Binary addition with visible carries, and overflow in the mark-scheme phrasing. Seven stages from theory to mastery. Click any stage card to dive in — or continue where you left off.
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Add two 8-bit binary numbers using column-by-column carrying, and recognise when a result overflows the 8-bit range. Cambridge IGCSE 0478 Paper 1.
Once you can read and write binary (1.1A), the next step is to add two binary numbers — column by column, right to left, carrying just like denary but at a 2-column boundary instead of 10. Cambridge test this every session on Paper 1, always with the phrase "show all your working".
Adding two 8-bit numbers can produce a result that needs 9 bits — the extra bit has nowhere to go, so an overflow occurs. Recognising overflow and explaining it in the mark-scheme's exact phrasing is the second half of this lesson.
Note: logical binary shifts and two's complement negative numbers are covered in 1.2B Shifts & Negative Numbers. This lesson (1.2A) covers addition and overflow only.
| Term | Definition |
|---|---|
| Binary addition | Adding two binary numbers column by column, right to left, using carry rules. |
| Carry | When a column's sum is 2 or 3, a 1 is passed to the next column left. |
| Sum | The digit written in the current column (0 or 1 after any carries). |
| Column method | The right-to-left, column-by-column approach — the required method. |
| Overflow | An error that occurs when the result is too large to represent in the number of bits available. |
| Register | A fixed-size storage location. An 8-bit register holds 8 bits — no more, no less. |
| MSB / 9th bit | Most Significant Bit. When an 8-bit addition needs a 9th bit, that bit is lost. |
| 8-bit range | 0 to 255 (unsigned). Result > 255 = overflow. |
| Show working | Cambridge award method marks per step — must be shown in binary, not denary. |
| Rule | Sum | Carry to next column |
|---|---|---|
| 0 + 0 | 0 | 0 (no carry) |
| 0 + 1 | 1 | 0 (no carry) |
| 1 + 0 | 1 | 0 (no carry) |
| 1 + 1 | 0 | 1 (carry) |
| 1 + 1 + 1 | 1 | 1 (carry) |
| Bits available | Max unsigned value | Overflow occurs when… |
|---|---|---|
| 4-bit | 15 (1111) | result > 15 |
| 8-bit | 255 (11111111) | result > 255 |
| 16-bit | 65535 (1111 1111 1111 1111) | result > 65535 |
| Step | What happens |
|---|---|
| 1 | Line up the two numbers, right-aligned |
| 2 | Start at the rightmost column (LSB, place value 1) |
| 3 | Apply the rule for that column, including any incoming carry |
| 4 | Write any outgoing carry above the next column left |
| 5 | Move one column left. Repeat until all columns processed |
| 6 | If a final carry falls off the left edge in an 8-bit register → overflow |
Add 00110101 + 00100011.
Add 10011011 + 00010011.
Add 11111111 + 00000001.
Converting both numbers to denary, adding in denary, then converting back does NOT demonstrate binary addition. Cambridge require binary working with carries visible. Method marks lost.
Cited: 2023 examiner report — "some candidates converted the values to denary, added the values and converted the answer back to binary. Candidates should note that this is not [the required method]"Even if the final answer is correct, you lose method marks if the carries aren't visible above the appropriate columns. Write them explicitly.
Cited: 2025 examiner report — "candidates need to make sure they are clearly showing where the carries are located"If a question asks whether overflow occurs, you must explicitly state so. Just giving a wrong 8-bit answer without flagging overflow scores zero.
Cited: 2023 examiner report — "many candidates were able to recognise that an overflow error occurs when the value is larger than…"Saying "the answer is too big" is not enough. The mark-scheme wants "too large to represent in the number of bits available". Reference the bits.
Cited: 2025 examiner report — recurring definition phrasingThere is no digit 2 in binary. 1 + 1 = 10 (write 0, carry 1). Students familiar with denary sometimes write "2" and lose the mark.
In a column with two 1s AND a carry-in of 1, the sum is 3 (denary) = 11 (binary). Write 1, carry 1. Students often write 10 instead.
If the question asks for the answer in binary, give binary. Some candidates give a denary answer at the end even though their working was in binary. Read the wording.
Cited: 2023–2025 recurring wording trapWrite the two numbers with LSBs aligned to the right. A one-column misalignment cascades through every place value and produces a completely wrong answer.
Cited: recurring exam technique flagWhen the addition produces a 9th bit, don't just write the last 8 bits and move on. Explicitly identify this as overflow — that's the mark-scheme point.
Cited: recurring in overflow-explanation questionsThese are the exact phrasings Cambridge accept. Learn them.
