🔢 Chapter 1 · Topic 3 · Paper 1 & 2

Binary Addition

Adding in a world with only 0 and 1 — and what happens when things go wrong

🔥 01 · Did You Know?

Every time your CPU performs arithmetic — adding your shopping basket total, calculating a game character's position, rendering a frame of video — it does so using binary addition. A modern processor performs this operation billions of times per second. The very first electronic binary adder was built in 1938 by Konrad Zuse in his parents' living room in Berlin, using telephone relays. Today, an iPhone chip contains 16 billion transistors, each acting as a microscopic binary switch — all performing the same fundamental operation you're about to learn. And critically: when addition produces a result too large for the available bits, computers make serious errors. In 1996, the Ariane 5 rocket exploded 40 seconds after launch because a 64-bit number was forced into a 16-bit register — a binary overflow, exactly what Topic 4 covers next. First: master the addition itself.

02 · Core Concept

The Four Binary Addition Rules

In denary addition you learn times tables. In binary, there are only four rules to memorise — because there are only two possible digits. Everything else, no matter how complex, is built from these four.

0 + 0
Zero plus zero
Write 0

Carry 0
= 0
0 + 1
Zero plus one (or 1 + 0)
Write 1

Carry 0
= 1
1 + 1
One plus one — binary 2!
Write 0

Carry 1
= 10 (binary)
1+1+1
With a carry in — three 1s
Write 1

Carry 1
= 11 (binary)
🧠 Memory trick: Think of it like carrying in denary. When denary goes past 9 you carry 1. In binary, when you go past 1 you carry 1 — it happens much sooner!

The Column-by-Column Process

Binary addition works right to left, exactly like denary. Always start at the Least Significant Bit (LSB — rightmost). A carry produced in one column is added into the next column to the left.

1

Write number A above number B, aligning all bits by position. Add a blank carry row above them.

2

Start at the rightmost column. Add: top bit + bottom bit + any incoming carry. Apply the four rules above.

3

Write the result bit below the divider. If a carry is generated, write it above the next column to the left.

4

Repeat for every column, moving left. If a carry comes out of the leftmost (MSB) column — overflow has occurred.

A Quick Example — 45 + 18 = 63

carry
0
0
1
1
1
0
0
0
A = 45
0
0
1
0
1
1
0
1
B = 18
0
0
0
1
0
0
1
0
= 63
0
0
1
1
1
1
1
1

45 = 00101101 · 18 = 00010010 · Result = 00111111 = 63 ✓

⚠️ Common Exam Mistakes

❌

Forgetting to include the carry when adding a column — always check if there's an incoming carry before applying the rules. Missing a carry produces a wrong result in every subsequent column.

❌

Writing "2" instead of "10" — in binary, 1 + 1 = 10. There is no digit 2. If you write 2 in a result, you've left denary mode.

❌

Working left to right — always start from the rightmost (LSB) column. You cannot know the carry for column 5 until you've completed column 6.

❌

Ignoring overflow — if a carry comes out of the MSB column, the answer stored in 8 bits is wrong. You must state: "overflow has occurred."

🏆
Cambridge Exam Tip: Always write a separate carry row in your working. Examiners explicitly award marks for correct carry bits — even if your final answer has an arithmetic slip. A question asking for binary addition is typically worth 3–4 marks: 1 for method, 1–2 for carries, 1 for correct result.
⚡ 03 · Interactive Tool
Step-Through Binary Adder
// Toggle bits or type numbers · Add · Step through each column to see the carry logic live
Quick examples:
51 + 21
76 + 50
178 + 78
overflow!
127 + 1
255 + 1
overflow!
85 + 42
A First number — click bits or type below:
128
64
32
16
8
4
2
1
or type: = 0
B Second number — click bits or type below:
128
64
32
16
8
4
2
1
or type: = 0
🚨
OVERFLOW DETECTED! A carry was generated out of the MSB (bit 7) column. This carry bit is lost — the 8-bit result stored is incorrect. This is binary overflow. (Topic 4 covers this in full.)
04 · Worked Example

Adding 156 + 73 in Binary

This example includes multiple carries and goes right to the edge of 8-bit capacity. Try each step before revealing.

