Binary addition, subtraction, overflow and underflow

Binary addition rules

• 0 + 0 = 0
• 0 + 1 = 1
• 1 + 1 = 10 (write 0, carry 1)
• 1 + 1 + 1 = 11 (write 1, carry 1)

Example (unsigned): 0110 1101 (109) + 0011 1010 (58) = 1010 0111 (167). Always show the carries in an exam.

Subtraction with two's complement

To work out A - B, find the two's complement of B (invert, add 1) and add it to A. Ignore any carry out of the MSB.

Example: -37 - 21 in 8 bits, worked as -37 + (-21): 1101 1011 + 1110 1011 = 1100 0110 with a carry out of 1, which is discarded. 1100 0110 = -58. Correct.

Overflow and underflow

• Overflow happens when the result of a calculation is too large (in size) to be represented in the bits available. Unsigned: there is a carry out of the MSB. Signed: adding two numbers with the same sign gives a result with the opposite sign. Example: 100 + 60 in 8-bit two's complement gives 1010 0000, which reads as -96, not 160.
• Underflow happens when a result is too small (too close to zero) to be represented accurately in the bits available, for example a very small fractional number divided by a large one, or a right shift of a fixed-point value, giving a result smaller than the smallest step, so it is stored as 0 or loses precision. A result that is too negative, such as -100 - 60 in 8-bit two's complement, is overflow, not underflow: it is too large in size.
• The CPU sets the carry or overflow flag in the status register so programs can detect the problem. Undetected overflow gives wrong results.

Worked example

Calculate 45 - 19 using 8-bit two's complement.

1. 45 = 0010 1101
2. 19 = 0001 0011, so -19 = 1110 1101
3. Add: 0010 1101 + 1110 1101 = 0001 1010, carry out 1 discarded

Answer: 0001 1010 = 26

Check yourself

1. Add the unsigned 8-bit numbers 0011 0110 and 0100 1011. Give the 8-bit result.
   Show answer

   10000001. 54 + 75 = 129 = 1000 0001.

2. What is the denary value of 0101 1101 + 0010 0111 (unsigned)?
   Show answer

   132. 93 + 39 = 132.

3. Calculate 30 - 47 using 8-bit two's complement. Give the 8-bit result.
   Show answer

   11101111. 30 = 0001 1110; -47 = 1101 0001; sum = 1110 1111 = -17.

4. In 8-bit two's complement, 0101 1010 + 0011 0010 is calculated. What denary value does the 8-bit result appear to represent?
   Show answer

   -116. The sum pattern is 1000 1100, whose MSB is 1, so it reads as -116. 140 is out of range: overflow.

5. Two positive two's complement numbers are added and the result has an MSB of 1. What has happened?
   • Underflow
   • Overflow
   • A correct negative result
   • A logical shift
   Show answer

   Overflow. Two positives cannot make a negative; the result was too large.

6. When adding two unsigned 8-bit numbers, what shows that overflow has happened?
   • A 0 in the least significant bit
   • The result is even
   • The result has more 1s than 0s
   • A carry out of the most significant bit
   Show answer

   A carry out of the most significant bit. The ninth bit cannot be stored.

7. Which is an example of underflow?
   • Adding 1 to 255 in an unsigned byte
   • A fractional result is so close to zero that it is stored as 0
   • Converting binary to hexadecimal
   • Shifting a pattern one place left
   Show answer

   A fractional result is so close to zero that it is stored as 0. Underflow: too small to represent. Adding 1 to 255 is overflow.

8. Using 8-bit unsigned binary, what is the largest number that can be added to 200 without causing overflow?
   Show answer

   55. 200 + x must be at most 255, so x = 55.

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Sources

• Pearson Edexcel International Advanced Subsidiary/Advanced Level in Computer Science specification (2026)
• Pearson Edexcel IAL Computer Science IAS sample assessment materials (Unit 1, WCP01)

Last updated 8 November 2026. Spotted a mistake? Tell us.