Dividing By Zero Is Illegal/Undefined-WHY?

  • Development of mathematics, Closer and closer, Number system, Positive number, Division by zero is undefined, Negative infinity

Your math teachers have undoubtedly told you that dividing by zero is illegal—that is, dividing by zero is “undefined,” and we should avoid this action.

But why is dividing by zero such a crime in mathematics?

Let’s take one example and investigate:

Consider the following expression:         a + b = c.
Let's do some algebra and let’s see what happens.
Subtract the c on both sides:         a + b – c = 0.
Now, let’s multiply both sides by 3, noticing that anything times zero is zero:         3(a + b – c) = 3 ∙ 0 = 0.
Instead of using 3, we could also have done this for another number, say, 4:         4(a + b – c) = 4 ∙ 0 = 0.
Okay, so far, so good. Let's write an easy identity:         0 = 0.
Since both of these expressions are equal to 0, we can equate them as:         3(a + b – c) = 4(a + b – c).
Both sides share the same factor:         a + b – c ;
so let's simplify here and divide both sides of the above equation by this common factor:         (a + b – c)
This leaves us with     3 = 4, which is an absurdity!

What just happened?
In fact, the same steps can be used to show that any number equals any other number, not just 3 = 4. Of course, this is all wrong!

Where did we err?
We divided both sides of the equation by      a + b – c.
Notice that      a + b – c = 0,
So actually we divided by zero. That is where the fault lies.

It follows that if dividing by zero can lead to nonsense like      3 = 4,
then there is good reason why this division is forbidden.

Let’s consider another example that can be used to emphasize this point is the following:

Begin by letting:         a = b.
Then multiply both sides of this equality by a to get:         a2 = ab.
Then subtract b2 from both sides of this equality to get:         a2 – b2 = ab – b2.
This can be factored as follows:         (a + b)(a – b) = b(a – b).
By dividing both sides by (a – b), we get:         a + b = b.
However, since a = b, by substitution, we can get:         2b = b.
If we now divide both sides by b, we get the ridiculous result:         2 = 1.
Perhaps by now you will see why this happened. At the point that we divided both sides of the equality by (a – b), we actually divided by zero, [since a = b so (a-b)=0].


Here you have further evidence why dividing by zero is outlawed in mathematics!