## CONVERSION OF OTHER BASED TO DECIMAL SYSTEM

To convert numbers in other bases to denary system, expand the given number in powers of its base and evaluate.

Examples

1) Convert the following numbers to base ten

(i) 1001_{2} (ii) 255_{eight} (iii) 354_{16}

(i) 1001_{2} = 1 x 2^{3} + 0 x 2^{2} + 0 x 2^{1} + 1 x 2^{0}

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

= 8 + 0 + 0 + 1

= 9_{ten}

(ii) 255_{eight} = 2 x 8^{2} + 5 x 8^{1} + 5 x 8^{0}

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

= 128 + 40 + 5

= 173_{ten}

(iii) 354_{16} = 3 x 16^{2} + 5 x 16^{1} + 4 x 16^{0}

= 3 x 256 + 5 x 16 + 4 x 1

= 768 + 80 + 4

= 852_{ten}

## CHANGING FROM BINARY TO OCTAL AND HEXADECIMAL

To convert from a number system to another one (not denary), it is usual to convert to base ten and then convert the base ten number to the new base number.

However, binary numbers can be converted to octal and hexadecimal numbers because of the fact that 2^{3} = 8 and 2^{2} = 16.

Examples

(1) Convert 110110_{two} to base 8, base 16

(note 2^{3} = 8 and 2^{4} = 16)

(i) 110110_{2} = (110_{2}) (110_{2})

= 66_{eight} (110_{2} = 6_{ten})

(ii) 110110_{2} = (0011_{2}) (0110_{2})

= 36_{hex} (0011_{2} = 3_{ten})

2) Convert 1110110_{two} to base 16

1110110 = (01110110)_{two}

= 76_{16}

3) Convert 1000101_{two} to base eight

1000101_{two} = (001)(000)(101)

= 103_{eight}

4) Convert 62_{eight} to base two

62_{eight} = 6 2

110 010

= 110010_{two}

5) Change A03_{16} = A 0 3

1010 0000 0011

= 10100000011_{two}