Long Multiplication Calculator

What this calculator does

Long multiplication is the column method you learned at school for multiplying two bigger numbers by hand, the one where you build up a few partial answers and add them together. This tool does exactly that, and shows you the parts, not just the final number.

Give it two whole numbers and it lays out each partial product and the total, so you can follow the working or check your own. It runs right here in the browser.

Using the calculator

1. Type the first number, the multiplier.
2. Type the second number, the multiplicand.
3. Press Calculate.

It shows the partial products stacked up, then adds them for the final answer. Reset clears both boxes.

How long multiplication works

The whole idea is to break one number down by its digits, multiply by each piece, and add. You take the top number and multiply it by each digit of the bottom number in turn, one at a time.

The catch, and the bit people forget, is place value. A digit sitting in the tens column is not really a 4, it is a 40, so the partial product for it has to be shifted up by that amount (in practice, a zero on the end). Work through the bottom number's digits from the right, giving each partial product its proper place value, then add the whole stack. That sum is your answer.

A worked example, step by step

Take 23 × 45. The bottom number, 45, is really 40 and 5, so there are two partial products.

1. Multiply by the units digit, 5: 23 × 5 = 115.
2. Multiply by the tens digit, 4, and remember it is really 40: 23 × 40 = 920.
3. Add the two partial products: 115 + 920 = 1035.

So 23 × 45 = 1035. The tool sets it out the same way, the 115 and the 920 stacked with the total underneath, which is exactly the layout you would write by hand.

If one of your numbers is negative, work out the sizes the normal way and then put the minus sign on the final answer. A negative times a positive is negative.

Why splitting it up works

Breaking the bottom number apart is not a trick, it is just the distributive property in action. Multiplying by 45 is the same as multiplying by 40 and by 5 and adding the two results, because 45 is 40 plus 5. Written out, 23 × 45 = 23 × (40 + 5) = 23 × 40 + 23 × 5. That is why the partial products always add up to the right answer. Every long multiplication you have ever done was quietly using this.

Questions people ask

How do you do long multiplication?

Multiply the top number by each digit of the bottom number in turn, give each partial product its correct place value, then add them all up.

What are the partial products?

They are the separate results from multiplying by each digit, like 23 × 5 and 23 × 40. Added together, they give the final product.

Why do I add a zero for the tens digit?

Because a digit in the tens column stands for that many tens, not units. The 4 in 45 is really 40, so its partial product is ten times bigger, which the extra zero accounts for.

Does it work with negative numbers?

Yes. It multiplies the sizes and then applies the sign, so a negative times a positive comes out negative.

Why show the steps instead of just the answer?

So you can follow the method, check your own working, or use it to learn. Seeing the partial products is the whole point of long multiplication.

References

A note on where this comes from. Long multiplication is the standard place-value algorithm for multiplying whole numbers, and it works because of the distributive property: multiplying by a number is the same as multiplying by each of its place-value parts and adding the results. For further reading, see Multiplication algorithm.

1. The distributive property, a × (b + c) = a × b + a × c, which is why the partial products add up to the full result.
2. Place value, which sets how far each partial product is shifted before the parts are added.