# Tagged Questions

Seemingly flawless arguments are often presented to prove obvious fallacies (such as 0=1). This tag is the appropriate place to ask "Where is the proof wrong?" when encountering such proof.

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### Proof Involving Imaginary Number: Where's the wrong one? [duplicate]

Here are the propositions: $$i=\sqrt{-1}$$ $$i^2=-1$$ $$(i)(i)=-1$$ $$\sqrt{-1}\sqrt{-1}=-1$$ $$\sqrt{(-1)(-1)}=-1$$ $$\sqrt{1}=-1$$ There's an error in the propositions above. I think it's in the ...
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### Failure of an elementary 'proof' of Fermat's Last Theorem?

Can someone explain to me why this does not constitute a proof of Fermat's Last Theorem, please? Basically, using something I've read online, it appears you can write an equation for $(a, b, c)$ to ...
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### Help Debugging a Bogus Proof

We want to prove the standard fact that a smooth function $u :R^2 \to R$ with $\nabla u = 0$ everywhere in some connected open set $\Omega$ is constant in that set. I'm comfortable with the usual ...
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### Is this logic of solving $\frac{0}{0}$ correct [duplicate]

I have seen such proofs many times and was unable to prove where it was wrong (not a math person-my bad). For example the following proof for $\frac{0}{0} = 2$ looks like it is proven correctly - but ...
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### Fake proofs using matrices

Having gone through the 16-page-list of questions using the tag (fake-proofs), and going though Best Fake Proofs? (A M.SE April Fools Day collection) and https://en.wikipedia.org/wiki/...
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### Exponential decay 'proof' that $.\overline{9}\neq 1$?

I have doubts about $.\overline{9}$ being equal to 1 due to the following proof: To get a decimal containing $c$ 9's after the decimal point, the equation f(c) = $1-10^{-c}$ can be used. For ...
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### Infinite summation: $x+x+x+x+… =2$?

One of my favourite little math problems is this $x^{x^{x^{x^{...}}}}=2$ The solution to it is quite simple. An infinite tower of x's is equal to 2, and above the first x there is still an infinite ...
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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 $n\ge 0$. ...
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### What is wrong with this reasoning when calculating circle perimeter? [duplicate]

Looking at the following image, which was posted on the internet: Could someone tell me what is wrong? It seems true for the first 4 small images. But, when it comes to infinitesimal length, ...
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### Contradiction in value of $\psi (1)$

What is the value of $\psi (1)$ ? If we take the definition in terms of derivative of Gamma function, we get $\psi (1) = \dfrac{\Gamma'(1)}{\Gamma(1)} = -\gamma$. But, if we consider the series ...
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### What is wrong with the given proof? [duplicate]

Here is the proof they gave: Start with the statement $a = b$. Multiply both sides by $b$ to get $ab = b²$. Subtract $a²$ from both sides to get $ab − a² = b² − a²$. Factor the left and right sides of ...
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### Alternative triangle inequality proof

I have looked everywhere for confirmation of this proof of the triangle inequality with no success. Prove the triangle inequality: $$\vert x + y \vert \leq \vert x \vert + \vert y \vert.$$ Proof: ...
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### Another $1=2$ proof [duplicate]

So a friend shows me this : $x^4= x^2+x^2+ \cdots +x^2$ ( i.e. $x^2$ added $x^2$ times) Now take the derivative of both side; $4x^3 = 2x + 2x + \cdots + 2x$; So $4x^3 = 2x^3 \cdots$(1) And ...
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I have found various proofs of the result but I have come up with something very different and I wonder whether it is a valid argument: Let $W$ be an algebraic set. Let $I=\mathcal{I}(W)$. We have $I=... 2answers 34 views ### Vacuous statements and explosion So my understanding of vacuous statements is as follows: For any statement$P$, the statement$(\forall x \in \emptyset)(P(x))$. This can be argued as follows: Assume for contradiction$\neg [(\forall ...
Let $n$ be the largest positive integer. Since $n ≥ 1$, multiplying both sides by $n$ implies that $n^2 ≥ n$. But since $n$ is the biggest positive integer, it is also true that $n^2 ≤ n$. It follows ...
Question: Consider n independent tosses of a $k$-sided fair dice. Let $X_i$ be the number of tosses that result in $i$. What is the covariance $\mathrm{cov}(X_1,X_2)$ of $X_1$ and $X_2$. \begin{...