Base ‘b’ number
• 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
i  n 1
b x
i
i 0
i
for i = 0 to n – 1
for an n digit quantity
• Commonly used base are 2, 3, 8, 10, 16, ...
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Bases 2, 8, and 16 are related
Binary
0000
0001
0010
0011
0100
0101
0110
0111
1000
1001
1010
1011
1100
1101
1110
1111
Decimal
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Octal
0
1
2
3
4
5
6
7
10
11
12
13
14
15
16
17
Hexadecimal
0
1
2
3
4
5
6
7
8
9
A
B
C
D
E
F
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Conversion
• 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
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More on Conversion
• 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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