# Number system (Decimal, Binary, Octal, Hexadecimal)

**Number System:**

A number system is a system in which different notations are used for representing numbers of given set. Depending upon the number of distinct notations used, there are are Decimal (base 10), Binary (base 2), Octal (base 8) and Hexadecimal (base 16) number system.

**Decimal** (base 10): let start with the familiar number system “Decimal”. It is a base 10 number system in which we use 10 distinct notations starting with 0. The notations used in Decimal are 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. If more notations required then we increase the number of digits with carryover. For 1 more than 9 is represented by 10 which means 2 digits are used; 1 is carry and 0 is starting of the count.

1+1=2, 8+1=9, 9+1=10, 10+1=11, 19+1= 20.

Counting Sequence:

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21…………….

**Binary** (base 2): Two distinct notions (0, 1) are used. So, if these two notation are used and need to represent more, we make a carry. For 1 more than 1 is represent by 10 which means 2 digits are used; 1 is carry and 0 is starting of the count on first digit.

0+1=1, 1+1=10, 10+1=11, 11+1=100, 100+1=101

Counting Sequence:

0, 1, 10, 11, 100, 101, 110, 111, 1000, 1001, 1010, 1011, 1100, 1101, 1110, 111…………

Only two notations (0, 1) need to be used to represent number.

**Octal **(base 8) : Octal is a base 8 number system in which we use 8 distinct notations. The notations used in Octal are 0, 1, 2, 3, 4, 5, 6 and 7. If 1 is added to 7 then the representation of number is 10 where first number 1 is carry and 0 is initial start of count on first digit.

1+1= 2, 5+1=6, 6+1=7, 7+1=10, 11+1=12, 16+1=17, 17+1=20

All the above numbers are base 8 number.

Counting Sequence:

0, 1, 2, 3, 4, 5, 6, 7, 10, 11, 12, 13, 14, 15, 16, 17, 20…..

Only 8 notations (0, 1, 2, 3, 4, 5, 6, 7) need to be used to represent number.

**Hexadecimal**: Hexadecimal is a base 16 number system in which we use 16 distinct notations. The notations in Hexadecimal are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E and F. In the notation A= decimal 10, B = decimal 11 C=12, D=13, E=14 and F= decimal 15. After F, the next number is 10 because all the unique notations have been used so make carry and 0 is initial start of count on first digit.

1+1=2 6+1=7, 9+1=A, C+1=D.

Counting Sequence:

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A. B. C. D. E. F. 10. 11. 12. ……18, 19, 1A, 1B, 1C, 1D, 1E, 1F, 20, 21…..

**Base Conversion:**

Number from one number system can be converted to other number base system.

**Binary to Decimal:**

For example:

=64+32+8+4+1=109 decimal Answer

**Octal to Decimal**

**Hexadecimal to Decimal**

= 44619 decimal Answer