2.2 Concepts of division of whole numbers  (Page 2/2)

Letting $Q$ represent a possible quotient, we get

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

Converting to the corresponding multiplication form, we have

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

Since $Q\times 0=0$ , $(\text{any nonzero whole number})=0$ . But this is absurd. This would mean that $6=0$ , or $\text{37}=0$ . A nonzero whole number cannot equal 0! Thus,

$\frac{\text{any nonzero whole number}}{0}$ does not name a number

Division by zero is undefined

Division by and into zero (zero as a dividend and divisor: $\frac{0}{0}$ )

We are now curious about zero divided by zero $\left(\frac{0}{0}\right)$ . If we let $Q$ represent a potential quotient, we get

$\frac{0}{0}=Q$

Converting to the multiplication form,

$0=Q\times 0$

This results in

$0=0$

This is a statement that is true regardless of the number used in place of $Q$ . For example,

$\frac{0}{0}=5$ , since $0=5\times 0$ .

$\frac{0}{0}=\text{31}$ , since $0=\text{31}\times 0$ .

$\frac{0}{0}=\text{286}$ , since $0=\text{286}\times 0$ .

A unique quotient cannot be determined.

Indeterminant

$\frac{0}{0}$ Is indeterminant

Sample set b

Perform, if possible, each division.

Practice set b

Perform, if possible, the following divisions.

Calculators

Divisions can also be performed using a calculator.

Sample set c

Practice set c

Use a calculator to perform each division.

Exercises

For the following problems, determine the quotients (if possi­ble). You may use a calculator to check the result.

Exercises for review