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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I'm missing something about transfinite arithmetic using Fundamental theorem of arithmetic

Ok, So I know this proof is wrong, but I can't find the error. The fundamental theorem of arithmetic says that every natural number can be uniquely expressed as a product of primes. I usually see this explained with multistep. (Definition) So the…
Donneaux
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Why does integration by parts yield $0=1$ here?

I tried doing integration by parts on this integral: $$I = \int \frac{1}{x\log{x}}dx$$ With these substitutions: $$u = \frac{1}{\log{x}} \rightarrow du = -\frac{1}{x\log^2{x}}$$ $$dv = \frac{1}{x}dx \rightarrow v=\log{x}$$ We get: $$\begin{align} I…
ihato
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Fake proof that the digits of $a$ squared then reversed equal the digits of $a$ reversed then squared

Apologies for pasting a screenshot but it was the fastest way for me to ask the question since it's rather long. I don't understand why the part where they say $(x^2 \vee 2xy \vee y^2) > 9$ Why do at least one of them have to be greater than…
Carpetfizz
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What is the mistake?

$$1=1$$ $$\Rightarrow\frac{-1}{1}=\frac{1}{-1}$$ $$\Rightarrow \sqrt{\frac{-1}{1}}=\sqrt{\frac{1}{-1}}$$ $$\Rightarrow\frac{i}{1}=\frac{1}{i}$$ $$\Rightarrow\frac{i}{2}=\frac{1}{2i}$$ $$\Rightarrow\frac{i}{2}+\frac{3}{2i} = \frac{1}{2i}…
user389162
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Possible fake proof of $1= -1$

Possible Duplicate: -1 is not 1, so where is the mistake? Simple Complex Number Problem: 1 = -1 Well, I remembered this after having Algebra II a year ago, is it possible that this is a valid proof that $1 = -1$? $$ 1 = \sqrt{1} =…
Rivasa
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Square root of 1 is (not) -1

Possible Duplicate: $i^2$ why is it $-1$ when you can show it is $1$? I was thinking on the following line of thoughts: $1 = \sqrt{1} = \sqrt{-1 \cdot -1} = \sqrt{-1} \cdot \sqrt{-1} = i^2 = -1$ Of course this is not true, but I was wondering…
Lotte Laat
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What's wrong 1 pound=1 penny?

Wheres the mistake? 1 pound = 100 penny = 10 penny x 10 penny = 1/10 pound x 1/10 pound = 1/100 pound = 1 penny => 1 pound = 1 penny I feel its wrong in $3^{rd}$ step, but it isn't clear to me!!
Display name
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Where is the problem here:$-1=(-1)^1=(-1)^\frac{2}{2}=({(-1)}^{2})^{1/2}=\sqrt{1}=1$?

Is there someone show me Why this is not true ? $$-1=(-1)^1=(-1)^\frac{2}{2}=({(-1)}^{2})^{1/2}=\sqrt{1}=1$$ then :$$-1=1$$ Thank you for any help
zeraoulia rafik
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Does $\frac{0}{0}$ really equal $1$?

If we agree that $\textbf{(a) }\dfrac{x}{x}=1$, $\textbf{(b) }\dfrac{0}{x}=0$, and that $\textbf{(c) }\dfrac{x}{0}=\infty^{\large\dagger}$, and let us suppose $z=0$: $$\begin{align*} z&=0&&\text{given.}\\ \dfrac{z}{z}&=\dfrac{0}{z}&&\text{divide…
Conor O'Brien
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What is wrong with this proof that 3 is less than 1?

What is wrong with this proof? Theorem. 3 is less than 1. Proof. Every number is either less than 1 or greater than 1 or equals 1. Let $c$ be an arbitrary number. Therefore, it is less than 1 or greater than 1 or equals 1. Suppose it is less than 1.…
user182280
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How can be 1 is equal to 2?

It may be a silly question. But I don't know it. So I'm questioning. Recently I've got a proof that proves 1=2. Is there any fault in the proof? If so then what is the fault??
Raihan Khalil
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Problem: use the well ordering principle to show that all positive rational numbers can be written in lowest terms

This problem involves pointing out the unjustified inference/logic error in the following proof that all positive rational numbers can be written in "lowest terms" that is as a ratio of positive integers with no common prime factor. Bogus proof:…
user3761743
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What is wrong in this proof that $\pi=2$ or $x=2$?

Let us consider the number $$\Large\pi^{\pi^\pi}=\pi^{\pi\cdot\pi}=\pi^{\pi^2}$$ As the bases are equal, the exponents must be equal, So $$\pi=2$$ You can take any $x$ instead of $\pi$. What is wrong in this proof?
Kartik
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Finding the number of solutions for $\left|x\right|+\left|x-3\right|=3$

$$\left|x\right|+\left|x-3\right|=3$$ Couple of ways to solve this. One way is to square both sides and then continue. Another way is to consider the function as a piece-wise function $$2x-3=3 \text{ or x = 3,} \text{ when x > 3}$$ $$x-x+3=3 \text{…
Adil Mohammed
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Fake proof: 2=4 using nested exponents

Let's solve the (real) equation $$x^{x^{x^{\dots}}}=2\qquad (x>0).$$ If we call $E(x):=x^{x^{x^{\dots}}}$, it holds that $x^{E(x)}=E(x)$, but our equation says that $E(x)=2$, so it holds that $x^2=2$, whence $x=\sqrt2$. Noew, lets solve…
Tito Eliatron
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