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Numeral system

A numeral is a symbol that represents a number — for example, 2 or 1729. A collection of such symbols is known as a numeral system or system of numeration (or, less formally, a number system — although the latter term technically has a different meaning)

Please see our list of number names and symbols for a table of numerals and corresponding number names in different languages.

The Natural numbers

Given a set of at least two digits, any natural number can be uniquely represented as a string of digits, with the leftmost digit being non-zero. That is, if ${\displaystyle D}$ is a set of ${\displaystyle n}$ digits, digits in ${\displaystyle D}$ can be used to represent (in base-${\displaystyle n}$) any natural number. The idea behind such representation follows from the following proposition:

Let ${\displaystyle n}$ be a natural number greater than 1. For any non-zero natural number M, there exists a unique natural number ${\displaystyle N}$ (representing the number of digits) and a finite sequence ${\displaystyle d_1,d_2,\dots,d_N}$ (representing the digits themselves) such that:

• ${\displaystyle d_j < n}$ for each ${\displaystyle j}$;
• ${\displaystyle d_N \ne 0}$, so that the leading digit is non-zero;
• ${\displaystyle M = \sum_{j=0}^N d_j n^j}$

Then as ${\displaystyle D}$ has ${\displaystyle n}$ digits, there is a one-to-one pairing between the digits and the set of natural numbers that are less than ${\displaystyle n}$.

Using the equality ${\displaystyle M = \sum_{j=0}^N d_j n^j}$, we can conceptualize the representation as the concatenation: ${\displaystyle D_N D_{N-1} \dots D_2 D_1 D_0}$, where each ${\displaystyle D_j}$ is the digit in ${\displaystyle D}$ representing ${\displaystyle d_j}$.

Common Numeral Systems

• The decimal system, using ${\displaystyle D = \left\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\right\}}$, is default decimal system used in everyday life.
• The binary system, using ${\displaystyle D = \left\{0, 1\right\}}$, used in computer science.
• The hexadecimal system, using ${\displaystyle D = \left\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F\right\}}$, also used in computer science.