Long multiplication extends tables work so that numbers bigger than 10 can be multiplied without using a calculator. There are a number of ways to do this.
The traditional method is demonstrated in the example below. This method is very versatile and can handle decimals as well as whole numbers. In the box on the right you can enter your own multiplications. Watch as the solution unfolds step by step.
Let's look at doing the sum 12 × 394, which was randomly generated when you loaded the page.
Of course, we could simply keep adding 394s together until we have 12 lots of 394, but that could take a very long time. Instead, we use the following method:
Step 1: Set the multiplication out as follows.
Note that the number with the smaller number of digits goes at the bottom.
Step 2: Multiply 394 by 2.
The result of 2 × 394 is shown in bold.
Step 3: Next, multiply 394 by 10. This is the same as multiplying 394 by 1 and by 10. We place a zero to the right and then write down the result of 1 × 394.
The result of 1 × 394 is shown in bold and the additional zero has been shown in blue.
Step 4: Finally, add these two rows together to give the final answer.
The final answer for 12 × 394 is 4728.
These techniques can be extended to numbers with any number of digits and to numbers involving decimals. For example, if the sum were 1.2 × 3.94, notice that there are 3 digits after the decimal point in total in the sum.
The answer would also have three digits after the decimal point, so instead of 4728,
1.2 × 3.94 = 4.728
Using the same rules for numbers with decimal points:
1.2 × 39.4 = 47.28
12 × 3.94 = 47.28
1.2 × 0.394 = 0.4728
If you refresh this page or press F5, a different long multiplication will be generated. We suggest you try this a number of times and then enter your own in the box on the right until you are familiar with the method.
Let's look at doing the sum 39 × 164, which was randomly generated when you loaded the page.
Of course, we could simply keep adding 164s together until we have 39 lots of 164, but that could take a very long time. Instead, we use the following method:
Step 1: Set the multiplication out as follows.
Note that the number with the smaller number of digits goes at the bottom.
Step 2: Multiply 164 by 9.
The result of 9 × 164 is shown in bold.
Step 3: Next, multiply 164 by 30. This is the same as multiplying 164 by 3 and by 10. We place a zero to the right and then write down the result of 3 × 164.
The result of 3 × 164 is shown in bold and the additional zero has been shown in blue.
Step 4: Finally, add these two rows together to give the final answer.
The final answer for 39 × 164 is 6396.
These techniques can be extended to numbers with any number of digits and to numbers involving decimals. For example, if the sum were 3.9 × 1.64, notice that there are 3 digits after the decimal point in total in the sum.
The answer would also have three digits after the decimal point, so instead of 6396,
3.9 × 1.64 = 6.396
Using the same rules for numbers with decimal points:
3.9 × 16.4 = 63.96
39 × 1.64 = 63.96
Source: Mark Riedel, mathsonline.org