📋 Question: Add the binary numbers 10011100 (156) and 01001001 (73). Give your answer in binary. Show all working including the carry row. [4]
1
📋 Set Up — Align the Numbers
Write A over B, aligned by column. Add the carry row above. What are the denary values as a check?
▶ Click to reveal
carry: _ _ _ _ _ _ _ _ A=156: 1 0 0 1 1 1 0 0 B= 73: 0 1 0 0 1 0 0 1 ───────────────── result: ? ? ? ? ? ? ? ? Denary check: 156 + 73 = 229 229 ≤ 255 → no overflow expected ✓
2
➡ Columns 1–4 (Right Side)
Work right to left through the first four columns. Track every carry carefully.
▶ Click to reveal
Col 1 (1s): 0 + 1 + carry 0 = 1 → write 1, carry 0 Col 2 (2s): 0 + 0 + carry 0 = 0 → write 0, carry 0 Col 3 (4s): 1 + 0 + carry 0 = 1 → write 1, carry 0 Col 4 (8s): 1 + 1 + carry 0 = 10 → write 0, carry 1 Result so far (right 4 bits): ...0101
3
⬅ Columns 5–8 (Left Side)
Continue left with the carry from column 4. What carries chain through here?
▶ Click to reveal
Col 5 (16s): 1 + 1 + carry 1 = 11 → write 1, carry 1 Col 6 (32s): 0 + 0 + carry 1 = 1 → write 1, carry 0 Col 7 (64s): 0 + 1 + carry 0 = 1 → write 1, carry 0 Col 8 (128s): 1 + 0 + carry 0 = 1 → write 1, carry 0 No carry out of MSB → No overflow ✓
4
✅ Final Answer
Assemble the result and verify against the denary sum.
▶ Click to reveal
carry: 0 0 1 1 0 0 0 0 A=156: 1 0 0 1 1 1 0 0 B= 73: 0 1 0 0 1 0 0 1 ───────────────── result: 1 1 1 0 0 1 0 1 Binary result = 11100101 Denary verify: 128+64+32+4+1 = 229 ✓ (156+73=229 ✓)
✍️
4-mark questions: Examiners typically award — [1] carry row shown, [1] at least 4 correct carry bits, [1] correct result bits, [1] final answer stated. Even if you drop a carry, writing all your working means you can still score 3 out of 4.
05 · Exam-Style Questions

Cambridge-Style Practice

Show all working including the carry row — then reveal the marking scheme.

Question 1
3 marks
Add the following 8-bit binary numbers. Show the carry row clearly in your working.
A: 0 1 0 0 1 1 0 1 (77) B: 0 0 1 1 0 1 1 0 (54)
✓Carry row shown with correct carries: 00001110 (carries in columns 2, 3, 4)[1]
✓At least 6 of 8 result bits correct[1]
✓Correct final result: 01111111 (77 + 54 = 131 — verify: 64+32+16+8+4+2+1=127… wait: 01111111 = 127, but 77+54=131 = 10000011)[1]
ℹWorking: 77=01001101, 54=00110110. Carries: col1:0+1=1, col2:0+1=1 carry0, col3:1+1=10 write0 carry1, col4:1+0+c1=10 write0 carry1, col5:0+1+c1=10 write0 carry1, col6:0+1+c1=10 write0 carry1, col7:1+0+c1=10 write0 carry1, col8:0+0+c1=1. Result: 10000011 = 131 ✓
Question 2
4 marks
A student adds two binary numbers and produces the result below. Identify and explain the error. State what should have been written.
A: 1 0 1 1 0 0 1 0 (178) B: 0 1 1 0 1 1 0 1 (109) carry: 0 1 1 0 1 0 0 0 ───────────────── R: 0 0 0 1 1 1 1 1 ← student's answer
✓The student's carry row is incorrect — they have missed the carry out of the MSB column[1]
✓Correct carry row: 11101100 (carry out of MSB = 1)[1]
✓Correct 8-bit result bits would be 00011111, but this is wrong because 178 + 109 = 287 > 255[1]
✓Overflow has occurred — a carry was produced out of the MSB. The result cannot be correctly stored in 8 bits.[1]
Question 3
1 mark
In binary addition, what is the result of 1 + 1 + 1 (one plus one plus a carry of one)? State the bit written and the carry generated.
✓Write 1, carry 1 — because 1+1+1 = 3 = 11 in binary (write the units bit = 1, carry the twos bit = 1)[1]
Question 4
3 marks
Add the 8-bit binary numbers 11001000 (200) and 01000110 (70). State whether overflow occurs and justify your answer.
✓Carry row: 11001000 (carries out of cols 4,5,6,7 and MSB)[1]
✓8-bit result: 00001110 (the carry out of MSB is lost)[1]
✓Overflow occurs — 200 + 70 = 270 which exceeds 255. A carry was produced out of the MSB column. The stored result (00001110 = 14) is incorrect.[1]
06 · Live Scored Quiz

5-Question Challenge

5 questions on binary addition rules, carries, and overflow. Earn your Topic 3 badge!

Score:
0 / 5
🥉
Topic 3 Complete — Binary Adder Certified!
+50 XP · Chapter 1 · Data Representation