Number System in Mathematics

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Introduction
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 civilizations. | 11| Undecimal| |
12| Duodecimal| |
13| Tridecimal| The Maya calendar.|
14| Tetradecimal| Programming for the HP 9100A/B calculator and image a processing applications.| 15| Pentadecimal| Telephony routing over IP and the Huli language.| 16| Hexadecimal| Human-friendly representation (hex dump) of binary data and Base16 encoding.| 20| Vigesimal| |

24| Tetravigesimal| |
26| Hexavigesimal| |
27| Septemvigesimal| Telefol and Oksapmin languages.|
30| Trigesimal| |
32| Duotrigesimal| Base32 encoding and the Ngiti language.| 36| Hexatridecimal| Base36 encoding.|
60| Sexagesimal| The Babylonian numerals positional numeral system.| 64| Tetrasexagesimal| Base64 encoding.|
85| | Ascii85 encoding.|
|

b. Non- standard form :
i. Bijective numeration
Base| Name| Usage|
1| Unary| Tally marks.|
10| Decimal without a zero| |
ii. Signed-digit representation
Base| Name| Usage|
2| Non-adjacent form| |
3| Balanced ternary| Ternary computers.|
iii. Negative bases
The common names of the negative base numeral systems are formed using the prefix nega-, giving names such as: Base| Name| Usage|
−2| Negabinary| |
−3| Negaternary| |
−10| Negadecimal| |
iv. Complex bases
Base| Name| Usage|
2i| Quater-imaginary base| |
−1 ± i| Twindragon base| Twindragon fractal shape.|
v. Non-integer bases
Base| Name| Usage|
φ| Golden ratio base| Early Beta encoder. |
e| Base e| |
π| Base π| |
√2| Base √2| |
vi. Other
* Nullary
* Mixed radix

III. Most used Number systems:
In the modern world, we use following number systems out of above two forms of number systems: 1) Binary Number System
2) Decimal Number System
3) Octal Number System
4) Hexadecimal Number System

Explanation of number systems:

I. Binary
a. What is the binary system?
The word “binary”...
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