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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Question from Stewart's Calculus regarding proof of independence of path and conservative vector fields.

Please look over this proof. In the proof, it says: "Notice that the first of these integrals does not depend on $x$, so..." How is that so? $C_1$ does depend on $x$. How/Why does it not ...
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1answer
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For $f(\theta)= e^{\theta}$. Is it true that $\hat{f}(n)(1-in)=0$ for all $n\in \mathbb Z.$?

(This is motivated from the following question) Fact: If $f \in C^1(\mathbb{T})$, then the Fourier coefficients $\widehat{f'}(n)$ of the derivative $f′$ can be expressed in terms of the Fourier ...
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3answers
135 views

$1^x = 1^y$ and $x,y$ belongs to Real Numbers.

$1^x = 1^y$, and $x,y \in \mathbb{R}$. Following the rule, same base has powers equal every $x$ should be equal to every $y$. $$1^x = 1^y$$ $$x = y$$ What went wrong?
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what are the problems with the followings “equations”?

what are the problems with the followings "equations"? A) In the complex number field consider the following: $-1=i^2=(i^4)^{\frac{1}{2}}=1^{\frac{1}{2}}=1$. B) In $\Bbb R$, ...
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“Proof” that $0=1$ using an integral. [duplicate]

I saw the following: $$\begin{align} \int \tan x \ \mathrm{d}x &= \int \sin x \sec x \ \mathrm{d}x \\ \int \tan x \ \mathrm{d}x &= -\cos x \sec x - \int - \cos x \sec x \tan x \ \mathrm{d}x ...
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0answers
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Derivative of conditional expectation

Let $\left( {{X_t}:t \in \left[ 0 \right.\left. {, + \infty } \right\rangle } \right)$ be a continuous time Markov chain on a probability space $\left( {\Omega ,\mathcal{F},\mathbb{P}} \right)$ with a ...
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1answer
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Question on series $\frac {\Gamma'(z)}{\Gamma(z)}$

Prove that: $$\frac {2\Gamma'(2z)}{\Gamma(2z)}-\frac {\Gamma'(z)}{\Gamma(z)}-\frac {\Gamma \prime(z+\frac{1}{2})}{\Gamma(z+\frac{1}{2})} =2 \log 2$$ But I obtain this equal zero: $$\frac ...
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1answer
77 views

Why is this $0 = 1$ proof wrong? [duplicate]

0 = 0 + 0 + 0 + ... 0 = (1 - 1) + (1 - 1) + (1 - 1) + ... 0 = 1 - 1 + 1 - 1 + 1 - 1 + ... 0 = 1 + (-1 + 1) + (-1 + 1) + (-1 + ... 0 = 1 + 0 + 0 + 0 + ... 0 = 1 I ...
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438 views

What is wrong with this induction proof?

What is wrong with this "proof" by strong induction? "Theorem": For every non-negative integer $n, 5n = 0$. Basis Step: $5(0) = 0$ Inductive Step: Suppose that $5j = 0$ for all non-negative integers ...
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Prove $-(-a) = a$

Let $F$ be a field and $a \in F$. Prove $-(-a) = a$. So we want to show that $(-a) + (-(-a)) = 0$, since inverses are unique (I successfully proved that inverses are unique in an earlier problem ...
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117 views

What is wrong with the following “proof” that $e=1$?

Let's analyze this expression $\lim_{n\rightarrow\infty} (1+\frac{1}{n})^n$ It's the definition of $e$ which, as we know is not equal to $1$. So what is wrong with the following "logic": As ...
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1answer
34 views

Determine whether $\phi$ is a homomorphism

Let $\phi: \mathbb{Z}_6 \rightarrow \mathbb{Z}_2$ be given by $\phi(x)=$the remainder of $x$ when divided by $2$, as in the division algorithm. Let $\phi: \mathbb{Z}_9 \rightarrow ...
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614 views

What's wrong with this proof that commutativity is implied by the other field axioms?

I seem to have found a proof that the commutativity of $+$ follows from the other field axioms. It is as follows: Let $(k,+,\cdot)$ be a structure satisfying all field axioms except commutativity of ...
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2answers
57 views

Proof by induction failure if assumption is wrong?

I never got a clear answer to this question in college. What happens in an induction proof if the assumption is wrong? For example, suppose we try to prove that $n^5$ > n! for n >= $2$ so we start ...
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1answer
51 views

Is this a proper way to prove simple geometrical result?

