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 x, where x are positive numbers, approach 0), the quotients of 1/x are getting larger and larger. On the other hand, as x approaches 0 from the left (as x, where x are negative numbers, approach 0), 1/x is getting smaller and smaller. Hence, there is no single number that 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.
- 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