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GCSE Computing – Representation of data in computer systems: numbers

Candidates should be able to: 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.

Slide 1

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Converting 8-bit binary numbers into positive denary whole numbers (0-255)

There are 256 different 8-bit binary numbers:00000000 to 11111111

Each bit represents a different power of 2.

One simple method of conversion from binary is therefore to add these powers of 2 for each non-zero bit (1).

For example:

8-bit binary 10011101 therefore converts to denary 157(128 + 16 + 8 + 4 + 1).

Slide 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

128 64 32 16 8 4 2 1

1 0 0 1 1 1 0 1

128 0 0 16 8 4 0 1

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Converting positive denary whole numbers (0-255) into 8-bit binary numbers: method 1

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.

Example 1:157 converts to -

Example 2:156 converts to -

Example 3:45 converts to –

Note, the 2 extra 0 bits were added to convert the number into an 8-bit binary number.

Slide 3

1 2 4 9 19 39 78 157

1 0 0 1 1 1 0 1

1 2 4 9 19 39 78 156

1 0 0 1 1 1 0 0

1 2 5 11 22 45

0 0 1 0 1 1 0 1

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Converting positive denary whole numbers (0-255) into 8-bit binary numbers – method 2

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).

Example 1:

157 converts to -

Example 2:

45 converts to -

Slide 4

128

64 32 16 8 4 2 1

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