbers (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 (ℤ): Positive and negative counting numbers, as well as zero:{…, -2, -1, 0, 1, 2,…}.

Rational numbers (ℚ): Numbers that can be expressed as a ratio of an integer to a non-zero integer. All integers are rational, but the converse is not true.

Real numbers (ℝ): Numbers that have decimal representations that have a finite or infinite sequence of digits to the right of the decimal point. All rational numbers are real, but the converse is not true.

Irrational numbers (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 (ℂ): Includes real numbers, imaginary numbers, and sums and differences of real and imaginary numbers.