How does long multiplication work?

Long multiplication multiplies the top number by each digit of the bottom number separately, starting from the ones digit and moving left. Each partial product is written below the previous one, shifted one position to the left for each successive digit. The partial products are then added together to give the final answer.

What are carries in long multiplication?

When two digits multiply to give a result of 10 or more, the tens digit is carried over to the next column on the left. For example, 7 times 8 equals 56: write down 6 and carry 5. The carry is added to the next column's product. This is the same carry mechanism as in regular column addition.

Can long multiplication handle large numbers?

The traditional long multiplication algorithm works for any size of integer as long as you have enough paper (or, in this case, screen space). This calculator shows the working for up to 6-digit by 6-digit multiplications. For very large numbers beyond JavaScript's safe integer range, use the large number calculator instead.

Long Multiplication Calculator

Long multiplication is the standard written algorithm for multiplying multi-digit numbers without a calculator. You learned it at primary school but may not have practised it since. The method works by multiplying the top number by each individual digit of the bottom number, one at a time from right to left, writing down the partial products with each successive one shifted one place to the left, then adding all the partial products together. The shifting accounts for the place value of each digit: the ones digit of the bottom number contributes once, the tens digit contributes ten times (hence shifting one left), the hundreds digit contributes a hundred times, and so on. Carries arise when multiplying two digits together gives a number of 10 or more; the tens part is noted above the next column and added in. This calculator takes any two positive integers and displays the full long multiplication working: the multiplicand, the multiplier, each partial product labelled by the digit that produced it, and the final sum. It also shows the traditional carry marks above the top row for each partial product row. The defaults use 137 times 23, a classic textbook example that gives 3,151.

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First number

Second number (multiplier)

3,151

137 x 23

Enter whole numbers. Long multiplication steps are shown for numbers up to 9 digits each.

How it works

The calculator takes the digits of the multiplier from right to left. For each digit d at position i (0-indexed from the right), it multiplies d by the multiplicand and appends i zeros to get the partial product. The partial products are summed to give the final answer. The display replicates the traditional layout: multiplicand on top, multiplier below a line, then each partial product with the appropriate offset, and a double line before the total.

Worked example

137 times 23. Ones digit of 23 is 3: 3 times 137 = 411. Tens digit is 2: 2 times 137 = 274, shifted one place left = 2740. Add partial products: 411 plus 2740 = 3,151. Working: 3 times 7 = 21, write 1 carry 2; 3 times 3 = 9 plus 2 = 11, write 1 carry 1; 3 times 1 = 3 plus 1 = 4. First partial product is 411. Then 2 times 7 = 14, write 4 carry 1; 2 times 3 = 6 plus 1 = 7; 2 times 1 = 2. Second partial product is 274. Total: 3,151. These match the defaults pre-filled above.

Related calculators

Large Number Calculator: arithmetic on arbitrarily large integers using string methods.
Cross Multiplication: solve proportions a/b = c/d for any unknown.
Completing the Square: rewrite a quadratic in vertex form.

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