How to do long addition
Long addition, also called column addition, is the written way to add numbers too big to add in your head. You add one column at a time, so every step is a small sum you already know.
The rule
Line the numbers up on the right and add one column at a time, starting with the ones. If a column adds up to 10 or more, write the ones digit and carry the tens into the next column on the left.
How to do it
- Write the numbers one under the other, lined up on the right, ones under ones and tens under tens. Draw a line underneath.
- Add the digits in the ones column. If the total is 9 or less, write it under the line.
- If the total is 10 or more, write only its ones digit under the line, and write the tens digit small above the next column to the left. That is the carry.
- Move one column left and add its digits, plus any carry sitting above it.
- Keep going until every column is done. If there is a carry left over at the end, it goes in a new column of its own on the left.
Worked examples
What is 324 + 152?
| 3 | 2 | 4 | |
| + | 1 | 5 | 2 |
| 4 | 7 | 6 |
4 + 2 = 6, 2 + 5 = 7 and 3 + 1 = 4. No column reaches 10, so there is nothing to carry, and 324 + 152 = 476.
What is 478 + 356?
| 1 | 1 | ||
| 4 | 7 | 8 | |
| + | 3 | 5 | 6 |
| 8 | 3 | 4 |
8 + 6 = 14, so write 4 and carry 1. 7 + 5 + 1 = 13, so write 3 and carry 1. 4 + 3 + 1 = 8. 478 + 356 = 834.
What is 765 + 487?
| 1 | 1 | |||
| 7 | 6 | 5 | ||
| + | 4 | 8 | 7 | |
| 1 | 2 | 5 | 2 |
Every column carries, and the last carry has nowhere to go but a new column on the left. 7 + 4 + 1 = 12, so the answer has four digits. 765 + 487 = 1252.
What is 586 + 79 + 47?
| 2 | 2 | ||
| 5 | 8 | 6 | |
| 7 | 9 | ||
| + | 4 | 7 | |
| 7 | 1 | 2 |
Three numbers, and two of them shorter. The ones make 6 + 9 + 7 = 22: write 2 and carry 2, not 1. The tens make 8 + 7 + 4 + 2 = 21: write 1 and carry 2 again. Then 5 + 2 = 7, and 586 + 79 + 47 = 712.
Expanded column addition, and why it is the same thing
Many British schools teach an expanded method first. You add each column on its own line, writing the whole total, and add those at the end: for 478 + 356, the ones make 8 + 6 = 14, the tens make 70 + 50 = 120 and the hundreds make 400 + 300 = 700. Then 14 + 120 + 700 = 834.
Long addition is the same sums written shorter. Instead of writing 14 in full, you keep the 4 and carry the 1 ten into the tens column straight away. Fewer lines, the same additions underneath.
Adding in your head
For small sums a mental method is often quicker. To add 38 + 25, add the tens and then the ones (30 + 20 = 50, 8 + 5 = 13, 50 + 13 = 63), or round and adjust: 40 + 25 = 65, then take away the 2 you added, which is 63. Long addition is for when there are too many digits to hold in your head.
Why it works
Each column holds one place value: ones, tens, hundreds. Adding a column adds things of the same size, so in 478 + 356 the tens column is 7 tens and 5 tens, not 7 and 5.
A carry is ten of one place swapped for one of the next. 8 + 6 is 14 ones, which is 1 ten and 4 ones. The 4 ones stay in the ones column, and the 1 ten moves to the tens column, where it belongs.
That is also why you start on the right. A carry always moves left, so a column is only finished once everything to its right has been added. Start on the left and you have to go back and change digits you already wrote.
A quick estimate catches most slips: 478 + 356 is close to 500 + 350 = 850, so 834 is believable and 734 or 8314 are not.
Where mistakes happen
- Forgetting the carry. Write it above the next column as soon as you make it, so it is there when you get to that column.
- Writing the whole column total. In 478 + 356 the ones make 14, but only the 4 goes under the line; writing 14 there pushes every other digit out of place.
- Lining the numbers up on the left. With 586 + 79, the 9 goes under the 6, not under the 5, so that every column holds one place value.
- Carrying 1 when the column made 20 or more. Adding three or more numbers can give a total like 22, and then the carry is 2.
Questions people ask
Does it matter which number goes on top?
No. 478 + 356 and 356 + 478 give the same answer, and so does any order of three or more numbers. Putting the longest number on top makes the columns easier to line up.
How do I check my answer?
Estimate first by rounding the numbers, and the answer should be close. For an exact check, take one of the numbers away from the answer and you should get the other. Long subtraction does that in columns too.
Why do you start from the right?
Because carries move to the left. Starting with the ones means each carry is ready before you reach the column it belongs to.
What if the numbers have different lengths?
Line them up on the right, ones under ones, and treat any empty place as a 0. In 586 + 79 the hundreds column is just the 5, plus any carry.