# Number Bases: Binary, Octal, Denary & Hexadecimal

Number bases refer to ways of counting numbers. Counting started way back in the ancient times when began counting first, with his fingers. He counts in tens maybe because he has ten fingers and this is called decimal system of counting. There are different bases of counting,

## Different number bases/system

Binary system

Octal system

Denary/decimal system

## Binary System

The word BI means two, so binary combination means numbers made up of a combination of only two numbers. It also refers to numbers in base 2.  The available digits in a binary system where 0 means off and 1 means ON.

## Octal System

This is counting in eight i.e. base 8. It has 0,1,2,3,,4,5,6,7 digits.

## Denary/Decimal

This is counting in tens. They are also called decimal systems. The decimal system has the following digits 0,1,2,3,4,5,6,7,8,9

This system deals with numbers in base 16. It has the following digits 0,1,2,3,4,5,6,7,8,9, A, B, C, D.E, F (A=10, B=11, C=12, D=13, E=14 and F=15).

## CONVERSION FROM BASE TEN TO OTHER BASES

To convert a number in a decimal system to other bases, the continuous division of the number by the new base number is used.

Convert 17ten to base 2

2         17

2          8  r  1

2          4  r  0                             17ten = 100012

2          2  r  0

2          1  r  0

0  r  1

Convert 58ten to base 2

2          58

2          29  r  0

2          14  r  1

2          7  r  0                          58ten = 1110102

2          3  r  1

2          1  r  1

0  r  1

Convert 248ten to octal

8          248

8          31   r  0

8          3    r   7              248ten = 370eight

0    r  3

Convert 312ten to base 16

16       312

16       19   r  8

16       1     r  3                   312ten = 13816

0         r  1

16       935

16       58    r 7

16       3      r A                    935ten = 3A716

0      r 3

## CONVERSION OF OTHER BASED TO DECIMAL SYSTEM

To convert numbers in other bases to a 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) 255eight (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

= 9ten

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

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

= 128 + 40 + 5

= 173ten

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

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

= 768 + 80 + 4

= 852ten

## 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 110110two to base 8, base 16

(note 23 = 8 and 24 = 16)

(i) 1101102 = (1102) (1102)

= 66eight (1102 = 6ten)

(ii) 1101102 = (00112) (01102)

= 36hex (00112 = 3ten)

2) Convert 1110110two to base 16

1110110 = (01110110)two

= 7616

3) Convert 1000101two to base eight

1000101two = (001)(000)(101)

= 103eight

4) Convert 62eight to base two

62eight = 6                  2

110            010

= 110010two

5)     Change A0316 = A            0           3

1010    0000     0011

= 10100000011two

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