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Division: Whole Numbers

Division is a mathematical operation, written using the symbol ÷ , that can be thought of in two ways: a ÷ b is the size of each group when a objects are divided into b groups of equal size, OR a ÷ b is the number of groups when a objects are divided into groups of b objects each.

For example, 20 ÷ 4 can be found by dividing 20 dots into 4 groups of equal size.

Math diagram

We find that each of the four groups contains 5 dots, so 20 ÷ 4 = 5 .

Alternately, we can find 20 ÷ 4 by forming groups of 4 dots each and then counting how many groups there are:

Math diagram

There are 5 groups.

Division is the inverse operation of multiplication. That is,

a ÷ b = c if and only if c × b = a .

Here a is called the dividend , b is called the divisor , and c (the result) is called the quotient .

Division by 0 is undefined; to see why, substitute b = 0 in the above equations. Since c × 0 = 0 no matter what the value of c , so a must also equal 0 ; and if a and b are both 0 , c can equal anything!

Division with whole numbers can result in a remainder. For example, if we divide 20 ÷ 6 by dividing 20 into groups of 6 , we get 3 groups with 2 left over:

Math diagram

Sometimes we write 20 ÷ 6 = 3 R 2 , where 2 is the remainder .

Or, we can write a single number answer as a fraction or decimal.

Long Division

To divide a multi-digit number by a one-digit number, we can use long division.

Example 1:

Divide. 496 ÷ 8

Place the dividend inside the division symbol, the divisor on the outside the division symbol.

8 496

Here, 8 cannot be contained in 4 , so consider the next digit also. There are 6 eights in 49 , so write 6 at the tens place of the quotient.

Multiply 6 by the divisor 8 and subtract.

6 8 49 6 48 _ 1

Now again 8 cannot be contained in 1 , so bring down the next digit 6 .

6 8 49 6 48 _ 16

There 2 eights in 16 , so write 2 at the ones place.

Multiply 2 by the divisor 8 and subtract.

6 2 8 49 6 48 _ 16 16 _ 0

To divide a multi-digit number by a multi-digit number, the process is similar.

Example 2:

Divide 1036 by 32 . Place the dividend inside the division symbol, the divisor on the outside the division symbol.

32 1036

Here, 32 cannot be contained in 1 , consider the next digit, still 32 cannot be contained in 10 . So, consider the next digit also. There are 3 thirty twos in 103 , so write 3 at the tens place of the quotient.

Multiply 3 by the divisor 32 and subtract.

3 32 103 6 96 _ 7

Now again 32 cannot be contained in 7 , so bring down the next digit 6 .

3 32 1036 96 _ 76

There 2 thirty twos in 76 , so write 2 at the ones place.

Multiply 2 by the divisor 32 and subtract.

Here the remainder is 12 .

3 2 32 103 6 96 _ 76 64 _ 12

Unlike addition and multiplication, for real numbers , the division operation is not commutative . That is, order matters: 40 ÷ 8 = 5 , but 8 ÷ 40 = 1 5 (a fractional value).

Similarly, division is not associative ; that is, grouping matters. For instance,

( 80 ÷ 10 ) ÷ 2 = 8 ÷ 2 = 4 , but

80 ÷ ( 10 ÷ 2 ) = 80 ÷ 5 = 16 .

 

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