GCSE · Computer Science · AQA · Spec 8525

Binary addition

You already know how to add in columns. Binary uses the same method, but 1 + 1 already needs a carry.

Binary addition · column by column

Watch a carry travel

Adding 00111011 and 00101111 (59 + 47). Start at the right-hand column and work left, one column per step.

Before each Next, predict the row: what gets written, and what gets carried?

Place	Top + bottom	Carry in	Total	Write	Carry out	Answer so far
1s	1 + 1	0	10	0	1	_______0

Output

Step 1: Start on the right. 1 + 1 makes two, and there's no single binary digit for two. So write 0 and carry 1 into the 2s column, where it's worth exactly the two you made.

1 / 8

Press Next for one column at a time, or Play to watch the carry travel left.

Step 1 of 8: Start on the right. 1 + 1 makes two, and there's no single binary digit for two. So write 0 and carry 1 into the 2s column, where it's worth exactly the two you made..

Why 1 + 1 isn't 2

What really happens in the column?

Two columns from a binary addition. The first holds 1 + 1. The next holds 1 + 1 plus a 1 carried in from the first.

Which of these is closest to what you think happens?

The first column gives 2 and the next gives 3. Adding is adding, whatever the number system.The first column gives 0, because 1 + 1 is too big for binary. Nothing needs to be carried.The first gives 10 and the next gives 111, because you write down all three 1s.The first gives 10: write 0, carry 1. The next gives 11: write 1, carry 1.

How sure are you?

I'm sureI think soI'm guessing

The first column gives 2 and the next gives 3. Adding is adding, whatever the number system.

This is a common misconception

The amounts are right: those columns do hold two and three. But binary can't write them as 2 and 3, because it only has the digits 0 and 1. Think about denary: there's no single digit for ten, so 9 + 1 is written 10, meaning one ten and no units. Binary runs out after 1 instead of after 9. So two is written 10 (one 2 and no 1s), and three is 11 (one 2 and one 1).

The first column gives 0, because 1 + 1 is too big for binary. Nothing needs to be carried.

This is a common misconception

The column does show 0, but the two hasn't vanished. It moves into the next column as a carried 1, where it's worth a whole 2. If you write the 0 and forget the carry, you throw away a value you really added. The next column comes out wrong, and often more columns after it.

The first gives 10 and the next gives 111, because you write down all three 1s.

This is a common misconception

The first column is right. But 111 in binary is seven, and three 1s are only worth three. Three is 11: one 2 and one 1. A column holds only one digit, so write the right-hand 1 of 11 in the column and carry the left-hand 1 to the next column.

The first gives 10: write 0, carry 1. The next gives 11: write 1, carry 1.

This matches the evidence

Exactly. Binary works like denary, but with only two digits. A column that makes two writes 0 and carries 1, and a column that makes three writes 1 and carries 1. Add the two easy facts, 0 + 0 = 0 and 0 + 1 = 1, and you have every fact binary addition ever needs.

Four facts run every binary sum: 0 + 0 = 0, 0 + 1 = 1, 1 + 1 = 10 (write 0, carry 1) and 1 + 1 + 1 = 11 (write 1, carry 1). They come from the same counting rule as denary. Binary just runs out of digits after 1 instead of after 9.

Check it in denary

Prove the answer is right

8-bit binary 00000000 = 0 in denary

Denary0

Build 01101010, then try 11111111

The addition at the top gave 01101010 for 59 + 47. Tap the bits to build 01101010 and read its denary value. If it isn't 106, a carry went wrong somewhere. Then switch every bit on: 11111111 is 255, the biggest total 8 bits can hold, so every addition here stays at or below 255.

Your turn · three numbers

Fill in the missing columns

Add 01010100, 00001110 and 00010101.

Stack the three numbers so the columns line up: 01010100, 00001110 and 00010101. Start in the 1s column on the right.
1s and 2s columns: each holds a single 1 and nothing is carried in. Write 1 in each, carry nothing. Answer so far: …11
missing step

Which line is step 3?

4s column: 1 + 1 + 1 = 11. Write 1, carry 1. Answer so far: …1114s column: 1 + 1 + 1 = 10. Write 0, carry 1. Answer so far: …0114s column: 1 + 1 + 1 = 11. Write 1, carry nothing. Answer so far: …111

Spot the slip

Where did this answer go wrong?

