GCSE Computer Science · Data Representation

Binary addition and overflow explained

Learn eight-bit binary arithmetic with worked sums, practical overflow examples and independent questions.

8-bit examples · original GCSE practice

Why do we add binary numbers?

Binary uses only 0 and 1. Computer hardware represents values using fixed numbers of bits; for an unsigned 8-bit value the range is 0 to 255. Binary addition uses carrying, just like denary addition, except a column containing 1 + 1 becomes binary 10: write 0 and carry 1.

Worked example: 13 + 7 = 20

Convert the two denary numbers into eight-bit binary:

    00001101   (13)
  + 00000111   (7)
  ----------
    00010100   (20)

From right to left: 1 + 1 = 10, so write 0 and carry 1. On the next columns, include the carried 1 before moving left. The final result is 00010100, which represents 16 + 4 = 20.

What is overflow in an unsigned 8-bit number?

Overflow happens if the result is larger than the maximum value representable in the chosen number of bits. An unsigned byte can represent 0 to 255. Adding one to 255 needs nine bits:

     11111111   (255)
   + 00000001   (1)
   ----------
    100000000   (256, needs 9 bits)

If only eight bits are stored, the retained pattern is 00000000 and the carry out is lost. That is unsigned overflow; it is not evidence that 255 + 1 really equals zero.

Signed interpretation is different

In eight-bit two's-complement representation, the range is −128 to +127. The pattern 01111111 represents +127. Adding 00000001 gives 10000000, which is interpreted as −128. That is signed overflow because the true result, +128, is outside the signed range. A carry-out flag alone is not the correct test for signed overflow.

Always check whether an exam question specifies unsigned binary, two's complement or a fixed bit-width before deciding if overflow has occurred.

Three original self-check questions

Add 00000101 and 00000011. Give the answer as eight bits.
Add unsigned 11111110 and 00000010. What is the real sum, and what eight-bit result remains?
Can the signed eight-bit value +100 be added to +50 without signed overflow?

Reveal the worked answers

5 + 3 = 8, so 00001000.
254 + 2 = 256, written as 100000000. An eight-bit register retains 00000000; unsigned overflow occurs.
No. 150 exceeds the largest signed eight-bit value, +127. The eight-bit pattern would represent a negative number, not +150.

How to check your exam working

Identify the number of bits and whether the values are signed or unsigned.
Line the bits up by place value and work from right to left.
Write carry bits carefully and include them in the next column.
Convert the result to denary as a reasonableness check.
Explain overflow using the range of values that can be stored.