Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [fake-proofs]

Seemingly flawless arguments are often presented to prove obvious fallacies (such as 0=1). This is the appropriate tag to use when asking "Where is the proof wrong?" about proofs of such obvious fallacies.

An example of a fake proof is $$1=\sqrt{-1\cdot-1}=\sqrt{-1}\sqrt{-1}=i^2=-1$$ which fails because $\sqrt{xy}=\sqrt x\sqrt y$ does not hold if $x$ or $y$ is negative. Sometimes the proof may be presented as a puzzle, the challenge being to identify the flaw.

For asking about identifying flaws in general proofs ("spot the mistake"); the tag should instead be used.

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All natural numbers are equal.

I saw the following "theorem" and its "proof". I can't explain well why the argument is wrong. Could you give me clear explanation so that kids can understand. Theorem: All natural numbers are equal. Let $a, b \in \mathbb{N}$, then a=b. Proof by…
user65175
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Axiom of Choice: Where does my argument for proving the axiom of choice fail? Help me understand why this is an axiom, and not a theorem.

In terms of purely set theory, the axiom of choice says that for any set $A$, its power set (with empty set removed) has a choice function, i.e. there exists a function $f\colon \mathcal{P}^*(A)\rightarrow A$ such that for any subset $S$ of $A$,…
p Groups
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Zero divided by zero must be equal to zero

What is wrong with the following argument (if you don't involve ring theory)? Proposition 1: $\frac{0}{0} = 0$ Proof: Suppose that $\frac{0}{0}$ is not equal to $0$ $\frac{0}{0}$ is not equal to $0 \Rightarrow \frac{0}{0} = x$ , some $x$ not equal…
Lennart
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Proof of 1 = 0 by Mathematical Induction on Limits?

I got stuck with a problem that pop up in my mind while learning limits. I am still a high school student. Define $P(m)$ to be the statement: $\quad \lim\limits_{n\to\infty}(\underbrace{\frac{1}{n}+\frac{1}{n}+\cdots+\frac{1}{n}}_{m})=0$ The…
Charles Bao
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What is wrong with this fake proof $e^i = 1$?

$$e^{i} = e^{i2\pi/2\pi} = (e^{2\pi i})^{1/(2\pi)} = 1^{1/(2\pi )} = 1$$ Obviously, one of my algebraic manipulations is not valid.
user58789
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Bad Fraction Reduction That Actually Works

$$\frac{16}{64}=\frac{1\rlap{/}6}{\rlap{/}64}=\frac{1}{4}$$ This is certainly not a correct technique for reducing fractions to lowest terms, but it happens to work in this case, and I believe there are other such examples. Is there a systematic…
Isaac
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What is the Todd's function in Atiyah's paper?

In terms of purported proof of Atiyah's Riemann Hypothesis, my question is what is the Todd function that seems to be very important in the proof of Riemann's Hypothesis?
Jose Garcia
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explaining the derivative of $x^x$

You set the following exercise to your calculus class: Q1. Differentiate $y(x) = x^x$. A student submits the following solution: Let $g(a)=a^x$ and $f(x)=x$. Then $y(x) = g(f(x))$, so by the chain rule, $y'(x) = f'(x) g'(f(x)) = 1 \cdot x \cdot…
Matthew Towers
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How do I find a flaw in this false proof that $7n = 0$ for all natural numbers?

This is my last homework problem and I've been looking at it for a while. I cannot nail down what is wrong with this proof even though its obvious it is wrong based on its conclusion. Here it is: Find the flaw in the following bogus proof by strong…
user378071
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Why $\sqrt{-1 \times -1} \neq \sqrt{-1}^2$?

We know $$i^2=-1 $$then why does this happen? $$ i^2 = \sqrt{-1}\times\sqrt{-1} $$ $$ =\sqrt{-1\times-1} $$ $$ =\sqrt{1} $$ $$ = 1 $$ EDIT: I see this has been dealt with before but at least with this answer I'm not making the fundamental mistake of…
Greg
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Proof that the real numbers are countable: Help with why this is wrong

I was just thinking about this recently, and I thought of a possible bijection between the natural numbers and the real numbers. First, take the numbers between zero and one, exclusive. The following sequence of real numbers is suggested so that…
analysisj
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Using Integration By Parts results in 0 = 1

I've run into a strange situation while trying to apply Integration By Parts, and I can't seem to come up with an explanation. I start with the following equation: $$\int \frac{1}{f} \frac{df}{dx} dx$$ I let: $$u = \frac{1}{f} \text{ and } dv =…
Michael Oliver
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Questions on "All Horse are the Same Color" Proof by Complete Induction

I'm bugged by the following that's summarized on p. 109 of this PDF. False theorem: All horses are the same color. Proof by induction: $\fbox{$P(n)$ is the statement: In every set of horses of size $n$, all $n$ horses are the same color.}$ …
MVTC
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Why is this 'Proof' by induction not valid?

I am trying to understand why induction is valid. For instance why would this 'proof' not be valid under the principle of proof by induction ? : $$ \sum_{k=1}^{\infty} \frac{1}{k} \lt \infty$$ because using induction on the statement $$S(n) = …
user3203476
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Can't find the flaw in the reasoning for this proof by induction?

I was looking over this problem and I'm not sure what's wrong with this proof by induction. Here is the question: Find the flaw in this induction proof. Claim $3n=0$ for all $n\ge 0$. Base for $n=0$, $3n=3(0)=0$ Assume Induction Hypothesis: $3k…
user262291
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