A student adds 01101011 and 00010110. Pick the line where the working first goes wrong.

A student's working — which line goes wrong?

Line up the numbers: 01101011 + 00010110. Start in the 1s column.
1s column: 1 + 0 = 1. Write 1.
2s column: 1 + 1 = 10. Write 0.
4s column: 0 + 1 = 1. Write 1.
8s column: 1 + 0 = 1. Write 1.
16s column: 0 + 1 = 1. Write 1.
32s and 64s columns: 1 + 0 = 1 each time. Write 1, 1.
128s column: 0 + 0 = 0. Write 0.
Answer: 01111101

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

1 + 1 = 10

The same column method as denary. Binary just runs out of digits sooner, so the carry comes sooner.

What you need to know

Binary has only two digits, 0 and 1, so it carries as soon as a column makes two, just as denary carries when a column makes ten.
The four facts: 0 + 0 = 0, 0 + 1 = 1, 1 + 1 = 10 (write 0, carry 1) and 1 + 1 + 1 = 11 (write 1, carry 1).
Work from the right-hand column to the left, and add every carry into the next column along.
You'll be asked to add up to three binary numbers of up to 8 bits each. In those additions, no column has more than three 1s to add, and the answer fits in 8 bits.

The big picture

Binary addition is the column method you already use for denary: start at the right, add each column, and carry into the next column to the left. The only twist is that binary has just two digits, 0 and 1, so a column that makes two can't hold it. 1 + 1 becomes 10 (write 0, carry 1), and 1 + 1 + 1 becomes 11 (write 1, carry 1). Get the carries right and you can add up to three 8-bit numbers, then check the answer by converting to denary.

Key points

1Same method as denary: line up the columns, start on the right, carry to the left.

21 + 1 = 10: write 0 in the column, carry 1 to the next.

31 + 1 + 1 = 11: write 1 in the column, carry 1 to the next.

4A carry is worth one of the column it lands in, which is two of the column it came from.

5One dropped carry can make every bit to its left wrong.

6Check a sum in denary: convert the numbers and add them. The binary answer must convert to the same total.

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Worked example

Problem

Add the binary numbers 00111111 and 01011101. Give your answer in binary, then check it in denary.

⚠ Watch out

Writing the 0 from 1 + 1 = 10 and forgetting to carry the 1. That column's bit looks fine, but the next column is now missing its carry, and the error can run on through every column to the left.

🧠

Memory hook

Denary runs out of digits after 9, so 9 + 1 = 10. Binary runs out after 1, so 1 + 1 = 10. Same trick, just sooner.

✓

Check yourself

Add 00011100 and 00010100 in binary. Which column makes your first carry? Then convert all three numbers to denary to check your answer.

Flashcards

(9)

What are the four binary addition facts?

0 + 0 = 0, 0 + 1 = 1, 1 + 1 = 10 and 1 + 1 + 1 = 11.

Why does 1 + 1 give 10 in binary?

Binary has no digit 2. Two is one 2 and no 1s, so the column shows 0 and a 1 is carried to the 2s column, just as 9 + 1 in denary gives 0 and carries 1.

A column totals 10 or 11. What do you write, and what do you carry?

10: write 0, carry 1. 11: write 1, carry 1. The right-hand digit stays in the column and the left-hand 1 moves to the next column left.

Which column do you start a binary addition in?

The right-hand column (the 1s), then work left one column at a time, because carries only ever move left.

A carry lands in the 8s column. What is it worth?

8. It is one 8, made from two 4s in the column it came from.

What goes wrong if you drop a carry?

The answer comes out too small by the value of that carry, and every bit to its left can come out wrong.

How can you check a binary addition?

Convert each number to denary and add them. The binary answer converted to denary must give the same total.

What is the biggest value an 8-bit answer can hold?

11111111, which is 255 in denary.

Adding three binary numbers: what changes about the method?

Nothing. You add every bit in the column plus any carry. In the additions you'll be set, a column never has more than three 1s to add.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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Watch a carry travel

What really happens in the column?

Prove the answer is right

Fill in the missing columns

Where did this answer go wrong?

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