Did you find the flaw in the logic?

Problems like this come up all the time, and every single one of them has the same fatal flaw: *division by zero.*

When we think about dividing, we usually state the problem as, “What is 48 divided by 12?” But a better way to phrase that question is, “How many groups do I get if I split 48 up into sets of 12?”

How do you split 48 up into sets of 0? Well, you can’t. The idea doesn’t make sense, and this isn’t always obvious when we phrase the question like, “What is 48 divided by zero?”

Rote memorization tells us that dividing by zero doesn’t work, but that means we don’t often look for how it becomes a problem in actual mathematics.

In the proof above, the problematic leap in logic comes right between these two lines.

**(A + B)(A – B) = B(A – B)
**This is some fun factoring!

**A + B = B
**Canceling common factors.

Canceling is such a basic tool that we often forget to write out an important step involving division.

Since we divided both sides by the same term and that term shows up in both numerators, we can cancel them, leaving the problematic **A + B = B**.

To see why this is an error, we need to plug in a number. Both values are the same, so let’s go with 7:

Both of these fractions have zero on the bottom. This equation has become illogical, but without pausing to appreciate why, it’s easy to skip this step and move on to a false conclusion.