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How to Do Long Multiplication (Mathematics Lesson)

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 - including decimals - can be multiplied in this way.

How to Do Long Multiplication

General Approach:

  • Write the numbers you wish to multiply, one underneath the other, the larger number above the smaller number.




  • Select the right most digit of the bottom number (in the units column).

    Multiply this with each digit of the top number in turn, starting on the right and moving left.

    Write the answer of each multiplication below the line, under the column of the top digit.



    Note: If the answer is 10 or more, the tens digit will have to be carried to the column to the left.

  • Move one digit to the left on the bottom number (to the tens column).

    Because we are multiply by a tens digit, the answer we get from multiplying by this digit will be tens times more. To show this, write a 0 in the answer row (below the previous answer):




  • Multiply the tens digit of the bottom number with each digit of the top number in turn, starting on the right and moving left.

    Write the answer of each multiplication below the line, to the left of the zero. (Note: Each digit in the answer will be one column to the left of the digit of the top number that produced it.)



    Note: If the answer is 10 or more, the tens digit will have to be carried to the column to the left.

  • Use long addition to add the two numbers between the lines:




The solution to 43 × 21 is 903.

Another Real Example of How to Do Long Multiplication

Question: What is the answer to the multiplication below?:



  • Write the numbers you wish to multiply, one underneath the other, the larger number above the smaller number.




  • Select the right most digit of the bottom number (in the units column).

    Multiply this with each digit of the top number in turn, starting on the right and moving left.

    Write the answer of each multiplication below the line, under the column of the top digit.

    Note: If the answer is 10 or more, the tens digit will have to be carried to the column to the left.



  • Move one digit to the left on the bottom number (to the tens column).

    Because we are multiply by a tens digit, the answer we get from multiplying by this digit will be tens times more. To show this, write a 0 in the answer row (below the previous answer):




  • Multiply the tens digit of the bottom number with each digit of the top number in turn, starting on the right and moving left.

    Write the answer of each multiplication below the line, to the left of the zero. (Note: Each digit in the answer will be one column to the left of the digit of the top number that produced it.)



  • Use long addition to add the two numbers between the lines:




The solution to 25 × 14 is 350.

Another Real Example of How to Do Long Multiplication

It is possible to multiply decimals together.

The slider below shows another example of how to do long multiplication:
Interactive Widget
Here is an interactive widget to help you learn about long multiplication.
Interactive Test
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Note

PARTS OF A MULTIPLICATION



  • The parts you multiply together are factors.

  • The result of multiplying the numbers together is the product.

ORDER OF MULTIPLICATION

The order in which numbers are multiplied does not matter.

For example:



If the 2 and 3 are swapped around, the product is the same:



This is the commutative property of multiplication - changing the order does not change the result.

Note

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,



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.

PLACE VALUE 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:



Note: 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, 13, 14, 15, 16, 17, 18, 19

A 1 in the tens place value is 10 times bigger than a 1 in the units column.

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 product is greater than 9, a digit is placed below the column to its left: