# Number System

**Binary-Decimal Conversion**

= 1 Ã— 2

^{4}+ 1 Ã— 2

^{3}+ 1 Ã— 2

^{2}+ 1 Ã— 2

^{1}+ 1 Ã— 2

^{0}

= 16 + 8 + 4 + 2 + 1 = ( 31 )

_{10}

Decimal-to Binary Conversion

Thus, (25)

_{10}= (11001)

_{2}

Read down to up

Fraction part

*i.e.,*0.5

_{10}= 0.1

_{2}

Therefore 25.5

_{10}= 11001.1

_{2}

Decimal-Octal Conversion

This can be achieved by dividing the given decimal number repeatedly by 8, until a quotient of 0 is obtained.

_{10}.

**Division Generated Remainder**

444 / 8

55/ 8 4

6/8 7

0/8 6

On reading the remainders from bottom to top, the decimal (444)

_{10}(674)

_{8}. Now, fractional conversion

Multiplication Generated Integer

0.4998 = 3.992 3

0.992 8 = 7.936 7

0.936 8 = 7.488 7

0.4888 = 3.904 3

The process gets terminated when significant digits are acquired. Thus, octal equivalent is (444.499)

_{10 }= (674.3773)

_{8}

Octal-Binary Conversion

It can be explained through the following example: To convert (377)_{8} into binary, replace each significant digit by its 3-bit binary equivalent.

(377)_{8 }= 3 7 7 = 011 111 111

Thus, (377)_{8 }= (011111111)_{2}

Binary-Hexadecimal Conversion

e.g., (10100110111110)_{2} = (0010 1001 1011 1110)_{2} = (2 9 B E)_{16} Ã— 1

Hexadecimal-Binary Conversion

It can be explained through an example. To convert (1D5)_{16} into binary, replace each significant digit by its 4-bit binary equivalent.

(1D5)_{16 }= 1 D 5

= 0001 1101 0101

Thus,** **(1D5)_{16} =** **(000111010101)_{2}