**Table of content**

## Decimal to number system conversion

To convert a decimal number system to a binary number system division method is used. The given Decimal number is divided by 2 until the quotient is less than 2. To convert the fraction number to binary multiply the fraction with the base (2) to get a new integer and a fraction. Remove the integer from the result and the multiplication with the new fraction is repeated till the result becomes zero.

**Example 1**: (25)_{10} = () _{2}

25_{10} = 11001_{2}

**Example 2**: (34.45)_{10} = (?)_{2}

Whole number Part:

**Decimal Part: **

0.45_{10}=0.01110_{2 }

The bunary equivallent of (34.45)_{10}= (100010.01110)_{2}

## Decimal to Octal number system conversion

To convert a decimal number system to an Octal number system division method is used. The given

Decimal number is divided by octal number system’s base 8 until the quotient is less than 8. To convert the fraction number to octal multiply the fraction with the base of octal number (8) to get a new integer and a fraction. Remove the integer from the result and the multiplication with the new fraction is repeated till the result becomes zero.

**Example 3**: (144.25)_{10} = (?)_{8 }

**Whole number Part:**

144_{10} = 220_{8}

**Decimal Part:**

0.25_{10}=0.20_{8 }

(144.25)_{10}= (220.20)_{8}

## Decimal to Hexadecimal number system conversion

To convert a decimal number system to a Hexadecimal number system division method is used. The given Decimal number is divided by 16 until the quotient is less than 16. To convert the fraction number to Hexadecimal multiply the fraction with the base of hexadecimal number (16) to get a new integer and a fraction. Remove the integer from the result and the multiplication with the new fraction is repeated till the result becomes zero.

**Example 4**: (100)_{10} = (?)_{16}

100_{10} = 64_{16}

## Binary to Decimal number system conversion

To convert a Binary number system to Decimal number system multiplication method is used. The given binary number is multiplied by the increasing powers of 2 based on the bit position

**Example 6:** (110110)_{2} = (?)_{10}

Graphical Method:

**Equation Method: **

= 0 × 2^{0 }+ 1 × 2^{1 }+ 1 × 2^{2 }+ 0 × 2^{3 }+ 1 × 2^{4 }+ 1 × 2^{5}

= 0 + 2 + 4 + 0 + 16 + 32

= 54

∴ (110110)_{2} = (54)_{10}

**Example 7:** (11110.0101)_{2} = (?)_{10}

**Graphical Method: **

**Equation Method:**

=1×2-4 + 0×2-3 + 1×2-2 + 0×2-1 + 0×20 + 1×21 +1×22 +1×23 +1×24

=0.0625+0+0.250+0+0+2+4+8+16

=54.3125

∴ (11110.0101)_{2} = (54.3125)_{10}

## Binary to Octal number system conversion

To convert a Binary number system to octal number system the given binary number is split into three bit groups. For this group the equivalent octal values are written

**Example 8:** (11110101.0101110)_{2} = (?)_{8}

## Binary to Hexadecimal number system conversion

To convert a Binary number system to Hexadecimal number system the given binary number is split into four bit groups. For this group the equivalent octal values are written

**Example 9:** (11110101.0101110)_{2} = (?)_{16}

(11110101.0101110)_{2} = (F5.5C)_{16}

## Octal to Binary number system conversion

To convert the octal number system to binary system represent the values of each octal number as a three bit binary number. **Example 10: **(7405)_{8} = (?)_{2}

(7405)_{8} = (111100000101)_{2}

**Example 11**: (627.35)_{8} = (?)_{2}

(627.35)_{8} = (110010111011101)_{2}

## Hexadecimal to Binary number system conversion

To convert Hexadecimal number system to binary system represents the values of each Hexadecimal number as four bit binary number.

**Example 12: **(F405)_{8} = (?)_{2}

(F405)_{8} = (1111010000000101)_{2}

**Example 13:** (ACF.E1)_{8} = (?)_{2}

(ACF.E1)_{16} = (101011001111.11100001)_{2}

## Octal to Hexadecimal number system conversion

Convert the octal number to binary. Split the binary number into four bits and represent the equivalent hexadecimal system

**Example 14:** (405)_{8} = (?)_{16}

(405)_{8} = (100000101)_{2}

(405)_{8} = (105)_{16}