Long multiplication (column multiplication) is a method for finding the product of large numbers by splitting one of the numbers into its digits, multiplying by each digit separately, and shifting each answer row to the left according to the digit's place value — and finally adding all the rows together. For example 34 × 27 splits into 34 × 7 plus 34 × 20, and you add: 238 + 680 = 918. Once you understand the shifting principle, long multiplication becomes a clear, mechanical process, however many digits the numbers have.
What is long multiplication and why do we need it?
Long multiplication is the method you learn for multiplying numbers with two or more digits, when it is not possible (or too hard) to work it out in your head. The method is based on a simple idea: every two-digit number is really a sum of tens and ones — for example 27 is 20 + 7 — so the multiplication can be split into two smaller, simpler multiplications, which are then added. It is exactly the same principle that lies behind the distributive law (a × (b + c) = a × b + a × c), except that in long multiplication the calculation is set out in a column layout that is easy to follow without losing digits.
The method is usually taught in Year 5 of the national curriculum (multiplying up to four digits by a two-digit number), after pupils have learnt to multiply a two- or three-digit number by a single digit in Year 4 and are secure with the times tables and column addition. It is the basis for all multiplication of bigger numbers later on — three-digit multiplication, multiplying decimals and algebraic expansion at secondary school all work on exactly the same principle of splitting and shifting.
Long multiplication step by step: 34 × 27
Let's go through 34 × 27 step by step, exactly as you would write it on paper.
- Step 1 — set out the calculation in columns: write 34 above 27, with the digits lined up by place value (ones under ones, tens under tens). Draw a line under the two numbers.
- Step 2 — multiply by the ones digit (7): multiply the whole of 34 by 7, just like ordinary short multiplication: 7 × 4 = 28, write 8 and carry 2. Then 7 × 3 = 21, plus the 2 we carried = 23. The first row: 238.
- Step 3 — multiply by the tens digit (2): move to the tens digit of 27, which is 2 — but it really represents 20, not 2. So the second row starts from the tens place: write a 0 in the ones (as a place holder), then multiply 34 × 2: 2 × 4 = 8, 2 × 3 = 6. The second row: 680.
- Step 4 — add the two rows: line up 238 and 680 one under the other (with the 0 of 680 sitting in the ones), and add in columns: 238 + 680 = 918.
- Step 5 — check: you can verify by rounding — 34 is close to 30 and 27 is close to 30, so 30 × 30 = 900 — very close to 918, so the answer is sensible.
| Step | Operation | Result |
|---|---|---|
| 1 | 34 × 7 (the ones digit) | 238 |
| 2 | 34 × 20 (the tens digit, shifted left) | 680 |
| 3 | Add the rows: 238 + 680 | 918 |
Another example: 156 × 43
For a three-digit calculation you add another row, but the principle is exactly the same. Multiply 156 by each digit of 43 separately, shift each row according to the digit's place, and add at the end.
- 156 × 3 (ones) = 468 — first row, no shift.
- 156 × 40 (tens) = 6,240 — second row, shifted one place to the left (or with a 0 in the ones).
- Add: 468 + 6,240 = 6,708.
The more digits there are in the bottom number, the more middle rows there are — but the process is always the same: multiply by a single digit, shift according to its place, and add all the rows at the end.
Common mistakes in long multiplication
- Forgetting the shift — writing the second row without moving one place to the left, which turns 34 × 20 into 34 × 2 by mistake. This is the most common error.
- Carrying wrongly — forgetting to add the number carried from the previous multiplication to the next digit.
- Mixing up the order of the digits — multiplying by the tens before the ones, which causes confusion with the shift. Always start with the ones digit at the bottom.
- A mistake in the final addition — after the rows are right, pupils sometimes slip up in the simple addition of them. It is worth checking the addition separately.
- Columns not lined up — writing the digits out of line, which makes you add a ones digit to a tens digit by mistake.
Practice
The best way to practise long multiplication is to build up gradually — first two-digit by one-digit (like 34 × 7), then two-digit by two-digit (like 34 × 27), and only then three-digit. An interactive long multiplication calculator shows all the steps exactly as described here, so you can check a calculation you have already done yourself and see exactly where the mistake was.
Frequently asked questions
Long multiplication explained — what is the basic principle?
Split one of the numbers into its digits (tens and ones), multiply the other number by each digit separately, shift each answer row to the left according to the digit's place (tens shifted one place, hundreds two places), and add all the rows at the end.
How do you do long multiplication for 34 × 27?
Multiply 34 × 7 = 238 (first row), then 34 × 2 = 68 but shifted one place to the left because it is tens, that is 680 (second row), and add 238 + 680 = 918.
Why do you shift the second row in long multiplication?
Because the second digit of the bottom number represents tens, not ones. Multiplying by it is really multiplying by 20 (for example) and not by 2, so the result is ten times bigger and needs to move one place to the left to show that correctly.
What is the difference between long multiplication and the column method?
They are two names for exactly the same method — multiplying numbers with two or more digits by setting them out vertically (in columns), multiplying digit by digit, shifting by place value, and adding the rows. 'Long multiplication' and 'the column method' are used interchangeably in schools in England; the grid method is a different (earlier) layout of the same idea.
Type in a calculation and see all the steps exactly as in the guide — great for checking homework.
Try the long multiplication calculator ←