Binary Addition & Overflow Errors

Arithmetic addition in binary follows positional place-value rules identical to denary addition, but with a base of 2 where $1 + 1 = 10_2$ (0 with a carry of 1).

Learning Objectives

• Add two positive 8-bit binary integers with accuracy.
• Identify carry bits and explain the causes and consequences of overflow errors in computer registers.

Core Syllabus Theory

1. The Four Fundamental Binary Addition Rules

When adding two binary digits in any column, four possible combinations exist:

1. $0 + 0 = 0$
2. $0 + 1 = 1$
3. $1 + 1 = 0 \text{ (carry 1 into the next column to the left)}$
4. $1 + 1 + 1 = 1 \text{ (carry 1 into the next column to the left)}$

Always perform binary addition starting from the Least Significant Bit (LSB) on the far right and work column by column towards the Most Significant Bit (MSB) on the far left.

2. Step-by-Step Worked Example (Without Overflow)

Add $01011010_2$ ($90_{10}$) and $00110111_2$ ($55_{10}$):

Checking in denary: $90 + 55 = 145$. The result $10010001_2 = 128 + 16 + 1 = 145_{10}$. The calculation is completely correct and fits cleanly within the 8-bit register.

3. Overflow Errors in Hardware Registers

An overflow error occurs when the result of a binary arithmetic operation exceeds the maximum representable value for the allocated hardware register size.

In an unsigned 8-bit register, the positional place values allow numbers from $0$ up to $2^8 - 1 = 255_{10}$ ($11111111_2$).

Worked Example: Addition With Overflow

Add $11001000_2$ ($200_{10}$) and $01010000_2$ ($80_{10}$):

Consequences of Overflow:

1. Lost Carry Bit: The 8-bit hardware register has no physical flip-flop/latch to store the 9th bit. The carry bit is discarded or pushed into an overflow flag register.
2. Corrupted Result: The accumulator register only holds the lower 8 bits: $00011000_2 = 16 + 8 = 24_{10}$. Instead of the correct sum $280$, the computer computes $24$!
3. Software Impact: Unless software explicitly inspects the CPU overflow flag, this arithmetic corruption can cause unpredictable bugs, incorrect game physics, navigation errors, or system failure.

Exam Tips & Common Pitfalls

Note: Show Every Carry Bit: In Cambridge IGCSE Paper 1, questions frequently award a dedicated mark for showing your carry bits clearly above each column. Never compute binary addition mentally without writing down carries. Spotting Overflow: If adding two 8-bit numbers produces a 1 in the carry-out from the 128 column, an overflow error has occurred. State explicitly that the result cannot fit into the 8-bit register.

Key Terminology Glossary

• Carry Bit: A bit generated when the sum of two or more bits exceeds the base value (base 2).
• Most Significant Bit (MSB): The leftmost bit in a binary number, carrying the highest positional place value.
• Least Significant Bit (LSB): The rightmost bit in a binary number, carrying the lowest positional place value ($1$).
• Overflow Error: An error that occurs when an arithmetic calculation produces a result too large to be held in the allocated register size.

Written practice and self-review

Two 8-bit unsigned binary integers are added together in a computer CPU: 11001000 (denary 200) and 01010000 (denary 80). Explain why an overflow error occurs during this operation, and describe what happens to the numerical value stored inside the 8-bit accumulator register.

Write an explanation before checking the theory. Check that each claim answers the question, uses the correct concept and explains why it applies. Discuss uncertain points with a teacher. The structured exercise below checks specific facts; it does not automatically mark this explanation.