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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4answers
132 views

imaginary number $i$ equals $-6/3.4641$? [duplicate]

$$-4^3 = -64$$ so the third root of $-64$ should be $-4$ than. $$\sqrt[3]{-64} = -4$$ But if you calculate the third root of -64 with WolframAlpha( ...
1
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1answer
34 views

Alternative proof of the Riemann Sum Theorem using Mean Value Theorem for Integrals.

I've been reviewing proofs for a couple of calculus theorems and as I was trying to recall the proof of the Riemann Sum Theorem which uses Lower Sums and Upper Sums I came up with an idea to prove it ...
1
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2answers
66 views

“Proof” that $(2n)!$ is divisible by $2^n 5^{n-3}$ for $n\ge3$

Please explain, as clearly as possible, what is wrong with the following "proof" by induction that $\hspace{1.4 in}$$(2n)!$ is divisible by $2^n 5^{n-3}$ for $n\ge3$. (There clearly must be an ...
0
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3answers
59 views

Where's the foolish part ? Prove that 0/0 = 2 [duplicate]

I've been across this on the web: \begin{align} \frac{0}{0} & = \frac{100-100}{100-100} \\ & = \frac{10^2-10^2}{10(10-10)} \\ & = \frac{(10-10)(10+10)}{(10)(10-10)} \\ & = ...
4
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5answers
116 views

Find a mistake type of math problems [closed]

I am interested in the problems where the formulation of the problem has some kind of mistake in it and as a consequence gives unexpected answer. Can't explain it better than this example: For ...
0
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1answer
67 views

Find the mistake of the inductive proof for $r^n=1$

Find the mistake in the following proof that purports to show that every nonnegative integer power of every nonzero real number is 1. Let r be any nonzero real number and let the property P(n) ...
2
votes
2answers
65 views

What is the fallacy of this proof?

I recently was working with square roots and came across this- $({\sqrt -1})$$=-1^\frac12$$=-1^\frac24$$=(-1^2)^\frac14$$=1^\frac14$$=1$ I understand that this is not true,but despite repeated ...
3
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0answers
48 views

Debunking an elementary proof of FLT

José Cayolla: Fermat's Last Theorem admits an infinity of proving ways and two corollaries. arXiv:1507.06989 [math.GM] I don't usually devote so much time to "crackpot papers", but I have a ...
0
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2answers
55 views

Please explain what's wrong with the proof that every group element is its own inverse.

What is wrong with my proof here? Proof: Let $a, b$ be elements of a group and let $aa = b$. Through manipulation, we see that $$a = ba^{-1}$$ $$b^{-1}a = a^{-1}$$ $$b^{-1}aa^{-1} = e$$ $$b^{-1} = ...
0
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3answers
49 views

Bogus set theory proof

I'm having trouble figuring out where I went wrong in this proof. I think it's to do with my understanding of things like $\cup$ and $\cap$ in that I don't really have a solid understanding of what ...
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votes
2answers
131 views

Does square difference prove that 1 = 2?

I was mathematically shown 1 = 2 by a function that states the following $$x^2-x^2 = x^2-x^2 $$ $$x(x-x)=(x-x)(x+x)$$ dividing by $(x-x)$ we get... $$x=x+x$$ $$ x=2x$$ $$1=2$$ I can see ...
2
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1answer
91 views

Mistakes in $\lim_{a\to \infty}(a^2 - a) = - \frac{1}{6}$?

One can say, using Ramanujan summation or the zeta function regularization, that the sum $\sum_{k=1}^{\infty} k=- \frac{1}{12}$. Using this result I've gotten a very confusing and counterintuitive ...
2
votes
2answers
52 views

What exactly is $\cap$-stable here?

From my lecture notes: Theorem: Let $(\Omega, \mathcal A, P)$ be a probability space, $A \in \mathcal A, \mathcal M := \{ M_1, \ldots, M_n \} \subset \mathcal A$. The following statements are ...
7
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3answers
300 views

What is wrong with this proof of $0=1$?

I am trying to understand what is wrong with the proof posted here that $0=1$ (source): Given any $x$, we have (by using the substitution $u=x^2/y$) ...
2
votes
0answers
28 views

$C^0([0,1])$ is separable – or isn‘t it?

Using Bernstein polynoms it can be proven that $(X, \|\cdot\|) := (C^0([0,1]), \|\cdot\|_{C^0([0,1])})$ is a seperable vector space. However, here is my “proof” that this space is not seperable: ...
2
votes
2answers
31 views

A Proof of a False Result: If $U$ is $T$-invariant, then so is $U^\perp$.

$\newcommand{\ab}[1]{\langle #1\rangle}\DeclareMathOperator{\tr}{trace}\newcommand{\mc}{\mathcal}$ I have a "proof" of the following wrong fact: Let $T$ be a linear operator on a finite ...
0
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1answer
66 views

Finding the error in this induction proof [duplicate]

Claim: If $n$ belongs to $\mathbb{N}$, and $p$ and $q$ are natural numbers with maximum $n$, then $p=q$. Let $S$ be the subset of the natural numbers for which the claim is true. $1$ belongs to $S$, ...
7
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2answers
319 views

Was Smullyan really wrong?

