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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3
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2answers
106 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. ...
1
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1answer
30 views

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 ...
2
votes
5answers
120 views

'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 ...
2
votes
1answer
71 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 $0$. ...
-2
votes
2answers
59 views

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 ...
2
votes
1answer
40 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 ...
15
votes
1answer
346 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 ...
0
votes
0answers
37 views

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 ...
-2
votes
1answer
54 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: $a^2 – b^2 = ab – b^2$. ...
4
votes
0answers
53 views

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) \\ ...
11
votes
5answers
397 views

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 $\,f(x)=f(y),\,$ ...
1
vote
2answers
194 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, ...
25
votes
2answers
1k views

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 ...
4
votes
3answers
360 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 ...
0
votes
2answers
107 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.
8
votes
6answers
504 views

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 ...
3
votes
3answers
163 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 ...
1
vote
2answers
63 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 ...
1
vote
1answer
84 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 ...
4
votes
1answer
425 views

Fake Proof of Prime Number Theorem

In David M. Burton's book on Elementary Number Theory I have found the following words, ... The first demonstrable progress toward comparing $\pi(x)$ with $\dfrac {x}{\ln x}$ was made by ... P. L. ...
5
votes
2answers
97 views

Why does this $u$-substitution zero out my integral?

Here's how I understand $u$-substitution working for an integral. Essentially, it involves substitution of differential expressions, allowing you to cancel out terms of the integrand. When we change ...
2
votes
2answers
58 views

Equal perimeters of squares and right angled isosceles triangles

Consider a square ABCD having length l and breadth. Now start folding the sides AB and AC so that the figure becomes something like this $$$$ All the vertical and horizontal folds/stairs are equal in ...
0
votes
0answers
31 views

Mistake with proof

Is there an error in the following proof? Proposition: $\forall r \in {N} ; r \neq 1$ , then $\exists$ $n \in {Z}$ such that $$2^{1/n} < r$$ Proof: Let $n$ be any integer with $$n > 1/ log_2 ...
0
votes
1answer
40 views

Incorrect combinatorial argument- 5 card hand with at least 3 red cards

How many 5 card hands can be made with at least three red cards? Of course, we're using a standard deck of 52. I know how to answer this, but I frequently see this argument, producing a different ...
0
votes
4answers
114 views

Best basic algebra examples to show students that proof by example is not sufficient [duplicate]

Often, students will try to 'prove' a propositon by checking some examples and 'concluding' that it will be true for all $n \in N$. I'm looking for some good, non-trivial examples from highschool ...
1
vote
1answer
147 views

Is $\pi = 4$ really? [duplicate]

Can anyone explain what's wrong with this?
5
votes
1answer
53 views

Complement of a countable open cover of the rationals

Suppose you take an open interval I of length 1, divide it into countable sub-intervals (I/2, I/4, etc.), and cover each rational with one of the sub-intervals. Since all the rationals are covered, ...
3
votes
2answers
69 views

Where is the error in my proof?

I have this excercise. I am able to solve it, but the problem is that I can solve it without using the last part of information of the existence of the u-vector. That makes me afraid that my proof is ...
2
votes
3answers
147 views

Proof that $\sqrt{x}=-\sqrt{x}$ [duplicate]

$\sqrt{x}=\sqrt{1\cdot x}=\sqrt{(-1)^2\cdot x} = \sqrt{(-1)^2} \cdot \sqrt{x} = (-1) \cdot \sqrt{x}=-\sqrt{x}$ The idea popped into my head while I was evaluating an integral. I have a feeling that I ...
2
votes
1answer
68 views

How to Prove it 4.1 ex.10

Prove that for any sets A, B, C, and D, if A × B and C × D are disjoint, then either A and C are disjoint or B and D are disjoint. Proof(someones). Suppose (A X B) and (C X D) are disjoint. Let (x,y) ...
1
vote
4answers
430 views

Proof of Drinker paradox [duplicate]

I searched all over the internet but didn't find a formal proof for this paradox, so here is my attempt: $\exists x[P(x)\implies \forall yP(y)]$ Let $x=x_0$. Thus $P(x_0)$ is given. Let $y$ be ...
20
votes
4answers
2k views

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} ...
6
votes
4answers
165 views

What did Johann Bernoulli wrong in his proof of $\ln z=\ln (-z)$?

