I have a friend who occasionally brings math problems to me to solve. Once in a while, he would bring a piece of paper, sit beside me and begin to scribble a few lines of algebra that “prove” that 1=2 or that 0/0=3.

Knowing that I am a math enthusiast, he often wants to test me and show me through any means necessary that math is broken. But every time, it isn’t math that is broken, it is his math that is broken.

Most of the time, his so-called “proofs” employed a division by zero and that was the problem. However, the last time he brought me a “proof”, everything was different. I couldn’t see any division by zero shenanigans and everything looked correct. Yet, he had managed to prove that 1=-1. I began to freak out. Did he finally manage to do it? Is math really broken?

Here’s what he did:

1=√1=√(-1)(-1)=√(-1)√(-1)=(i)(i)=i²=-1

And by so doing, he managed to prove that 1=-1.

It turns out that he violated a math rule in order to make this happen. Before I reveal what it is and why it’s so important, I suggest you try and figure it out yourself.

The Problem with The proof

In a nutshell, the problem with what my friends wrote was the portion where he stated that √(-1)(-1)=√(-1)√(-1).

He thought he was obeying a rule about multiplying square roots of numbers when he actually wasn’t. He was trying to apply the formula √(a)(b)=√(a)√(b). It’s called the Product Rule for Radicals.

Here’s where attentiveness to detail is important. The product rule is guaranteed to work only when both values a and b are non-negative. When this rule is stated in math texts, there is a preamble that makes this clear. It is stated that “if a and b are non-negative real numbers” then √(a)(b)=√(a)√(b).

The rule isn’t guaranteed to work if both numbers are negative, and that’s exactly why my friend was able to find a case where the product rule “failed”. He hadn’t disproved the product rule, he just misapplied it.

In fact, if one isn’t aware of this, it is possible to use the same approach to prove that x=-x for any positive real number x(in other words, all positive numbers are equal to their negative counterparts). That’s exactly why a and b need to be suitably constrained to avoid math from breaking.

Leonhard Euler explored this concept in his 1770 book The Elements of Algebra.

What’s the Lesson here?

This particular math problem taught me a valuable lesson: it’s extremely important to pay attention to the preambles and hypotheses of rules, laws and theorems. The importance of this lesson cannot be overemphasized.

When pressure is placed on students, it can get really tempting to just memorize the formulas in the theorems and ignore the hypotheses. The danger is that such behaviour can come back to bite us when a friend brings us a proof that 1=-1.

The good news is that math isn’t broken…yet. It’s just our understanding and memory that might be.