Multiplication: Whole Numbers

Multiplication can be thought of as repeated addition. So, if you multiply a number a by another number b , this is the same as adding the number a over and over again b times. (Or adding b over and over again a times). For example:

3 × 5 = 5 + 5 + 5 = 15 3 × 5 = 3 + 3 + 3 + 3 + 3 = 15

Another way to think of whole number multiplication a × b is to visualize objects arranged in a rectangle, with a rows and b columns.

3 × 5

Note that there are 15 dots in the figure.

The Standard Algorithm

To multiply a multi-digit number by a one-digit number using the standard algorithm, write the two numbers on top of each other, with the ones digits vertically aligned and the multi-digit number on top.

127 × 3 _

Multiply the ones digit of the top number by the bottom number. Write down the ones digit of the result. If the result is greater than 10 , carry the tens digit, as you do when adding.

Here, 7 × 3 = 21 , so

1 2 2 7 × 3 _ 1

Now multiply the tens digit of the top number by the bottom number, and add the carried digit to the result. Here, 2 × 3 = 6 , and then we add 2 to get 8 . Since 8 is less than 10 , we don't have to carry this time.

1 2 2 7 × 3 _ 8 1

Finally, multiply the hundreds digit of the top number by the bottom number. Here, 3 × 1 = 3 .

1 2 2 7 × 3 _ 3 8 1

So, 127 × 3 = 381 .

To multiply two multi-digit numbers , write the number with more digits on top. For example, to multiply 29 by 543 , we write

543 × 29 _

First multiply the top number by the ones digit of the bottom number, as explained above. 3 × 9 = 27 , so write down the 7 and carry the 2 :

5 4 2 3 × 2 9 _ 7

4 × 9 is 36, plus 2 is 38 , so write down the 8 and carry the 3 :

5 3 4 2 3 × 2 9 _ 8 7

5 × 9 is 45 , plus 3 is 48 . There are no more digits to carry, so write down 48 .

5 2 4 2 3 × 2 9 _ 4 8 8 7

Next, we need to multiply the top number by the tens digit of the bottom number. Since we're actually multiplying by 20 , not by 2 , we write down a 0 as a place holder.

5 4 3 × 29 _ 4887 0

3 × 2 is 6 , so write down a 6 .

5 4 3 × 2 9 _ 4887 6 0

4 × 2 is 8 , so write down an 8 .

5 4 3 × 2 9 _ 4887 8 6 0

5 × 2 is 10 , and there are no more digits to carry, so write down the 10 .

5 4 3 × 2 9 _ 4887 10 860

The final step is to add the two results.

5 4 3 × 29 _ 4887 + 10 860 _ 13947

So, 543 × 29 = 13947 .

Like addition, multiplication is commutative for real numbers (that is, a × b = b × a ; order does not matter) and associative (that is, ( a × b ) × c = a × ( b × c ) ; grouping does not matter.) See Properties of Multiplication for more.

 

d:T7258,

Multiplication: Whole Numbers

Multiplication can be thought of as repeated addition. So, if you multiply a number a by another number b , this is the same as adding the number a over and over again b times. (Or adding b over and over again a times). For example:

3 × 5 = 5 + 5 + 5 = 15 3 × 5 = 3 + 3 + 3 + 3 + 3 = 15

Another way to think of whole number multiplication a × b is to visualize objects arranged in a rectangle, with a rows and b columns.

3 × 5

Math diagram

Note that there are 15 dots in the figure.

The Standard Algorithm

To multiply a multi-digit number by a one-digit number using the standard algorithm, write the two numbers on top of each other, with the ones digits vertically aligned and the multi-digit number on top.

127 × 3 _

Multiply the ones digit of the top number by the bottom number. Write down the ones digit of the result. If the result is greater than 10 , carry the tens digit, as you do when adding.

Here, 7 × 3 = 21 , so

1 2 2 7 × 3 _ 1

Now multiply the tens digit of the top number by the bottom number, and add the carried digit to the result. Here, 2 × 3 = 6 , and then we add 2 to get 8 . Since 8 is less than 10 , we don't have to carry this time.

1 2 2 7 × 3 _ 8 1

Finally, multiply the hundreds digit of the top number by the bottom number. Here, 3 × 1 = 3 .

1 2 2 7 × 3 _ 3 8 1

So, 127 × 3 = 381 .

To multiply two multi-digit numbers , write the number with more digits on top. For example, to multiply 29 by 543 , we write

543 × 29 _

First multiply the top number by the ones digit of the bottom number, as explained above. 3 × 9 = 27 , so write down the 7 and carry the 2 :

5 4 2 3 × 2 9 _ 7

4 × 9 is 36, plus 2 is 38 , so write down the 8 and carry the 3 :

5 3 4 2 3 × 2 9 _ 8 7

5 × 9 is 45 , plus 3 is 48 . There are no more digits to carry, so write down 48 .

5 2 4 2 3 × 2 9 _ 4 8 8 7

Next, we need to multiply the top number by the tens digit of the bottom number. Since we're actually multiplying by 20 , not by 2 , we write down a 0 as a place holder.

5 4 3 × 29 _ 4887 0

3 × 2 is 6 , so write down a 6 .

5 4 3 × 2 9 _ 4887 6 0

4 × 2 is 8 , so write down an 8 .

5 4 3 × 2 9 _ 4887 8 6 0

5 × 2 is 10 , and there are no more digits to carry, so write down the 10 .

5 4 3 × 2 9 _ 4887 10 860

The final step is to add the two results.

5 4 3 × 29 _ 4887 + 10 860 _ 13947

So, 543 × 29 = 13947 .

Like addition, multiplication is commutative for real numbers (that is, a × b = b × a ; order does not matter) and associative (that is, ( a × b ) × c = a × ( b × c ) ; grouping does not matter.) See Properties of Multiplication for more.

 

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