What is Binary, Octal and Hexadecimal Number System

Binary and Octal Number System

The base of Binary Number system is 2 because it consists of two different  digits or symbols i.e. 0 and 1. The digits 0 and 1 are called binary digits  commonly abbreviated as bit. 

Thus a “bit” in computer  terminology means either a ‘0’ or a ‘1’.

Each position in a binary number represents a power of the base (2) . 

In this system, the rightmost position is the units (2⁰) position, the second position from the right is the 2’s (2¹) position and then 4’s (2²) position, 8’s (2³) position, 16’s (2⁴) position and so on.

Thus, the decimal equivalent of the binary number 11010 is:

           (1×2⁴) + (1×2³) + (0×2²) + (1×2¹) + (0×2⁰)

or       16+8+0+2+0

or       26

In order to represent any number in different number system, place that number in a bracket and indicates its base as a subscript.

Thus, we can write

(11010)₂ = (26)₁₀

A binary number consisting of n bits is called an n-bit number. Only 8 or (2)³different patterns of 0’s and 1’s are possible with a 3-bit number. 

3-bit Numbers with their Decimal Equivalents

Octal Number System

In the Octal Number system, the base is 8. There are only eight symbols or digits : 0,1,2,3,4,5,6,7

Each position in a octal number represents a power of the base 8. The decimal equivalent of the octal number 4075 is:

        (4×8³) + (0×8²) + (7×8¹) + (5×8⁰)

or     2048+0+56+5

or     2109

so,    (4075)₈ = (2109) ₁₀

Since there are only 8 digits in the octal number system , 3 bits (2³=8) are sufficient to represent any octal number in binary.

Hexadecimal Number System

In the hexadecimal number system, the base is 16. The first 10 digits are the digits of a decimal system 0,1,2,3,4,5,6,7,8,9. 

The remaining six digits are denoted by A,B,C,D,E, F representing the decimal values 10,11,12,13,14,15 respectively. The largest single digit is F or 15. Each position in a hexadecimal system represents  a power of base (16).

The decimal equivalent of the hexadecimal number 2BE is:

          (2×16²) + (B×16¹) + (E×16⁰)

or      (2×256) + (11×16) + (14×1)

or      522+176+14

or       712

Thus, (2BE)₁₆ = (712)₁₀

_{4-bits can represent any hexadecimal number in binary:}