Number System is a method of representing any particular quantity through arithmetic values. Number Systems are mathematical notations of sets of digits or symbols. Every Number System can be uniquely identified with the help of a base (or radix).
Base or Radix
Radix comes from a Latin word called “root”. The Base or Radix of a number system is a method of distinguishing the total number of different symbols or number present in a particular number system. A base - 10 number, also called Decimal Number, means there are 10 unique digits in Decimal Numbers. Those number range from 0 to base - 1. I.e, 0 1 2 3 4 5 6 7 8 9.
Likewise, a Number System with X base means it has X unique digits which lie in the range [0 , X - 1].
Positional Number System
The Positional Number System is a method of expressing numbers in which digits are arranged in a certain succession. It is also called Weighted Number System. Here, digits are used to represent any value in any place. The position of each digit has a place value. The sum of the product of each digit with its place value gives the number.
Let us take the number 132.
| Digit | Place Value |
|---|---|
| 1 | 100 |
| 3 | 10 |
| 2 | 1 |
Table: Digit and Place Value of Positional Number System
The number 132 can then be represented as:
132 = sum of (Digit x Place Value)
132 = 1 x 100 + 3 x 10 + 2 x 1
The four most common types of positional number systems are:
- Decimal Number System: Base - 10
- Binary Number System: Base - 2
- Octal Number System: Base - 8
- Hexadecimal Number System: Base - 16
Non-Positional Number System
The Non-Positional Number System is a method of expressing numbers in which symbols are arranged in a certain order. It is also called the Non-Weighted Number System. Here, symbols are combined to represent a number. This system is not much in practice since it is difficult to read. Large arithmetic calculations are not possible in this system. The sum of the values in Decimal gives the number.
Some examples of a Non-Positional Number System are gray code, Roman code, and excess-3 code.
In Roman code, the number 132 can be represented by CXXXII where,
| Symbol | Decimal (Base 10) Value |
|---|---|
| C | 100 |
| X | 10 |
| I | 1 |
Table: Symbol and Values of Roman Code
So it can be written as:
CXXXII = 100 + 10 + 10 + 10 + 1 + 1
CXXXII = 132
Some Popular Number System(s)
- Binary Number System
- Decimal Number System
- Octal Number System
- Hexademical Number System