I found this on Quora : Is there anything wrong in the steps illustrated ?
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1answer
100 views

What is the incorrect proof by Euler that $\pi = 0$ (or something like that)?

I seem to remember a proof by Euler, involving infinite series, which was really complex (for a maths hobbyist). I believe it was sent in a letter to someone, and that it ended up with $\pi = 0$ or ...
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2answers
43 views

Is this a valid method of proof?

We are given that $y = a + b$, and we want to prove that $y = a + c$ (using all the usual properties of numbers that we know from grade school). Does it suffice to set $a + b = a + c$, and by ...
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1answer
59 views

Spotting mistake: unnecessary given condition

I have solved the following problem without using a given premise. Could someone please spot whether I have done something wrong? Suppose we have a relation $\geq$ that is transitive, but not ...
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0answers
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Proof of law of reflection using Fermat's principle : are we really proving the law of reflection?

Before you skip reading this, let me tell you that this isn't a "how to derive the law of reflection using Fermat's principle" question. Also, I asked it on MSE instead of the physics site because ...
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1answer
47 views

Isn't that proof going the wrong way?

I'm currently working on the very well written book Understanding Analysis, by Stephen Abbott. But I found a proof that looks wrong, I think that it going the wrong way (showing that A $\implies$ B ...
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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 ...
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1answer
150 views

Did I construct an infinite set equal to $\{1\}$?

Okay, I'm trying to understand the argument that NJ Wildberger gives in the following video: https://www.youtube.com/watch?v=5CiiGdaYEPU He tries to explain why he things infinite sets don't make ...
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1answer
70 views

Find the mistake in this proof.

I need help with finding the mistake in this proof. Statement: All natural numbers are divisible by 3. Proof: Suppose, for the sake of contradiction, the statement were false. Let X be the set of ...
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2answers
281 views

Homomorphism problem gone wrong

Okay, so I'm working on a homework problem in abstract algebra, and I have found the solution already, what I want to know is why my initial line of reasoning didn't work - i..e, what have I done or ...
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1answer
50 views

Proving property for all predicates in first order logic

Let's consider language with predicate $P$ and following derivation $${{{[P[a/x]]^1} \over {P[a/x] \rightarrow P[a/x]}}\rightarrow I^1 \over {\forall_x (P(x) \rightarrow P(x))}}\forall I$$ Doesn't ...
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2answers
96 views

Proof that $\sqrt{4}\notin\mathbb{Q}$ of course wrong but where is the flaw?

Assume $$\eqalign{ \sqrt{4}\in\mathbb{Q}&\Longrightarrow(\exists a,b\in\mathbb{Z})\sqrt{4}=\frac{a}{b}\text{ and }\gcd(a,b)=1\\ &\Longrightarrow 4b^2=a^2\Longrightarrow a\text{ is even}\\ ...
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1answer
85 views

how $2x=x$ , related to differential calculus [duplicate]

can anybody please tell me what's happening here ? $$1^2=1$$ $$2^2=2+2$$ $$3^2=3+3+3$$ $$x^2 = x+x+\cdots+x \mbox{ ($x$ times)}$$ differentiating both the sides $$2x = 1 + 1 + \cdots+1 \mbox{ ...
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1answer
55 views

Apparently same probability questions with different answers.

I was reading A first course in probability by Sheldon Ross when and then I came up with this question. This is how he introduces the famous problem of points Independent trails, resulting in a ...
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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??
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Demonstration that 0 = 1 [duplicate]

I have been proposed this enigma, but can't solve it. So here it is: $$\begin{align} e^{2 \pi i n} &= 1 \quad \forall n \in \mathbb{N} && (\times e) \tag{0} \\ e^{2 \pi i n + 1} &= e ...
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326 views

What is the flaw in this proof that all triangles are isosceles?

What is the flaw in this "proof" that all triangles are isosceles? From the linked page: One well-known illustration of the logical fallacies to which Euclid's methods are vulnerable (or at least ...
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3answers
217 views

Proof of $x*0=0$ is invalid?

My professor told me today that while my logic was good and all my steps after the assumption were correct, my argument was invalid because I was assuming what I want to prove. I don't see how. ...
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1answer
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Why does this proof fail?[convergence of infinite sums]

An equivalent way of saying that a normed vector space is complete is saying that every absolutely convergent series, converges. Hence' in some normed vector-space(incomplete), there must be a ...
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'Proof ' that $\ln(x)$ converges

Where is the flaw in the following 'proof '? $$\lim_{x \to \infty}\left[\frac{\mathrm{d}}{\mathrm{d}x}\left\{\ln(x)\right\}\right]=\lim_{x \to \infty}\left[\frac{1}{x}\right]=0 \implies\lim_{x \to ...
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1answer
87 views

A possible incorrect application of Law of Large numbers

A friend left this teaser for me. He asked me to first compute: $$ \lim_{n \to \infty} \frac{\binom{2n}{n}}{2^{2n}}$$ Using Stirling's approximation (and another method), I got the answer as $0$. ...
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What's wrong with this induction based proof?

