What Is Long Multiplication? (Interactive Widget)
Use this interactive widget to see a step-by-step explanation of long multiplication.See similar widgets on long addition, long subtraction, and long division.
What Is Long Multiplication?
Long multiplication is a method for multiplying numbers.
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.
Step-by-Step:
1
Write the numbers you wish to multiply, one underneath the other.
2
Find the right most digit of the bottom number (in the units column).
3
Find the right most digit of the top number (in the units column).
4
Multiply the bottom digit (4) with the top digit (5).
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.
- Write the digit on the right underneath the column (beneath the line).
- Carry the left digit to the column to the left.
6
Move a digit to the left in the top number.
7
- Multiply the bottom digit (4) with the top digit (2).
2 × 4 = 8
- Add any carried numbers to the answer.
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.
- Write the digit on the right underneath the column (beneath the line).
- Carry the left digit to the column to the left.
9
Move a digit to the left in the top number.
There are no more digits to the left.
There are no more digits to the left.
10
Write the carried digit underneath the line.
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).
13
Find the right most digit of the top number (in the units column).
14
Multiply the bottom digit (1) with the top digit (5).
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.
16
Move a digit to the left in the top number.
17
Multiply the bottom digit (1) with the top digit (2).
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.
19
Move a digit to the left in the top number.
There are no more digits to the left.
There are no more digits to the left.
20
Use long addition to add the two numbers beneath the line.
Answer:
The solution to 25 × 14 is 350.Parts of a Multiplication
- 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,
123 consists of:
- 1 hundred
- 2 tens
- 3 units
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 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:
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 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.
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:
This page was written by Stephen Clarke.
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