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Why Zero Does Not Equal One

Any student of math usually at one point or another encounters a purported proof that zero equals one, perhaps given by a teacher who wants the student to find the flaw in logic. Such proofs often involve a division by zero, and must always contain some flaw. In this article we start by looking at two proposed proofs and examine their flaws. In the second half of this article, we will prove some interesting things by simply assuming that zero equals one, which is actually an interesting exercise. In fact, just about anything can be proven by assuming zero equal one; you would be hard pressed to find many things that can’t be proven this way. I am fairly certain that proving that an inequality is true is the only thing that can’t be done by making this assumption. If you don’t believe me, send me your ideas, and I may just post the proofs on my web site.

Proofs That Zero Equals One

Proof: Basic Algebra

Basic algebra proof

The error here is of course the division by a, since we said at the beginning that a = 0.

Proof: Calculus (Integration by Parts)

Integration by parts

There is in fact one of two errors here, depending on how you view this problem. If this were an indefinite integral, we would need to add a constant to the right-hand side, which would nullify our unexplained one. On the other hand, if this were a definite integral, we would need to take the difference of the function 1 between its endpoints to evaluate the integral, which difference is zero for any interval, so that we preserve the equality.

Proofs Assuming 0 = 1

Theorem 1: The Earth is Flat (Sorry, Columbus)

We can define the surface of the earth as having a curvature equal to f(x), where f is a function of position, x is the position on the surface, and 0 corresponds to no curvature. By taking our equation 0 = 1 and multiplying both sides by f(x), we find that 0 = f(x). Therefore, the Earth has a curvature equal to 0 at all points on its surface; therefore, it has no curvature. Hence, it is flat.

Theorem 2: I’m a Multimillionaire (But Don’t Tell the IRS)

I’m not sure how much money I have, so I’ll just call it m. If we take 0 = 1, and multiply each side by m, we have 0 = m. We also can multiply each side by $5,000,000, and we find that 0 = $5,000,000. By substitution, we have that m = $5,000,000. Thus, I have $5,000,000 and am a multimillionaire (Hurray!).

Theorem 3: The Sun Does Not Exist

There is one sun. Therefore, there are zero suns, i.e. the sun does not exist.

As you can see, these proofs are all quite trivial. Once you get the hang of it, you can see how any proof should progress, and you can go on to become a great pseudomathematician!