Numbers can be classified according to how they are represented or according to the properties that they have. Main types Natural numbers (): 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 sometimes called whole numbers. Integers (): Positive and negative counting numbers, as well as zero: {..., −3, −2, −1, 0, 1, 2, 3, ...}. Rational numbers (): Numbers that can be expressed as a ratio of an integer to a non-zero integer. All integers are rational, but there are rational numbers that are not integers, such as . Real numbers (): 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 (): Includes real numbers, imaginary numbers, and sums and differences of real and imaginary numbers. Hypercomplex numbers include various number-system extensions: quaternions (), octonions (), and other less common variants. -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 using base ten. Binary: The base-two numeral system 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: 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: 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, still occasionally used today, mostly in situations that do not require arithmetic operations. Tally marks: Usually used for counting things that increase by small amounts and