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How do you add binary numbers, what is overflow, and how does an arithmetic shift multiply or divide a binary number?

Adding two binary numbers using the carry rules, the meaning of overflow, and using arithmetic (binary) shifts to multiply and divide by powers of two.

A focused answer to the WJEC GCSE Computer Science Unit 1 content on binary arithmetic, covering adding two binary numbers with the carry rules, the meaning and cause of overflow, and using left and right arithmetic shifts to multiply and divide a binary number by powers of two.

Generated by Claude Opus 4.88 min answerUpdated 2026-06-15

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What this topic is asking
Adding binary numbers
Overflow
Arithmetic shifts
Why shifts are used
Try this

What this topic is asking

WJEC wants you to add two binary numbers using the carry rules, explain what overflow is and why it happens, and use an arithmetic (binary) shift to multiply or divide a binary number by powers of two. This is part of the Data representation and data types content in Unit 1 of WJEC GCSE Computer Science (3500).

Adding binary numbers

Overflow

Arithmetic shifts

Adding binary numbers and shifting to multiply

Add $00110101$ and $00011011$ , then use a left arithmetic shift to multiply the answer by $4$ .

step 1: Add the two binary numbers from the right

Column by column: $1+1=10$ (write $0$ , carry $1$ ); $0+1+1=10$ (write $0$ , carry $1$ ); $1+0+1=10$ (write $0$ , carry $1$ ); $0+1+1=10$ (write $0$ , carry $1$ ); $1+1+1=11$ (write $1$ , carry $1$ ); $1+0+1=10$ (write $0$ , carry $1$ ); $0+0+1=1$ ; $0+0=0$ . The result is $01010000$ .

step 2: Check the addition in denary

$00110101 = 53$ and $00011011 = 27$ , and $53 + 27 = 80$ . The answer $01010000 = 64 + 16 = 80$ , which agrees.

step 3: Multiply by 4 with a left shift of 2 places

Multiplying by $4$ is $2^2$ , so shift left by $2$ places. $01010000$ becomes $01000000$ once the top bits move off, but since the number fits, shifting $01010000$ left twice gives $0101000000$ , which needs more than $8$ bits.

step 4: Interpret the result and overflow risk

In denary the answer is $80 \times 4 = 320$ , which is larger than $255$ , so an $8$ -bit register would overflow and lose the top bits. This shows how a left shift can cause overflow if the result is too big for the available bits.

Why shifts are used

Multiplying and dividing by powers of two with a shift is much faster for a processor than full multiplication or division, so compilers often replace a multiplication by $2$ , $4$ or $8$ with a left shift, and a division by those values with a right shift.

Try this

Q1. Add the binary numbers $00001111$ and $00000001$ . [2 marks]

Cue. The carries ripple all the way up: the answer is $00010000$ ( $15 + 1 = 16$ ).

Q2. A binary number is shifted left by $3$ places. By what number has it been multiplied? [1 mark]

Cue. $2^3 = 8$ .

Exam-style practice questions

Practice questions written in the style of WJEC exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.

WJEC-style Unit 13 marksAdd the two

8

-bit binary numbers

01011010

and

00101110

. Show your working and give the

8

-bit answer.

Show worked answer →

A Unit 1 binary addition question. Add column by column from the right using the rules $0+0=0$ , $0+1=1$ , $1+1=10$ (write $0$ carry $1$ ), $1+1+1=11$ (write $1$ carry $1$ ). Working from the right: the columns give $0,0,0,1,0,0,0,1$ with carries handled, producing $10001000$ (2 marks for correct method and carries). The answer is $10001000$ , which checks as $90 + 46 = 136$ (1 mark for the correct result). Markers reward correct carrying. A common error is to forget to carry when two or three $1$ s meet, or to lose the leftmost carry.

WJEC-style Unit 12 marksExplain what is meant by overflow when adding binary numbers.

Show worked answer →

A Unit 1 explain question. Overflow happens when the result of a binary addition is too large to fit in the fixed number of bits available, for example when adding two $8$ -bit numbers gives a result that needs $9$ bits (1 mark). The extra carry bit cannot be stored, so it is lost and the stored answer is incorrect (1 mark). Markers reward the idea of a result exceeding the available bits and a carry being lost. A common error is to describe overflow as simply a wrong answer without saying that the result is too big for the number of bits.

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Sources & how we know this

WJEC GCSE Computer Science specification (3500) from 2017 — WJEC (2017)