Some people say, Johann Bernoulli has proven $\ln z=\ln (-z)$ in the following way $$\ln ((-z)^2 )=\ln(z^2)\;\;\;\Rightarrow\;\;\;2\ln(-z)=2\ln z\;\;\;\Rightarrow\;\;\;\ln (-z)=\ln z$$ While the ...
15
votes
3answers
1k views

Riddle with Pi = 3

This is a riddle someone posted on Google+, so please forgive it's triviality - I'm asking here because I just can't figure out what exactly is wrong, and it really bugs me ;) I think something is ...
-11
votes
4answers
157 views

A proof of $1=2$

$$2\times 8x=8x$$ therefore divide both sides by $8x$ we get $$2\times 8x/8x=8x/8x$$ therefore because $$8x/8x=1$$ we have $$2\times 1=1$$ so $1=2$. Is my proof correct?
18
votes
2answers
1k views

Proof derivative equals zero?

I know this must be wrong, but I am confused as to where the mathematical fallacy lies. Here is the 'proof': $$f '(x) = \lim_{ h\to0}\frac{f(x+h)-f(x)}{h}$$ L'Hopital's Rule (Previous limit was ...
2
votes
2answers
2k views

Contradiction: Prove 2+2 = 5 [duplicate]

While browsing I came across a weird proof which says 2 + 2 = 5. The proof is like this: After going through this for almost 30 minutes, I was not able to figure ...
8
votes
0answers
97 views

Complex Exponential False “Proof” That All Integers Are $0$

The following false "proof" is attributed to Thomas Clausen in 1827, and was stated on page 79 of Nahin's An Imaginary Tale. $e^{i2\pi n}=1$ for all integers $n$. So \begin{align*} ee^{i2\pi ...
1
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0answers
79 views

What is the flaw in this proof?

Below is a proof that straight lines cannot exist in the coordinate plane. Where is the flaw in its reasoning? It will be shown that the equation of a straight line leads to a mathematical ...
1
vote
3answers
122 views

What is the different between these two triangles? [duplicate]

What is the different between rigorous proof and proof based on intuition on this problem? It seems to me that these triangle are equivalent in area.
11
votes
7answers
1k views

What's wrong with these equations? [duplicate]

My friend Boris (Boryan) gave me a task, and completely refuses to give the answer what's wrong here. $$x^2=\overbrace{x+\cdots+x} ^{x\text{ times}}$$ $$(x^2)'=(x+\cdots+x)'$$ $$2x=1+\cdots+1$$ ...
0
votes
1answer
55 views

Find the Logical Inconsistency

Recently one of my friend came up with something which he claimed to be a proof of the famous Legendre Conjecture. Let me brief his argument. Statement of The Conjecture There exists at least ...
79
votes
20answers
14k views

Visually deceptive “proofs” which are mathematically wrong

Related: Visually stunning math concepts which are easy to explain Beside the wonderful examples above, there should also be counterexamples, where visually intuitive demonstrations are actually ...
1
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2answers
43 views

Are these subgroups of G only subgroups if G is abelian?

I am doing some exercises in a book I am reading. The exercises and my answers for them are as follows: Let $H$ be a subgroup of $G$, and let $K = \{x \in G: x^2 \in H\}$. Prove that $K$ is a ...
0
votes
3answers
76 views

In sphere $r \propto \frac{1}{A}$! How is this possible? What's the wrong here?

Surface area $A$ and volume $V$ of a sphere of radius $r$ are \begin{eqnarray} A=4\pi r^2,\\ V=\frac{4}{3} \pi r^3. \end{eqnarray} But then \begin{align} \frac{V}{A} & = \frac{r}{3}\\ ...
0
votes
1answer
49 views

Where exactly is the following process incorrect to yield an impossible answer

I was playing with my calculator and found some strange phenomena. $\cos(\tan(\tan(\tan(\pi/4)))) = 0.75686700166$ Verify here Now when we apply some inverses, then $\tan(\tan(\tan(\pi/4))) = ...
0
votes
1answer
43 views

Induction to prove that something is not true?

This is maybe a very basic question, but I have never seen it done before. Can you use induction to prove that something is not true? In particular if something does not hold in dimension n=1, can I ...
2
votes
1answer
100 views

Flaw in proof that a functional is not continuous

I am trying to show Consider $F\colon C[a,b] \rightarrow R$ ; $F(f)=f(t_0)$ where $a<t_0< b$ . Show that this linear functional is not continuous under $\|\cdot\|_1$ on C[a,b] I have ...
2
votes
2answers
62 views

What's wrong in this reasoning of $l_\infty$ separability?

While solving a problem, related to functional analysis, I've accidentally got a "proof" of $l_\infty$ being separable, tried to find a fault in it (as the result isn't true), but didn't succeed in ...
17
votes
8answers
4k views

Is Lewis Carroll's reasoning correct?

A bag contains 2 counters, as to which nothing is known except that each is either black or white. Ascertain their colours without taking them out of the bag. Carroll's solution: One is black, and ...