NUMBER BASES

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) 1001₂ (ii) 255_eight (iii) 354₁₆

(i) 1001₂ = 1 x 2³ + 0 x 2² + 0 x 2¹ + 1 x 2⁰

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

= 8 + 0 + 0 + 1

= 9_ten

(ii) 255_eight = 2 x 8² + 5 x 8¹ + 5 x 8⁰

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

= 128 + 40 + 5

= 173_ten

(iii)   354₁₆ = 3 x 16² + 5 x 16¹ + 4 x 16⁰

= 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 2³ = 8 and 2² = 16.

Examples

(1) Convert 110110_two to base 8, base 16

(note 2³ = 8 and 2⁴ = 16)

(i) 110110₂ = (110₂) (110₂)

= 66_eight (110₂ = 6_ten)

(ii) 110110₂ = (0011₂) (0110₂)

= 36_hex (0011₂ = 3_ten)

2) Convert 1110110_two to base 16

1110110 = (01110110)_two

= 76₁₆

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 A03₁₆ = A            0           3

1010    0000     0011

= 10100000011_two