Binary Arithmetic (Cambridge (CIE) A Level Computer Science): Revision Note
Exam code: 9618
Binary addition
What is binary addition?
Binary addition involves summing numbers in base-2, which uses only the digits 0 and 1
Like denary addition, start from the rightmost digit and move towards the left
Carrying over occurs when the sum of a column is greater than 1, passing the excess to the next left column
Example addition
Worked Example
Add the following two binary integers using binary addition. Show your working. [2]
1 0 0 1 1 0 1 0 + 0 0 1 0 1 1 1 1
Answer
carry | 1 | 1 | 1 | 1 | 1 | |||
|---|---|---|---|---|---|---|---|---|
1 | 0 | 0 | 1 | 1 | 0 | 1 | 0 | |
+ | 0 | 0 | 1 | 0 | 1 | 1 | 1 | 1 |
1 | 1 | 0 | 0 | 1 | 0 | 0 | 1 |
1 mark for working [1 mark]
1 mark for answer [1 mark]
Overflow
What is an overflow?
Overflow occurs when the sum of two binary numbers exceeds the given number of bits
In signed number representations, the leftmost bit often serves as the sign bit; overflow can flip this, incorrectly changing the sign of the result
Overflow generally leads to incorrect or unpredictable results as the extra bits are truncated or wrapped around
Binary subtraction
To carry out subtraction, the number being subtracted is converted into its negative equivalent using two's complement
The two numbers are then added together
Example
48 - 12
-128 | 64 | 32 | 16 | 8 | 4 | 2 | 1 | ||
|---|---|---|---|---|---|---|---|---|---|
0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | = 48 | |
0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | = 12 | |
find two's complement of -12 | |||||||||
0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | = 12 | |
invert | |||||||||
1 | 1 | 1 | 1 | 0 | 0 | 1 | 1 | ||
add 1 | |||||||||
1 | |||||||||
result | |||||||||
1 | 1 | 1 | 1 | 0 | 1 | 0 | 0 | = -12 | |
Add 48 and -12 | |||||||||
-128 | 64 | 32 | 16 | 8 | 4 | 2 | 1 | ||
0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | = 48 | |
1 | 1 | 1 | 1 | 0 | 1 | 0 | 0 | = -12 | |
carries | |||||||||
1 | 1 | 1 | 1 | ||||||
result | |||||||||
0 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | = 36 | |
48 - 12 = 36
The additional overflow bit is ignored leaving a result of 0010 0100, denary equivalent of 36, which is the correct answer
In two's complement arithmetic, the overflow bit does not contribute to the actual value of the operation but is more of a by-product of the method
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