GCSE Computing – Representation of data
in computer systems: numbers
Candidates should be able to:
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convert positive denary whole numbers (0-255) into 8-bit
binary numbers and vice versa
add two 8-bit binary integers and explain overflow errors
which may occur
convert positive denary whole numbers (0-255) into 2-digit
hexadecimal numbers and vice versa
convert between binary and hexadecimal equivalents of the
same number
explain the use of hexadecimal numbers to represent binary
numbers.
© GCSE Computing
Slide 1
Converting 8-bit binary numbers into positive
denary whole numbers (0-255)
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There are 256 different 8-bit binary numbers:
00000000 to 11111111
Each bit represents a different power of 2.
Denary equivalent
128
64
32
16
8
4
2
1
Equivalent power of 2
27
26
25
24
23
22
21
20
Binary bits
1
1
1
1
1
1
1
1
One simple method of conversion from binary is therefore to
add these powers of 2 for each non-zero bit (1).
For example: 128 64 32 16
8
4
2
1
1
0
0
1
1
1
0
1
128
0
0
16
8
4
0
1
8-bit binary 10011101 therefore converts to denary 157
(128 + 16 + 8 + 4 + 1).
© GCSE Computing
Slide 2
Converting positive denary whole numbers
(0-255) into 8-bit binary numbers: method 1
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One method is to repeatedly divide the denary number by 2,
placing the remainder (0 or 1) below the number and the
integer quotient to the left.
1
2
4
9
19
39
78 157
Example 1:
0
0
1
1
1
0
1
157 converts to - 1
Example 2:
156 converts to Example 3:
45 converts to –
1
2
4
9
19
39
78
156
1
0
0
1
1
1
0
0
1
2
5
11
22
45
1
0
1
1
0
1
0
0
Note, the 2 extra 0 bits were added to convert the number into an 8-bit
binary number.
© GCSE Computing
Slide 3
Converting positive denary whole numbers
(0-255) into 8-bit binary numbers – method 2
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Another method is to repeatedly subtract decreasing powers
of 2 from the denary number, starting with 27 (128) .
If the result is zero or positive, place 1 below the number,
then place the difference to the right. Otherwise place 0
below the number and copy the number to the right. Repeat
until you reach 20 (1).
128 64
32
16
8
4
2
1
Example 1:
157 converts to -
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Example 2:
45 converts to © GCSE Computing
157
29
29
29
13
5
1
1
1
0
0
1
1
1
0
1
128
64
32
16
8
4
2
1
45
45
45
13
13
5
1
1
0
0
1
0
1
1
0
1
Slide 4