Claim: $\forall x \in \mathbb{R^+} ,$ $ x^n=1 $ $where$ $ n\in \mathbb{N}$ Proof by induction on n: Basis step: $\forall x \in \mathbb{R^+} ,$ $ x^0=1 $ Induction Step: Let this holds for all ...
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1answer
46 views

Fake proof using mean value property

Let $f : \Delta \rightarrow \mathbb{C}$ be a holomorphic function on the unit disk. Let $u = |f|$. Assume that $u(0)=f(0)=0$. Question : since $u$ is harmonic, the mean value property should imply ...
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1answer
391 views

An outrageous way to derive a Laurent series: why does this work?

I had to compute a series expansion of $1/(e^{x}-1)$ about $x=0$, and in the course of its derivation, I made a couple of manipulations that are not allowed mathematically. Still, comparing the final ...
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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 ...
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1answer
60 views

The problem with this simple proof [duplicate]

1) Find the error in the following proof that $2 = 1$. Consider the equation $a = b$. Multiply both sides to obtain: $a^2 = ab$. Subtract $b^2$ from both sides to get: $a^2 – b^2 = ab – b^2$. ...
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Euler's proof of divergence of sum of reciprocals of primes

On Wikipedia at link currently is: \begin{align} \ln \left( \sum_{n=1}^\infty \frac{1}{n}\right) & {} = \ln\left( \prod_p \frac{1}{1-p^{-1}}\right) = -\sum_p \ln \left( 1-\frac{1}{p}\right) \\ ...
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Where's the problem with a false “proof”: $\;1^0 = 1^2 \overset{?}\implies 0 = 2$

What's wrong with this: $$\large 1^0=1^2$$ Since bases are same, therefore $$\large 0=2$$ My thinking: Since the function $\,f(x)=1^x\,$ is not one to one, therefore whenever $\,f(x)=f(y),\,$ ...
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222 views

Why aren't all real numbers equal to one another?

I know, stupid question. But humor me for a sec. First off, we know that all real numbers have two numbers which are infinitely close to them, right? That would seem to be, for any given value of x, ...
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Lamport claims there is an error in Kelley's proof of the Schroeder-Bernstein theorem. What is it?

In section 4.1 of his note How to write a proof, Leslie Lamport mentions an error in Kelley's exposition of the Schroeder-Bernstein theorem: Some twenty years ago, I decided to write a proof of ...
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3answers
521 views

Interesting Mathematical Fallacies [duplicate]

I recently volunteered to help with a summer math program at a local high school for which I thought would be a breeze. As it turns out, it isn't a program for those catching up (summer school) like I ...
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2answers
131 views

Why is this wrong (complex numbers and proving 1=-1)?

$$(e^{2πi})^{1/2}=1^{1/2}$$$$(e^{πi})=1$$ $$-1=1$$ I think it is due to not taking the principle value but please can someone explain why this is wrong in detial, thanks.
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Can every indefinite integral of a discontinuous function be written in a way that “proves” something false?

I just saw the following fake proof. $$\int \frac1x dx =\int 1\cdot \frac1x dx=x\frac1x+\int x \frac1{x^2} dx = 1+ \int \frac1x dx$$ Which would imply $1=0$, hence the fake proof tag. The ...
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175 views

Validity of a trigonometric proof that $2 = 0$.

I can't find where this proof goes wrong. We know $$\tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A\cdot\tan B}$$ so $$\begin{align} \tan(90^\circ-45^\circ) &= \frac{\tan 90^\circ -\tan ...
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2answers
73 views

Simplifying $x^i$ to real numbers

I have been studying power functions, and started to think about imaginary powers. Take the function $x^i$. Because I don't know how to multiply a number $i$ times, I tried to simplify the equation ...
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1answer
95 views

What is fundamentally wrong with this?

N tosses of a fair coin. There are $\binom{2+N-1}{N} = N+1$ ways to choose with replacement from {h, t}. So the probability of N heads is $\frac 1{N+1}$. Obviously not all ways are equivalent but is ...