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 padic numbers, etc.
II. Types of Number systems:
We can defined two types of number systems: standard form & nonstandard 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 ProtoUralic 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 Humanfriendly 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. Signeddigit representation
Base Name Usage
2 Nonadjacent 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 Quaterimaginary base 
−1 ± i Twindragon base Twindragon fractal shape.
v. Noninteger 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”...