# 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