EDIT: the OP has since edited the question fixing all the issues mentioned here. Yay! There was a question asked on Puzzling recently, titled ...
1
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1answer
40 views

Error in proof: Distribution of exponents for negative number [duplicate]

Here are steps of the "proof": $1=1$ $\Rightarrow 1=\sqrt{1}$ $\Rightarrow 1=\sqrt{-1\times-1}$ $\Rightarrow 1=\sqrt{-1}\times\sqrt{-1}$ $\Rightarrow 1=i\times i$ $\Rightarrow 1=-1$ At which ...
4
votes
5answers
147 views

Where does this argument showing there are uncountably many TMs fail?

This argument comes up once every while on Lambda the Ultimate. I want to know where the flaw is. Take a countable number of TMs all generating different bitstreams. Construct a Cantor TM which runs ...
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2answers
57 views

How can the ball reach the wall when it always has to travel halfway? [duplicate]

If I throw a ball at the wall, when it has travelled halfway, it still has half the distance to travel. As it continues, the fraction left to travel continues i.e. one quarter to go, one eighth to go, ...
1
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2answers
50 views

Is $\sum_{n=1}^\infty a_n\sin(nx)$ converges on $[\varepsilon, 2\pi-\varepsilon]$?

Let $a_n$, a sequence monotonically decreasing to $0$. Consider $$\sum_{n=1}^\infty a_n\sin(nx)$$ Is the series converges uniformly on $[\varepsilon, 2\pi-\varepsilon]$? ($\varepsilon ...
1
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2answers
58 views

Do we really need the constraint qualification?

I can't keep my fingers off Nocedal/Wright's Numerical Optimization (1999,1E) and I apologize. But maybe YOU can shed light on the question: Why does a point $x \in \mathbb{R}^n$ need to satisfy the ...
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1answer
44 views

Is this mathematical statement? [closed]

$\{\text{integers $n$ such that $n$ is even}\}$ It can be true/false so does that mean it's proposition/mathematical statement?
1
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1answer
65 views

Where is the problem here:$-1=(-1)^1=(-1)^\frac{2}{2}=({(-1)}^{2})^{1/2}=\sqrt{1}=1$? [duplicate]

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
8
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6answers
206 views

I was wondering, shouldn't the fraction $\frac {-2}{-1}$ be less than 1?

Because technically, the numerator is smaller than the denominator as $-2 < -1$ I know it's an extremely stupid question. I mean I know that I can just multiply $-1$ to the numerator and the ...
0
votes
2answers
72 views

Paradox - minus one equals one using square roots [duplicate]

I was looking on Howard Eves's book "An Introduction to the History of Mathematics" and I stumbled upon a demonstration on how $-1 = 1$. The demonstration follows: $$ \sqrt{-1} = \sqrt{-1} $$ $$ ...
0
votes
1answer
41 views

P(TT|T) in two coin tosses not 1/3?

So the question is, if two consequtive coin tosses occured and we know, in the aftermath, that at least of them resulted in a tails, what is the probability that they were both tails? The common ...
0
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3answers
89 views

What's wrong with this?

What's wrong with this : $$e^{i\pi} = -1$$ $$\therefore e^{2i\pi} = 1$$ $$\therefore log \left( e^{2i\pi} \right ) = log(1) = 0$$ $$\therefore 2i\pi = 0$$
0
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1answer
17 views

Using Union to prove a context-free language? [closed]

I am working through many examples and I seem to have confused myself and made all the questions rather trivial. If I have the CFLs, $L_1 = \{1^n 0^{mn} : n,m \in \Bbb N\}$ and $L_2 = \{1^m 0^n : ...
4
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1answer
118 views

What is wrong with this “proof” that there is no $\omega$th inaccessible cardinal?

"Theorem": There is no $\omega$th inaccessible cardinal. "Proof": Assume ZFC. Let $\kappa_n$ be the $n$-th inaccessible cardinal; since $V_\kappa$ is a model of ZFC for inaccessible $\kappa$, ...
0
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1answer
84 views

Mathematical Induction. Horses made me question my understanding

I recently read about the false inductive proof that all horses are the same colour. There are some mathSE threads about this already (MathSE_thread_1, MathSE_thread_2). After reading this, I now ...
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votes
4answers
173 views

Why is this proof of $i = -1$ wrong? [closed]

I know that $-7^2 = -49$ Therefore $\sqrt{-49} = -7$ Because $\sqrt{-1} = i$ we can then expand it to $\sqrt{-49} = -7 = 7i$ And therefore $-7 = 7i$, divide both sides by 7 and you get $-1 = i$ ...
0
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1answer
42 views

Check my logical argument for this proof.

if x is a real number $x \not =\ 1 $, then there exists y which is also a real number $ ((y+1) \div ( y-2) ) = x .$ Prove it's converse also. Logical Argument: given: $x \not = 1$ Goal: $ ...
6
votes
3answers
209 views

Where does the proof of $\sqrt 2$ is irrational break down when trying to prove the same for $\sqrt 4$?

