Number System in Mathematics

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I. Number Systems in Mathematics:
A Number system (or system of numeration) is a writing system for expressing numbers, that is a mathematical notation for representing number of a given set, using graphemes or symbols in a consistent manner. It can be seen as the context that allows the symbols "11" to be interpreted as the binary symbol for three, the decimal symbol for eleven, or a symbol for other numbers in different bases. Ideally, a number system will:

* Represent a useful set of numbers (e.g. all integers, or rational numbers) * Give every number represented a unique representation (or at least a standard representation) * Reflect the algebraic and arithmetic structure of the numbers. For example, the usual decimal representation of whole numbers gives every whole number a unique representation as a finite sequence of digits. However, when decimal representation is used for the rational or real numbers, such numbers in general have an infinite number of representations, for example 2.31 can also be written as 2.310, 2.3100000, 2.309999999... etc., all of which have the same meaning except for some scientific and other contexts where greater precision is implied by a larger number of figures shown. Number systems are known as numeral systems, but that name is ambiguous, as it could refer to different systems of numbers, such as the system of real numbers, the system of complex numbers, the system of p-adic numbers, etc.

II. Types of Number systems:
We can defined two types of number systems: standard form & non-standard form. a. Standard form of number systems:
Base| Name| Usage|
2| Binary| All modern digital computations.|
3| Ternary| |
4| Quaternary| Data transmission and Hilbert curves.|
5| Quinary| |
6| Senary| Diceware and the Ndom and Proto-Uralic languages.| 7| Septenary| |
8| Octal| Charles XII of Sweden.|
9| Nonary| |
10| Decimal| Most widely used by modern...
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