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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.
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