The number system is the system of counting and calculations. Number system is formed of characters called “Digits”. Each number is made up of these characters. The number of digits a system uses is called its base or Radix. For example the number system we use in our habitual life is called Decimal System.

There are four branches of the number system.

1) Decimal 2) Binary 3) Octal 4) Hexadecimal

**Decimal system:**

The number system which we use in our habitual life is based on ten digits. **0, 1, 2, 3, 4, 5, 6, 7, 8,** and **9. **Any numbers can be expressed as a combination of these ten digits. For example, 25 is the combination of two digits 2 and 5, 125 is the combination of three digits and so on. In this system, every digit of a number is a multiple of some power of 10.

Any decimal integer can be expressed as the sum of each of its digits times power of 10.

Since this number system has ten digits, this system is also called “Decimal Number System” Or “ Denary System” or “ Base ten system”.

For example, 8326 can be expressed as:

= 8 x 10^{3} + 2 x 10^{2} + 3 x 10^{1} + 6 x 10^{0}

= 8 x 1000 + 2 x 100 + 3 x 10 + 6 x 1

= 8000 + 200 + 30 + 6

**Position Weight**

1st 10^{0}= 1

2nd 10^{1}= 10

3rd 10^{2}= 100

4th 10^{3}= 1000

5th 10^{4} =10000

**Binary System:**

The binary number system is ideal for internal networking of electronic computers.

The Binary System is especially useful in the construction of computers because the digits can be represented by switches in “ON” or “OFF” positions.Computer circuitry represents data in a pattern of ON or OFF state of electric current. Because there are only two states, they are represented by 1 (for ON) and 0( for OFF). Binary system is based on two fundamental digits 0 and 1 to represent its numeric values there the base of this system is two.So, it is also called “Base two number system”.

The positional weight for the binary system are based on the powers of two.To visualized the positional values of the binary system the weights for the first five positions are given as under:

1st 2^{0}= 1

2nd 2^{1}= 2

3rd 2^{2}= 4

4th 2^{3}= 8

5th 2^{4} =16

Here, we have an example 100112 can be represented in decimal as:

=1 x 2^{4} + 0 x 2^{3} + 0 x 2^{2} + 1 x 2^{1} + 1 x 2^{0}

= 1 x 16 + 0 x 8 + 0 * 4 + 1 x 2 + 1 x 1

= 16 + 0 + 0 + 2 + 1

= 19

**The octal system:**

Octal is latin word for eight (8). Octal has eight fundamental digits as,** 0, 1, 2, 3, 4, 5, 6**, and **7**. So, the base of this system is eight. The weights of digits position are successive powers of eight as under :

**Position Weight**

1st 8^{0}= 1

2nd 8^{1}= 8

3rd 8^{2}= 64

4th 8^{3}= 512

5th 8^{4} = 4096

Here, we have an example 352_{8} can be expressed in decimal as:

= 3 x 8 ^{2} + 5 x 8 ^{1} + 2 x 8 ^{0}

= 3 x 64 + 5 x 8 + 2 x 1

= 192 +40 + 2

= 234 _{10}

**Hexadecimal system:**

Hexadecimal means 16, therefore this is a base -16 number system.It has 16 basic digits. The first ten digits are same as decimal (0 to 9) and the rest of the six digits are the first six letters of english alphabet (a, b, c, d, e, f).The letters A through F represent the decimal numbers 10 through 15.This system is often used in programming as a shortcut to the binary number system. The weight used in the hexadecimal system are the successive power of 16.

The weight of the first five digits positions are as under.

**Position Weight**

1st 16^{0} = 1

2nd 16^{1} = 16

3rd 16^{2} = 256

4th 16^{3} = 4096

5th 16^{4} = 65536

Here, we have an example 3AC8 can be expressed in decimal as:

= 3 x 16 ^{3} + A x 16 ^{2} + C x 16 ^{1} + 8 x 16 ^{0}

= 3 x 4096 + 10 x 256 + 12 x 16 + 8 x 1

= 12288 + 2560 + 192 + 8

= 15048 _{10}