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According to Wikipedia False proof

For example the reason validity fails may be a division by zero that is hidden by algebraic notation. There is a striking quality of the mathematical fallacy: as typically presented, it leads not only to an absurd result, but does so in a crafty or clever way.

The Wikipedia page gives examples of proofs along the lines $2=1$ and the primary source appears the book Maxwell, E. A. (1959), Fallacies in mathematics.

What are some examples of interesting false proofs?

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Is this a duplicate? – Bruce Westbury Apr 21 '12 at 17:19
the answers to this will turn out to replicate many of the responses to Gowers' famous question on "false beliefs", so I am not so sure if this question should remain open. – Suvrit Apr 22 '12 at 5:46
A false proof is not the same as a false belief. One can read a false proof, know for certain that the conclusion is false (so there is no false belief), and still have trouble pinpointing the error. – Steven Landsburg Apr 22 '12 at 15:36
Indeed, a false proof is not the same as a false belief, and at no point did I imply that! But I mentioned Gowers' question, because the top answer's "false proof" (Cayley-Hamilton) also occurred there as one of the answers. (Believing a "false proof" to be true, is a "false belief", and because of that, there is a strong chance of intersection between the two questions) :-) – Suvrit Apr 22 '12 at 22:27
I'm surprised no one has mentioned Stallings's false proof of the Poincare Conjecture, in his paper "How Not to Prove the Poincare Conjecture". – Steve D Apr 30 '12 at 22:13

34 Answers 34

I think nobody point to these interesting false proof:

Let $i=\sqrt{-1}$ be the complex number.

$1)$ $1=\sqrt{-1\times-1}=\sqrt{-1}\times\sqrt{-1}=i\times i=-1$.

$2)$ We know that $x^\frac{2}{6}=x^\frac{1}{3}\Rightarrow (\sqrt{x^2})^\frac{1}{6}=(\sqrt{x})^\frac{1}{3}$. Now, let $x=-1$ and so we have: $$(\sqrt{(-1)^2})^\frac{1}{6}=(\sqrt{-1})^\frac{1}{3}\Rightarrow1=-1.$$

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In S. Bosch's Algebra, exercise 3.4.2 is to find an error in the following existence proof of an algebraic closure of a field $K$ (my translation):
"Consider all algebraic extensions of $K$. Since for a totally ordered (w.r.t. inclusion) family $(K_i)_{i \in I}$ of algebraic extensions of $K$, the union $\bigcup_{i \in I} K_i$ is an algebraic extension of $K$, Zorn's lemma shows the existence of a maximal algebraic extension, i.e. of an algebraic closure of $K$."

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What is the right answer? Is it that "why is the collection of all algebraic extensions of a given field K a set?". – knsam Jun 11 '14 at 8:37
I think so, too. – Torsten Schoeneberg Jun 14 '14 at 10:37

Doron Zeilberger proved that P is equal to NP

Abstract: Using 3000 hours of CPU time on a CRAY machine, we settle the notorious P vs. NP problem in the affirmative, by presenting a “polynomial” time algorithm for the NP-complete subset sum problem. Alas the complexity of our algorithm is $O(n^{10^{10000}})$ (with the implied constant being larger than the Skewes number).

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This is 11 days too early. – Noam D. Elkies Mar 21 '13 at 15:24
@Noam: Actually, 11 days minus 4 years too early. – Lee Mosher Mar 21 '13 at 15:29

e is irrational.

Assume to the contrary that e were rational. Then e would be e-rational, a contradiction.

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It seems like mathematicians are completely and utterly incapable of understanding even the lowest level of humor (i.e. puns). – Joseph Van Name Jun 16 '13 at 9:45
I think it is more likely that at least 12 voters don't see puns as examples of interesting false proofs. In the event that you feel the absolute need to publicly communicate your displeasure, could you please find a way that doesn't involve insults? – S. Carnahan Jun 18 '13 at 2:35
I know for a fact that at least half of the 12 people down voted this answer because they went on my profile and looked for the question with the lowest score just so they can downvote it because they were irritated about a controversial answer that I gave to another question. And I don't see how a pun is an uninteresting example of a false proof yet cancelling out the 6's in 64/16 to get 64/16=4/1=4 is an interesting false proof. – Joseph Van Name Jun 18 '13 at 3:47
Anyways. I am proud of this answer regardless of how people vote! – Joseph Van Name Dec 18 '13 at 4:19

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