What Is Long Multiplication?

Long multiplication is a method for multiplying numbers.

long multiplication Long multiplication involves writing the numbers to be multiplied one underneath another, so the digits are in columns. Many numbers of any length can be multiplied in this way.

A Real Example of How to Do Long Multiplication

Doing long multiplication is easy.

Question

Multiply the numbers below. 25 times 14

Step-by-Step:

1

Write the numbers you wish to multiply, one underneath the other. 25_times_14_long_multiplication_step_1

2

Find the right most digit of the bottom number (in the units column). 25_times_14_long_multiplication_step_2

3

Find the right most digit of the top number (in the units column). 25_times_14_long_multiplication_step_3

4

Multiply the bottom digit (4) with the top digit (5). 25_times_14_long_multiplication_step_4
5 × 4 = 20

5

Check if the answer from Step 4 is 9 or less: No. 20 is not 9 or less.
  • If No, the answer will have two digits. 25 times 14 long multiplication step 5 1
  • Write the digit on the right underneath the column (beneath the line). 25 times 14 long multiplication step 5 2
  • Carry the left digit to the column to the left. 25 times 14 long multiplication step 5 3

6

Move a digit to the left in the top number. 25_times_14_long_multiplication_step_6

7

  • Multiply the bottom digit (4) with the top digit (2). 25 times 14 long multiplication step 7 1
    2 × 4 = 8
  • Add any carried numbers to the answer. 25 times 14 long multiplication step 7 2
    8 + 2 = 10

8

Check if the answer from Step 7 is 9 or less: No. 10 is not 9 or less.
  • If No, the answer will have two digits. 25 times 14 long multiplication step 8 1
  • Write the digit on the right underneath the column (beneath the line). 25 times 14 long multiplication step 8 2
  • Carry the left digit to the column to the left. 25 times 14 long multiplication step 8 3

9

Move a digit to the left in the top number. 25_times_14_long_multiplication_step_9 There are no more digits to the left.

10

Write the carried digit underneath the line. 25_times_14_long_multiplication_step_10

11

Write a 0 on the right in a new row underneath the line.

12

Move a digit to the left in the bottom number (in the tens column). 25_times_14_long_multiplication_step_12

13

Find the right most digit of the top number (in the units column). 25_times_14_long_multiplication_step_13

14

Multiply the bottom digit (1) with the top digit (5). 25_times_14_long_multiplication_step_14
5 × 1 = 5

15

Check if the answer from Step 4 is 9 or less: Yes. 5 is 9 or less.
  • If Yes, write the number beneath the line, to the left of the 0. 25_times_14_long_multiplication_step_15

16

Move a digit to the left in the top number. 25_times_14_long_multiplication_step_16

17

Multiply the bottom digit (1) with the top digit (2). 25_times_14_long_multiplication_step_17
2 × 1 = 2

18

Check if the answer from Step 17 is 9 or less: Yes. 2 is 9 or less.
  • If Yes, write the number below beneath the line. 25_times_14_long_multiplication_step_18

19

Move a digit to the left in the top number. 25_times_14_long_multiplication_step_19 There are no more digits to the left.

20

Use long addition to add the two numbers beneath the line. 25_times_14_long_multiplication_step_20

Answer:

The solution to 25 × 14 is 350.

Lesson Slides

The slider below shows another real example of how to do long multiplication.

Parts of a Multiplication

product explained
  • The numbers you multiply together are factors.
  • The result of multiplying the numbers is the product.

The Order of Multiplication

The order in which numbers are multiplied does not matter. For example:
2 × 3 = 6
If the 2 and 3 are swapped around, the product is the same:
3 × 2 = 6
This is the commutative property of multiplication - changing the order does not change the result.

Digits and Place Value

Numbers consist of digits. In a decimal, the digits can take values 0 through to 9. The value of the digits depend on its place value. The place value is the place in the number where the digit is. Place values include hundreds, tens and units. For example, place_value_explained 123 consists of:
  • 1 hundred
  • 2 tens
  • 3 units
That is: 123_place_value Each place value is 10 times bigger than that to its right. A hundred is 10 times a ten, a ten is 10 times a unit. The same system applies to the right of the decimal place: place_value_after_decimal_explained

Place Value and Columns in Long Multiplication

Long multiplication relies on place value. The digits of the top number are multiplied by digits of the bottom number. The right-most digit of the bottom number is used first, then one to the left, then the next left. Because of place value, each digit to the left is 10 times bigger than the digit to its right. When the digit to the left of the bottom number is used, each answer will be 10 times bigger than the answers generated by the right-most digit of the bottom number. To signify this, a 0 must be added to the end of the answer: place_value_long_multiplication_1_zero Adding a 0 makes each answer 10 times bigger in place value (10 is 10 times bigger than 1, 200 is 10 times bigger than 20 etc.) When the next left digit of the bottom number is used, two 0s must be added: place_value_long_multiplication_2_zeroes

Place Value and Carrying

Digits in a decimal system go from 0 through to 9. The numbers 0 through to 9 can be written just using the units place value.
0, 1, 2, 3, 4, 5, 6, 7, 8, 9
To write numbers after 10, the tens place value must be used:
10, 11, 12...
A 1 in the tens place value is 10 times bigger than a 1 in the units column. Similarly, the numbers up to 99 use the tens and units place values. After 100, the hundreds place value also has to be used:
100, 101, 102...
where, 100 is 10 tens. Which ever place value we are at, once the digit in that place value becomes greater than 9, we need represent the larger number by placing digits in the place value to the left. carrying_place_value This is why when doing long multiplication, if the numbers in any column multiply up to be greater than 9, a digit is placed below the column to its left: carrying_12
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This page was written by Stephen Clarke.