# 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 (${\displaystyle \mathbb {N} }$): The counting numbers {1, 2, 3, ...} are commonly called natural numbers; however, other definitions include 0, so that the non-negative integers {0, 1, 2, 3, ...} are also called natural numbers. Natural numbers including 0 are also called whole numbers.[1][2]
• Integers (${\displaystyle \mathbb {Z} }$): Positive and negative counting numbers, as well as zero: {..., −3, −2, −1, 0, 1, 2, 3, ...}.
• Rational numbers (${\displaystyle \mathbb {Q} }$): Numbers that can be expressed as a ratio of an integer to a non-zero integer.[3] All integers are rational, but there are rational numbers that are not integers, such as −2/9.
• Real numbers (${\displaystyle \mathbb {R} }$): 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 numbers: 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 numbers (${\displaystyle \mathbb {C} }$): Includes real numbers, imaginary numbers, and sums and differences of real and imaginary numbers.
• Hypercomplex numbers include various number-system extensions: quaternions (${\displaystyle \mathbb {H} }$), octonions (${\displaystyle \mathbb {O} }$), and other less common variants.[4]
• p-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.

## 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.

## Computability and definability

4. ^ Sedenions (${\displaystyle \mathbb {S} }$), trigintaduonions (${\displaystyle \mathbb {T} }$), tessarines, coquaternions, and biquaternions.