Binary and Decimal Numbers:
Binary and decimal numbers are two different systems used for counting and performing mathematical operations. The main difference between these two systems is the base of the number system.

Decimal Number System:
Decimal number system is also known as base-10 numbering system. In this system, there are 10 digits from 0 to 9. The position of each digit in a decimal number represents the power of 10. For example, the number “123” in decimal system represents:

1 x 10^2 + 2 x 10^1 + 3 x 10^0

= 100 + 20 + 3

= 123

Binary Number System:
Binary number system is also known as base-2 numbering system. In this system, there are only two digits, 0 and 1. The position of each digit in a binary number represents the power of 2. For example, the number “101” in binary system represents:

1 x 2^2 + 0 x 2^1 + 1 x 2^0

= 4 + 0 + 1

= 5

Conversion between Binary and Decimal Numbers:
Binary numbers can be converted to decimal numbers by multiplying each digit of the binary number with its corresponding power of 2 and adding the results. For example, the binary number “1010” can be converted to decimal number as follows:

1 x 2^3 + 0 x 2^2 + 1 x 2^1 + 0 x 2^0

= 8 + 0 + 2 + 0

= 10

Decimal numbers can be converted to binary numbers by dividing the decimal number by 2 and writing down the remainder. This process is repeated until the quotient becomes 0. The binary number is formed by writing down the remainders in reverse order. For example, the decimal number “13” can be converted to binary number as follows:

13 ÷ 2 = 6 remainder 1

6 ÷ 2 = 3 remainder 0

3 ÷ 2 = 1 remainder 1

1 ÷ 2 = 0 remainder 1

Binary number = 1101

Uses of Binary and Decimal Numbers:
Binary numbers are widely used in computer systems, as digital devices can only understand and process binary numbers. Decimal numbers, on the other hand, are used in everyday life for counting and performing arithmetic operations.