Numeral

See also the Wikipedia article:

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 $D$ is a set of $n$ digits, digits in $D$ can be used to represent (in base-$n$) any natural number. The idea behind such representation follows from the following proposition:

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

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

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

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

Common Numeral Systems

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