Answer these in your head, then tap to see the model answer. Aim for 5 or 6 right before moving on.
Binary addition works just like denary addition — but the column overflows at 2 instead of 10.
Think of each column as a stack. When the stack reaches 2, it collapses — write 0 in this column and carry 1 to the next. When the stack reaches 3, write 1 and carry 1.
Add 10011011 + 00010011. Right to left with all carries visible.
Right to left: 1+1=10 (write 0, carry 1) → 1+1+1=11 (write 1, carry 1) → 0+0+1=1 → 1+1=10 (write 0, carry 1) → 1+0+1=10 (write 0, carry 1) → 0+0+1=1 → 0+0=0 → 1+0=1. Result: 10101110 (174 in denary).
Converting to denary, adding, then converting back does not earn method marks. Cambridge want binary addition demonstrated with carries visible.
Cited: 2023 examiner report — verbatimOverflow occurs when the result of a binary addition is too large to be represented in the number of bits available.
For 8-bit: overflow happens when the result is greater than 255 — a 9th bit is needed but there's nowhere to store it.
11111111 + 00000001
= 255 + 1 = 256
256 cannot fit in 8 bits → overflow
The 9th bit is lost; the register shows 00000000 (incorrect).
"Too large to represent in the number of bits available" is the mark-scheme phrasing. Don't just say "the answer is too big" — reference the bits.
Cited: 2025 examiner focusWhat is the correct definition of overflow?
Add the two binary numbers. Type the 8-bit result.
Stretch: can you get 5 in a row without a mistake?
Look at the two binary numbers. Will adding them cause overflow in 8 bits?
Press New Question.
Stretch: predict overflow before adding — quick check by comparing to 128 (both > 128 = guaranteed overflow).
Which statement about binary addition is WRONG?
Press New Question.
Stretch: explain out loud why the wrong statement is wrong, using mark-scheme phrasing.
Type the missing values. Aim for 3 in a row without error.
Press New Question.
Stretch: after 3 correct, write out all 5 rules from memory in the notepad.
Answer as many binary-addition questions as you can in 60 seconds. Warm-up before Practice or Exam.
Press Start Sprint to begin.
Stretch: beat your personal best twice in a session.
Weak skills come round twice as often. No question repeats immediately. 20 questions across 8 skills.
Press New Question.
12 questions in Cambridge Paper 1 style. Every question cites its source. Write your answer, then click Mark for keyword scoring.
Press New Question to begin.
Press New Question.
Every trap below is drawn from a real Cambridge examiner report or a recurring pattern flagged across multiple sessions.
Does NOT demonstrate binary addition. Cambridge require binary working with visible carries.
Cited: 2023 examiner report — "some candidates converted the values to denary, added the values and converted the answer back to binary. Candidates should note that this is not [the required method]"Method marks lost even when the final answer is right. Write carries explicitly above the appropriate columns.
Cited: 2025 examiner report — "candidates need to make sure they are clearly showing where the carries are located"If a question asks about overflow, you must explicitly state whether it occurs. Just giving a wrong 8-bit answer without flagging overflow loses the mark.
Cited: 2023 examiner report — "candidates were able to recognise that an overflow error occurs when the value is larger than…"Saying "the answer is too big" isn't enough. Mark-scheme: "too large to represent in the number of bits available". Reference the bits.
Cited: 2025 examiner focusThere is no digit 2 in binary. 1 + 1 = 10 (write 0, carry 1).
Cited: recurring patternTwo 1s plus a carry-in of 1 = 3 = 11 in binary. Write 1, carry 1. Students often write 10 instead.
Cited: recurring multi-carry mistakeRead the question wording. If it says "give your answer in binary", the final answer must be binary — even if you check with denary.
Cited: 2023–2025 recurring wording trapRight-align the LSBs. A one-column misalignment cascades through every place value.
Cited: recurring exam-technique flagWhen the addition produces a 9th bit, don't just write the last 8 bits. Explicitly identify overflow — that's the mark-scheme point.
Cited: recurring in overflow-explanation questionsColumn sum = 2? Write 0, carry 1. Column sum = 3? Write 1, carry 1. That's the whole system.
8-bit max = 11111111 = 255. Any result > 255 = overflow. The 9th bit has nowhere to go.
Just like denary addition — start at the LSB (rightmost column) and work left, carrying as you go.
2- and 3-mark addition questions reward method. Even with an arithmetic slip, visible carries + correct method = partial marks. No working = all or nothing.
When defining overflow: mark-scheme phrase is "too large to represent in the number of bits available". Say it exactly.
If the question gives binary and asks for a binary answer, don't touch denary except to sanity-check yourself. Denary in the working = no method marks.
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