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J277 1.2.4 Binary addition and overflow

Part 4 of 5 · J277 1.2.4 · Number systems

Adding binary numbers directly with the column method, and what happens when the answer will not fit.

Students will:

  • recall the four binary addition facts
  • add two binary numbers using the column method
  • record carries clearly
  • explain what an overflow error is
  • recognise when an answer needs more than 8 bits

Inside: 6 explanation cells, 3 number grids, 2 multiple-choice questions, 1 fill-in-the-blanks cell and 2 written answers. 27 marks, about 60 minutes.

Series: J277 1.2.4 · Number systems, part 4 of 5.

Shared by Coding PathwayVerified teacher

  • 14 cells
  • About 60 minutes
  • CC BY-SA 4.0
  • Shared 17 Aug 2026
  • Updated 9 Sept 2026

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Number systems 4: binary addition and overflow

Binary addition lets us add binary numbers directly, without converting them to denary first. The column method will feel familiar: begin on the right, add one column at a time and carry into the next column when needed.

By the end of this worksheet, you should be able to:

  • recall the four binary addition facts
  • add two binary numbers using the column method
  • record carries clearly
  • explain what an overflow error is
  • recognise when an answer needs more than 8 bits

Work through the explanation, checks and applications in order. Use the final challenge to explain the method as well as use it.

1. Four facts to remember

These are the building blocks for every column:

  • 0 + 0 = 0
  • 0 + 1 = 1
  • 1 + 1 = 10: write 0 and carry 1
  • 1 + 1 + 1 = 11: write 1 and carry 1

The results 10 and 11 are binary. Binary 10 is denary 2, and binary 11 is denary 3.

Always begin with the column on the far right. Include any carry from the previous column before writing the new total.

Fill in the blanks4 marks
In binary, 0 + 0 = label 1 and 0 + 1 = label 2. The result of 1 + 1 is label 3, so write 0 and carry 1. The result of 1 + 1 + 1 is label 4, so write 1 and carry 1.
  • 11
  • 100
  • 0
  • 2
  • 10
  • 1

2. Start with four columns

Example: add 0101 + 0011.

Work from right to left:

  1. 1 + 1 = 10: write 0 and carry 1.
  2. 0 + 1 + carried 1 = 10: write 0 and carry 1.
  3. 1 + 0 + carried 1 = 10: write 0 and carry 1.
  4. 0 + 0 + carried 1 = 1.

The answer is 1000.

Record each carry above the column it enters. This makes your working easy to follow and helps you avoid dropping a carried 1.

Number systems6 marks

Add each pair of 4-bit binary numbers and show the carries.

Begin at the right. Write the answer bit for each column, then carry 1 into the next column when the total is 2 or 3.

a)Add these 4-bit binary numbers: 0011 + 0011.

Carries
First number
0011
Second number
0011
Total

b)Add these 4-bit binary numbers: 0011 + 0101.

Carries
First number
0011
Second number
0101
Total

c)Add these 4-bit binary numbers: 0111 + 0101.

Carries
First number
0111
Second number
0101
Total

3. Extend the same method to eight columns

Nothing about the method changes when the numbers have 8 bits. There are simply more columns to complete.

For each question:

  1. begin at the right
  2. add the two bits and any carry
  3. write the answer bit
  4. record a carry in the next column when needed
  5. continue until every column is complete

OCR can award one mark for the correct answer and another for showing the carries. Converting the numbers to denary may help you check afterwards, but it does not show binary column addition.

Number systems6 marks

Complete each 8-bit binary addition and show all carries.

Use the same right-to-left method. Check that every carry is written in the column it enters.

a)Add these 8-bit binary numbers: 01000101 + 00101111.

Carries
First number
01000101
Second number
00101111
Total

b)Add these 8-bit binary numbers: 00001011 + 10111110.

Carries
First number
00001011
Second number
10111110
Total

c)Add these 8-bit binary numbers: 00001001 + 11011010.

Carries
First number
00001001
Second number
11011010
Total
Multiple choice1 mark

Why should carries be shown in an OCR binary-addition question that asks for working?

  • AA carry must be written above every column, even when the column total is less than 2
  • BThe carries demonstrate the binary column method and may earn a working mark
  • CThe carries replace the need to write a final answer
  • DA carry changes a binary question into a denary question

4. What is an overflow error?

The largest value that can be represented using 8 bits in this topic is 255.

An overflow error occurs when the answer to an addition is greater than 255 and therefore needs more than 8 bits.

Example:

11111111 + 00000001 = 100000000

The complete answer has 9 bits. It cannot be represented using only the 8 bits available, so an overflow error has occurred.

A carry within the eight columns is a normal part of addition. Overflow occurs when the complete answer needs a ninth bit.

Number systems4 marks

Complete these two additions that cause overflow and show the carries in the 8-bit grid.

Each complete answer needs a ninth bit. In the grid, enter the rightmost eight answer bits. You will write and explain the complete 9-bit answer in the next task.

a)Add these 8-bit binary numbers: 10100100 + 01110000.

Carries
First number
10100100
Second number
01110000
Total

b)Add these 8-bit binary numbers: 11110000 + 01010101.

Carries
First number
11110000
Second number
01010101
Total
Written answer3 marks

Return to the first overflow addition, 10100100 + 01110000. Write the complete 9-bit answer, state its denary value and explain why an overflow error occurs.

Place the final carried 1 to the left of the eight answer bits shown in the grid. You can use denary to check the value, then explain why the complete answer does not fit in 8 bits.

Students type their answer here.

Multiple choice1 mark

Which statement best explains an overflow error in 8-bit binary addition?

  • AOverflow occurs whenever a carry is written anywhere in the working
  • BOverflow occurs when the answer contains more 0s than 1s
  • COverflow occurs when the complete answer is greater than 255 and needs more than 8 bits
  • DOverflow occurs whenever two 8-bit numbers are added
Written answer2 marks

Challenge: a student converts two binary numbers to denary, adds them, and converts the total back to binary. Their final answer is correct. Explain why they might earn only 1 of the 2 marks in an OCR question that says 'Complete the binary addition' and 'Show your working out'.

Show the addition and every carry before stating whether the complete result fits in eight bits.

Students type their answer here.

Closed-book review

Without looking back, complete these prompts:

  1. 1 + 1 in binary is...
  2. 1 + 1 + 1 in binary is...
  3. I begin binary addition at...
  4. I write a carry when...
  5. Showing carries matters because...
  6. An overflow error occurs when...

Then explain the difference between a carry inside the calculation and an answer that needs a ninth bit.