A review of number bases, positional value and how to convert between bases

1. Number base conversion
from denary (base10)
Peter R Breach
January 2016

2. Base 10 (Denary) numbers
Use the symbols
0, 1, 2, 3, 4, 5, 6, 7, 8, 9
The “value” of the digit is dependent on its position
Thousands
Hundreds
Tens
Units
1 3 4 5
1*1000 + 3 * 100 + 4* 10 + 5 * 1
Value comes from the number base (10) and its position
10^3 10^2 10^1 10^0

3. Base 2 (Binary) Numbers
Use the symbols
0, 1
The “value” of the digit is dependent on its position
Eights
Fours
Twos
Units
1 0 1 0
1*8 + 0 *4 +1 *2 + 0 *1
Value comes from the number base (2) and its position
2^3 2^2 2^1 2^0

4. Base 8 (Octal) Numbers
Use the symbols
0, 1, 2, 3, 4, 5, 6, 7
The “value” of the digit is dependent on its position
Five Hundred and twelves
Sixtyfours
Eights
Units
1 3 2 4
1*512 + 3 *64 +2 *8 + 4 *1
Value comes from the number base (8) and its position
8^3 8^2 8^1 8^0

5. Base 16 (Hexadecimal) Numbers
Use the symbols
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F
The “value” of the digit is dependent on its position
Four thousand and ninetysixes
Two hundred and fiftysixes
Sixteens
Units
A 0 2 2
10*4096 + 0*256+2*16 +2 *1
Value comes from the number base (16) and its position
16^3 16^2 16^1 16^0

6. How to know what base?
After the symbols for the number base, a subscript
symbol is used
7410 In normal usage this would be assumed
102
748
7416 or 74hex computing uses 0X74

7. Convert from Base10 to Base2
Use repeated division, using the target base
each remainder is the value in the base.
25 divide 2 12 remainder 1 least significant
12 divide 2 6 remainder 0
6 divide 2 3 remainder 0
3 divide 2 1 remainder 1
1 divide 2 0 remainder 1
25 10 becomes 11001 2

8. Convert from Base10 to Base8
Use repeated division, using the target base
each remainder is the value in the base.
46 divide 8 5 remainder 6 least significant
5 divide 8 0 remainder 5
46 10 becomes 56 8

9. Convert from Base10 to Base16
Use repeated division, using the target base
each remainder is the value in the base.
247 divide 16 15 remainder 7 least significant
15 divide 16 0 remainder 15 f (2 digit values become letters)
247 10 becomes f716 or 0x00f7

10. Hexadecimal, Octal and Binary
conversions
These rely on changing patterns.
Binary is the base level, the whole is a
single group
0110011010101101
Octal groups 3 binary digits
0 110 011 010 101 1010632558
Hexadecimal groups 4 binary digits
0110 0110 1010 1101 66AD16