I'm trying find where the common proof by contradiction that $\sqrt 2$ is irrational breaks down when trying to prove $\sqrt 4$ is irrational. Assume $\left(\frac pq\right)^2=4$ and $\gcd(p,q)=1$. I ...
4
votes
5answers
188 views

Is $\sqrt{x^2} = (\sqrt x)^2$? [duplicate]

Take $x=4$ for example: $ \sqrt{(4)^2} = \sqrt{16} = \pm4 $ However: $ (\sqrt{4})^2 = \sqrt{\pm2}$ Case 1: $ (-2)^2 = 4$ Case 2: $ (2)^2 = 4$ Solution : $+4$ How come the $ \sqrt{(4)^2} = \pm4$; ...
2
votes
1answer
55 views

Same Expansion for Different Functions! What's Wrong?

We know, the function $arctan(x)$ is defined as the real number $y$ such that $tan(y)=x$ and $-\frac{\pi}{2}\leq y\leq\frac{\pi}{2}$ From this definition we can derive the well-known ...
2
votes
0answers
277 views

An argument for “Brocard's problem has finite solution”

Brocard's problem is a problem in mathematics that asks to find integer values of n for which $$x^{2}-1=n!$$ http://en.wikipedia.org/wiki/Brocard%27s_problem. According to Brocard's problem ...
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votes
3answers
195 views

$1+1=0$ What am I doing wrong???! [duplicate]

Does someone know what I'm doing wrong? I'm struggling with this for a while now and I don't see what I do wrong! $$1+1=$$ $$1+\sqrt{1}=$$ $$1+\sqrt{-1*-1}=$$ $$1+\sqrt{-1}*\sqrt{-1}=$$ $$1+i*i=$$ ...
4
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1answer
78 views

Fake proof: Equivalence of norms

Good morning. I'm having a hard time finding what's wrong with the following argument. Let $f$ be any function in $C^{1}([0;1])$ and let $||f||$ and $N(f)$ be two norms defined as follows: $$||f|| = ...
15
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7answers
2k views

What is wrong with this putative proof?

So I've spent about an hour trying to figure out what is wrong with this proof. Could somebody clearly explain it to me? I don't need a counterexample. For some reason I was able to figure that out. ...
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0answers
48 views

Proof subtraction is not forward stable

I've been taught that the "subtraction operation" is not accurate/forward stable as the relative error can be arbitrary large. I tried to prove it formally but I end up with a contradiction. What ...
3
votes
1answer
112 views

What is wrong with this proof that $\text{card}(\mathcal{P}(\mathbb{N}))=\aleph_1$?

I'm reading books on set theory and I came up with the following 'proof' that $\text{card}(\mathcal{P}(\mathbb{N}))=\aleph_1$. What is wrong with it? I really cannot tell! By ...
1
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1answer
36 views

Is the set of vanishing $k$ derivatives of smooth functions in a null set dense in $W^{1,p}$?

Let $\Omega$ be an open set with compact closure denoted by $\overline{\Omega}$ and a null set $N\subset\Omega$ with respect the Lebesgue measure. Then consider the two sets ...
47
votes
10answers
1k views

Fake induction proofs

Question: Can you provide an example of a claim where the base case holds but there is a subtle flaw in the inductive step that leads to a fake proof of a clearly erroneous result? [Note: Please do ...
0
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3answers
64 views

E is countable $\longleftrightarrow$ there exist a surjection from $\mathbb{N}$ to E

The book I am using for my Introduction of Topology course is Principles of Topology by Fred H. Croom. I was given the following problem: Prove that a set $E$ is countable if and only if there ...
5
votes
4answers
1k views

Why is “All horses have the same color” considered a false proof by induction? [duplicate]

Upon reading of All horses have the same color "paradox", I began to wonder a couple of things. First of all, to me the inductive step seems flawed. Just because I have $n$ white horses, does not ...
15
votes
6answers
863 views

Induction - Examples where the induction step is correct but the base case is always wrong [duplicate]

I'd like to present to my students some induction examples that always satisfy the inductive step but never the base case. It could be for natural numbers, graphs or anything else.
1
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3answers
143 views

Why is $(-1)^3=(-1)^{6/2}=((-1)^6)^{1/2}=1^{1/2}=1$ wrong? [duplicate]

Why is this wrong? $$(-1)^3=(-1)^{6/2}=((-1)^6)^{1/2}=1^{1/2}=1$$ It seems logical but I know it's wrong.
1
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1answer
63 views

Explain what is wrong with the following “proof” by induction.

Basically, there's so much going on in this problem that I don't even understand it. I've read it about one million times, but it still isn't making sense to me. Any hints would be appreciated... ...