# Division by Zero

In studying mathematics, you have probably heard that division of zero is** undefined**. What does this mean?

Since we do not know exactly what is the answer when a number is divided by **0**, it is probably reasonable for us to examine the quotient of a number that is divided by a number that is close to **0**.

If we look at the number line, the numbers close to **0** are numbers numbers between –**1** and **1**.

For instance, several positive numbers close to **0** and less than **1 **are **0.1, 0.01, 0.001** and so on. Similarly, negative numbers close to **0** but greater than **– 1** are –**0.1, -0.01, -0.001** and so on.

The table and the numbers below shows the quotient 1**/***x* when **1** is divided by * x*, where the

*x*’s are numbers close to

**0**.

In the graph, as ** x** approaches 0 from the right (as

**, where**

*x***are positive numbers, approach 0), the quotients of**

*x***1/**

**are getting larger and larger. On the other hand, as**

*x***approaches**

*x***0**from the left (as

**, where**

*x***are negative numbers, approach 0),**

*x***1/**is getting smaller and smaller. Hence, there is no single number that

*x***1/x**approach as

**x**approaches

**0**. For this reason, we say that

**1/0**is undefined.

A simple analogy would also let us realize that allowing division by **0** will violate an important property of real numbers. For example **8/4 = 2** because **2 x 4 = 8**. Assuming division of **0** is allowed**.** If **5/0 = n**, then **n x 0 = 5**. Now, that violates the property of a real number that any number multiplied by **0** is equal to **0**.

Since division by **0** yields an answer which is not defined, the said operation is not allowed.

**Related Posts**

- An Intituive Introduction to Limits
- The Definition of Undefined
- Is 0.999… really equal to 1?
- Counting the Infinite: A Glimpse at Infinite Sets
- Counting the Real Numbers