How To Debunk Fake Math Proofs? - a = b; a*a = a*b; a*a - b*b = a*b - b*b; (a+b)*(a-b) = b*(a-b); a+b = b; 2b = b; 2 = 1

I recently came across a viral fake math proof, and it got me thinking. For the life of me, I could not figure out what makes such a phenomenon go viral.

Is it akin to a real-life magic trick, I wonder? We all know that a magic trick does not break the laws of physics that we are used to. However, there is a certain charm to it, for the end result appears to break the laws.

Some of us want to believe that extraordinary events are possible; events that break the laws of reality; events that are truly magic.

Some of us just want to look underneath the veil to figure out what the trick is all about and expose its underlying sleight of hand. I belong to this category.

Why don’t we start with the fake math proof in the title image?

The Oldest and Simplest Trick in the Book

We all know that 2 cannot be equal to 1. It is an illogical result. If someone simply claimed that 2 = 1, most of us would not bother holding a conversation with that person.

However, since the fake proof supposedly uses algebra to arrive at the (illogical) result, I guess that it has an air of seriousness to it. We are intrigued to look into the details — wondering where the trick lies.

To help debunk this fake proof, I have laid out the operation performed in each step below. Note that these steps were not provided in the original fake proof (I will revisit this point later).

How To Debunk Fake Math Proofs? — a = b; Muliplying by a on both sides -> a*a = a*b; Subtracting (b*b) from both sides -> a*a — b*b = a*b — b*b; Factoring out (a-b) on both sides -> (a+b)*(a-b) = b*(a-b); Dividing by (a-b) on both sides -> a+b = b; Substituting a = b -> 2b = b; Dividing by b on both sides -> 2 = 1
Fake math proof with labelled operations— Math illustrated by the author

Now, focus on the following step:

Factoring (a-b) out on both sides: (a+b)*(a-b) = b*(a-b)
Factoring out (a-b) — Math illustrated by the author

Since we started with the equation a = b, the expression (a – b) must equate to zero. Therefore, the above expression where both sides are multiplied by (a – b), essentially communicates to us that something multiplied by zero equals anything multiplied by zero.

This, by itself, is nothing that breaks the rules of math and is perfectly fine. It is what follows that subtly hides one of the oldest tricks in the book:

Division by (a-b) — Math illustrated by the author

We just established that the expression (a – b) must equate to zero. Therefore, when we divide by (a – b) on both sides, we are essentially dividing by zero on both sides.

In mathematics, division by zero is not allowed, as it leads to an undetermined result. To get a fundamental understanding of why this is and what goes on underneath, check out my essay that explores whether division by zero leads to infinity.

This is one of the oldest tricks in the book as far as fake math proofs are concerned: hide division by zero underneath a clever ruse.

There’s more going on here, though. Let us assume that division by zero did not take place and we somehow arrived at the result: a + b = b. This equation can be true only if ‘a’ equals zero. From the initial condition, we know that if ‘a’ equals zero, ‘b’ must also be equal to zero.

In other words, the equation applies to only one case (and is not a generalised result), which brings me to the next point. Algebra is not mere manipulation of symbols using mathematical operations. Algebra also involves strong reasoning.

A trick that fake math proofs often employ is to not mention any word of reasoning at all. Note that things started getting clearer as soon as I laid out each mathematical operation performed in each step.

Fake math proofs specifically avoid doing this so that they can better hide the rule-breaking steps inside operations that appear to be correct at first glance. When you write out the steps, things become clearer.

To go beyond algebra, let us look at a more advanced trick.

The Betrayal of Calculus

Take a look at the following mathematical proof:

How to debunk fake math proofs? — x*x = x + x+ … + x (x times); d(x*x)/dx = d(x + x+ … + x (x times))/dx; 2x = 1 + 1 + … + 1 (x times); 2x = x; 2 = 1
Proof using calculus — Math illustrated by the author

I know what you must be thinking; it makes me wonder too.

“Why do fake math proofs obsess over proving 2 equals 1?!”

I would say that this is a couple of notches more advanced than the fake math proof that (mis-)used algebra. You could write out all the mathematical steps for this one and still come out stumped.

The trick here lies in the phrasing on the right-hand side, where x is added to itself x times. There’s nothing wrong with this, as it simply deconstructs the multiplication operation that is involved in the squared exponentiation.

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But it tells just one side of the story. The opening condition that x² equals x added to itself x-times holds true only if x is a positive integer. The fake math proof skips laying out the “*conditions applied”.

While the left-hand side (x²) depicts a variable x, the right-hand side (x added to itself x times) treats ‘x’ as both a variable and a constant at the same time!

Calculus is a branch of mathematics that studies continuous change.

While the left-hand side is a continuously differentiable function, the right-hand side appears to be an arithmetic function where the variable also bounds the operation. Differentiating both sides and equating them is a gross misuse of calculus! And therein lies the sleight of hand.

Conclusion

While fake math proofs come in all shapes and sizes, they share a common pattern that you might spot. They all try to prove an illogical result using an approach that appears formal at first glance.

But when you look closer, you might spot that they leave out necessary information like specific conditions necessary for the opening statement to hold true or operations performed.

Typically, writing out the operation performed in each step is a good way to start cracking down on the rule-breaking step(s).

Any fake math proof is hiding at least one rule-breaking (illegal) step somewhere in there. The advanced ones just use higher layers of abstraction, while the simpler ones use tricks like division by zero to catch you out.

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