1

• In general a number system can have any base b

• the digit used are 0, 1, … , b-1

• The weight of ith place is bi

• The conversion formula from base b into decimal number is

• Commonly used base are 2, 3, 8, 10, 16, ...

Base ‘b’ number

for i = 0 to n – 1

for an n digit quantity

1

0

ni

i

iixb

2

Bases 2, 8, and 16 are related

Binary Decimal Octal Hexadecimal0000 0 0 00001 1 1 10010 2 2 20011 3 3 30100 4 4 40101 5 5 50110 6 6 60111 7 7 71000 8 10 81001 9 11 91010 10 12 A1011 11 13 B1100 12 14 C1101 13 15 D1110 14 16 E1111 15 17 F

3

• From binary to octal– make groups of 3 bits from right to left

01 110 1102 1668

• From octal to binary– make each digit as 3 bits sequence

2768 010 111 1102

• From binary to hexadecimal– make groups of 4 bits from right to left

0111 01102 7616

• From hexadecimal to binary– make each digit as 4 bits sequence

3716 0011 01112

Conversion

4

• Convert from base b1 to decimal

• Convert from decimal to base b2

• Direct conversion from base b1 and base b2

– we will not pursue this anymore• Decimal to Hexadecimal

– Divide by 16 recursively and collect digit from right to left from the remainders.

– Example 347110 = D8F16

• 3471 divided by 16 gives 216, remains 15 (F)

• 216 divided by 16 gives 13, remains 8 (8)

• 13 divided by 16 gives 0, remain 13 (D)

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