2.2 Concepts of division of whole numbers

Section overview

• Division
• Division into Zero (Zero As a Dividend: $\frac{0}{a}$ , )
• Division by Zero (Zero As a Divisor: $\frac{0}{a}$ , )
• Division by and into Zero (Zero As a Dividend and Divisor: $\frac{0}{0}$ )
• Calculators

Division

Division is a description of repeated subtraction.

In the process of division, the concern is how many times one number is contained in another number. For example, we might be interested in how many 5's are contained in 15. The word times is significant because it implies a relationship between division and multiplication.

There are several notations used to indicate division. Suppose $Q$ records the number of times 5 is contained in 15. We can indicate this by writing

$\underset{\text{5 into 15}}{\underbrace{\begin{array}{c}\hfill Q\\ \hfill 5\overline{)15}\end{array}}}$ $\underset{\text{15 divided by 5}}{\underbrace{\frac{15}{5}=Q}}$

$\underset{\text{15 divided by 5}}{\underbrace{15/5=Q}}$ $\underset{\text{15 divided by 5}}{\underbrace{15÷5=Q}}$

Each of these division notations describes the same number, represented here by the symbol $Q$ . Each notation also converts to the same multiplication form. It is $\text{15}=5\times Q$

In division,

Dividend

Divisor

Quotient

$\begin{array}{c}\hfill \text{quotient}\\ \hfill \text{divisor}\overline{)\text{dividend}}\end{array}$

$\frac{\text{dividend}}{\text{divisor}}=\text{quotient}$

$\text{dividend}/\text{divisor}=\text{quotient}\text{dividend}÷\text{divisor}=\text{quotient}$

Sample set a

Find the following quotients using multiplication facts.

Practice set a

Use multiplication facts to determine the following quotients.

Division into zero (zero as a dividend: $\frac{0}{a}$ , $a\ne 0$ )

Let's look at what happens when the dividend (the number being divided into) is zero, and the divisor (the number doing the dividing) is any whole number except zero. The question is

What number, if any, is $\frac{0}{\text{any nonzero whole number}}$ ?

Let's represent this unknown quotient by $Q$ . Then,

$\frac{0}{\text{any nonzero whole number}}=Q$

Converting this division problem to its corresponding multiplication problem, we get

$0=Q\times (\text{any nonzero whole number})$

From our knowledge of multiplication, we can understand that if the product of two whole numbers is zero, then one or both of the whole numbers must be zero. Since any nonzero whole number is certainly not zero, $Q$ must represent zero. Then,

$\frac{0}{\text{any nonzero whole number}}=0$

Zero divided by any nonzero whole number is zero

Division by zero (zero as a divisor: $\frac{\mathrm{a}}{0}$ , $a\ne 0$ )

Now we ask,

What number, if any, is $\frac{\text{any nonzero whole number}}{0}$ ?