Is 2=-2? Incorrect Mathematical Proofs

Introduction

We all have that moment where we find something that just… doesn’t work. For me, I had found a proof that 2 was equal to -2. Which can’t be right. If you don't understand it, that’s fine, I’ll explain it.

Proof

2 = 2

2 = 1 + 1

2 = √1 + √1

2 = √(-1 * -1) + √(-1 * -1)

2 = √-1 * √-1 + √-1 + √-1

2 = i * i + i * i

2 = -1 + -1

2 = -2

Obviously, this proof isn’t correct. So let’s look at this proof, then look at where I’ve gone wrong.

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Explaining the proof

I think we can all agree that 1+1 = 2, which is what the proof is based off of. The proof relies on defining what “one” is equal to. √1 = 1 because 1² = 1 (1*1) so 2 = √1 + √1. Now -1 * -1 = 1 because two negatives cancel out and make a positive. Since 1 *1 = 1, -1 * -1, two negatives cancel out, is 1*1 which is 1. Since we put brackets around (-1 * -1) we can use this to replace the 1 in the √, so it’s just 2 = √(-1 * -1) + √(-1 * -1).

The 5th and 6th line are where it gets a little more complicated. We’ve already established that 2 = √(-1*-1) + √(-1*-1). Now we can apply the rule that √(x * y) = √x * √y which can be proven as such:

√(x*y) = √x * √y

(√(x*y))² =(√x * √y)²

xy = xy

For example √(4*9) = √4 * √9 = 2*3 = 6. This is true because √(4*9) = √36 = 6 This means that √(-1 * -1) = √-1 * √-1. So 2 = √-1 * √-1 + √-1 * √-1.

√-1 has no real answer (-1² = 1), so we must turn to the complex realm. Mathematicians defined √-1 as equal to i.

Multiplication of i is a rotation 90º counterclockwise on this imaginary plane so i*i=-1. I won’t get into too much detail but all you need to know is that the √-1 is i. Now, let’s get back to the proof. Since we establised that 2 = √-1 * √-1 + √-1 * √-1, 2 = i*i + i*i which is the same 2 = -1 + -1 which is the same 2 = -2.

Where I went wrong

Most lines I was actually correct, the line where I went wrong was between line 4 and line 5.

2 = √(-1*-1) + √(-1*-1) to 2 = √-1 * √-1 + √-1 * √-1

You cannot distribute negative numbers this way. √xy = √x * √y is true, but not if both numbers are negative. Let’s take a look.

Applying the rule

√(4*9) = √36 = 6

√4 * √9 = 2 * 3 = 6

This all checks out becuase the rule √xy = √x *√y is true if both numbers are positive. It also checks out if one number is positive and the other is negative. Here’s where imaginary units come back into play, √-x = √x * i

√(4 * -9) = √-36 = 6i

√4 * √-9 = 2 * 3i = 6i

So why doesn’t it work for negative numbers?

If we ignore the fact the it makes 2 equal to -2, we can formulate a proof.

√xy = √x * √y, we know that if both are negative, x < 0, y < 0. Let’s set x = -a and y = -b. Therefore xy = ab

√x = √-a = √a * i

√y = √-b = √b * i

√x * √y = (√a * i)(√b * i) = √a * √b * i² = √a * √b * -1 = -√ab

Here arises the problem, we just proved that √x * √y = -√ab , but using algebra, we also know that √xy = √ab (square root both sides of the equation xy = ab). If √xy = √x * √y, then -√ab = √ab which means -1 = 1. This is why we cannot apply the rule to negative numbers.

Conclusion

It’s easy to make mistakes in math, following rules blindly, but when we do that, we sometimes reach conclusions that don’t make much sense. We can use these same methods to reach 2=0 or any other crazy results. I hope this article has helped you learn something and show that proofs don’t always work.

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