Division: Whole Numbers

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

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

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

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

There are $5$ groups.

Division is the inverse operation of multiplication. That is,

$a \div b = c$ if and only if $c \times b = a$ .

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 $c \times 0 = 0$ 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 $20 \div 6$ by dividing $20$ into groups of $6$ , we get $3$ groups with $2$ left over:

Sometimes we write $20 \div 6 = 3 R 2$ , 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. $496 \div 8$

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

$8496$

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.

$\begin{array}{l} \begin{matrix} 6 \\ 8496 \end{matrix} \\ \underline{48} \\ 1 \end{array}$

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

$\begin{array}{l} \begin{matrix} 6 \\ 8496 \end{matrix} \\ \underline{48} \\ 16 \end{array}$

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

Multiply $2$ by the divisor $8$ and subtract.

$\begin{array}{l} \begin{matrix} 62 \\ 8496 \end{matrix} \\ \underline{48} \\ 16 \\ \underline{16} \\ 0 \end{array}$

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.

$321036$

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.

$\begin{array}{l} \begin{matrix} 3 \\ 321036 \end{matrix} \\ \underline{96} \\ 7 \end{array}$

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

$\begin{array}{l} \begin{matrix} 3 \\ 321036 \end{matrix} \\ \underline{96} \\ 76 \end{array}$

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 $12$ .

$\begin{array}{l} \begin{matrix} 32 \\ 321036 \end{matrix} \\ \underline{96} \\ 76 \\ \underline{64} \\ 12 \end{array}$

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

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

$(80 \div 10) \div 2 = 8 \div 2 = 4, but$

$80 \div (10 \div 2) = 80 \div 5 = 16$ .

Division: Whole Numbers

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