Division Involving Zero

• For any real number a, $\frac{a}{0}$ is undefined
• For any non-zero real number a, $\frac{0}{a}$ = 0
• $\frac{0}{0}$ is indeterminate

For any real number a, $\frac{a}{0}$ is undefined

Dividing any real number by zero is undefined and sometimes taken as infinity. Division is splitting into equal parts or groups.

Let us consider an example: Suppose there are 12 ice cream cups and 4 friends want to share them. How do they divide the ice cream cups?

$\frac{12}{4}$ = 3; So they get 3 each: Now, let us try dividing the 12 ice cream cups among zero people. How much does each person get?

Does that question make sense? No, it doesn't.

We can't share among zero people, and we can't divide by 0.

Suppose we could get some number k by dividing any real number a by zero

Let us assume $\frac{a}{0}$ = k. Then k × 0 = a. There is no such number k which when multiplied by zero will give a. So k does not exist and therefore $\frac{a}{0}$ is said to be undefined.

For any non-zero real number a, $\frac{0}{a}$ = 0

If zero is divided by any non-zero real number a, we get 0 as the result. If zero items are divided among a number of people, share got by each person will be zero only

$\frac{0}{0}$ is indeterminate

Division of zero by zero is a quantity that cannot be found and is called indeterminate.

Find the value of $\frac{0}{5}$

Solution

Step 1:

For example, $\frac{3}{4}$ = 3 × $\frac{1}{4}$ .

Step 2:

Similarly, $\frac{0}{5}$ = 0 × $\frac{1}{5}$ = 0

as the product of zero and any number is zero.

Evaluate $\frac{7}{0}$

Solution

Step 1:

By definition, division of any number by zero is not defined.

Step 2:

So, $\frac{7}{0}$ is not defined.

Evaluate $\frac{0}{13}$

Solution

Step 1:

Zero divided by any number is zero.

Step 2:

So, $\frac{0}{13}$ = 0