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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0
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1answer
37 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 ...
74
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19answers
12k 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
37 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
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3answers
61 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
41 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
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1answer
34 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
93 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
55 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
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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 ...
0
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1answer
12 views

Diagonal Relation as a poset I - establishing the result by vacous truth

I have a problem with the logic behind the fact that, given a nonempty set X, the diagonal relation $D_X := \{ (x,x) : x \in X \}$ is a partial order on $X$. More specifically, my problem is with ...
0
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7answers
375 views

What is wrong in this proof?

\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)} \\ &= \frac{10+10}{10} \\ &= \frac{20}{10} \\ &= ...
1
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3answers
105 views

Multiplication of infinite series

Why multiplication of finite sums $(\sum_{i=0}^n a_i)(\sum_{i=0}^n b_i)=\sum_{i=0}^n (\sum_{j=0}^ia_jb_{i-j})$ (EDIT: This assumption was shown to be false) does not work in infinite case? I have ...
2
votes
1answer
34 views

Problems with the definition of transitive relation

Recently I found this problem, which made me realize I have some problems with relations that are vacuously transitive. Problem: Assume that $R$ is a relation on $A$ and define the relation $S$ as ...
0
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3answers
118 views

$\sqrt{-1}=1$?! Help me correct my proof [duplicate]

I know that this is definitely wrong, as $\sqrt{-1}$ is not defined in the real numbers. $\sqrt{-1}=i$. But I really want to know what is wrong with my proof. $$\sqrt{-1}=(-1)^{1/2}$$ ...
0
votes
1answer
36 views

Prove that $f^{-1}$ is continuous

Let $M$ be a compact metric space ($N$ is a metric space too) and let $f:M \to N$ be a continuous bijection. Prove that $f^{-1}$ is continuous. My proof. Let $A \subset M$ be closed. Then $A$ is ...
5
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1answer
70 views

Why does the same limit work in one case but fail in another?

The following questions has been bugging me since high-school calculus. Please help me find my peace once and for all: Consider a revolution solid generated by rotating a nice curve $f(x)$ around the ...
2
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0answers
39 views

Simple knapsack with arbitrary weights: Algorithm won't work, but my proof by induction doesn't agree.

We want to solve the simple knapsack problem: We're given a set of $n$ positive item weights, which are unique integers $\{w_1, \ldots , w_n\}$, and an integer $C > 0$, representing the capacity of ...
0
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3answers
94 views

Fake proof of $\mathbb{R}$ being countable

We know that $\mathbb{R}$ is second countable. Let $\{V_n\}$ be a countable basis. To every $a\in \mathbb{R}$ consider the interval $(-\infty,a)$. Now find the least $n$ such that $(-\infty,a)^c\cap ...
3
votes
3answers
88 views

False “proof” that $\mathbb{R}$ is countable

About to fall asleep, I came up with the following "proof" that $\mathbb{R}\cap[0,1]$ is countable: For each $k \in \mathbb{N}$, let the set $S_k$ be the set containing ever positive real number ...
4
votes
1answer
213 views

Problems with fake proofs of limit of sequences

I can hardly imagine an easier example of the fact that my understanding of the topic is more than rusty. I will divide the question in two parts to make the reading easier: 1) Background; 2) ...
0
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3answers
68 views

Baloney detection kit for Math

There are folks who claim they proved Fermat's Last Theorem, Riemman Hypothesis offering no more than a half a dozen pages (sometimes one or two pages) of proof, very few if any citations of previous ...
3
votes
2answers
172 views

Fallacious Induction Proof that All Integers are Equal [duplicate]

We're currently learning about induction in my real analysis course. I came up with the following proof that is obviously false but cannot quite figure out why it fails. Here it is... "Any set $S$ of ...
1
vote
1answer
54 views

What's wrong with this argument for $NP \ne EXP$?

Let $\{M_i\}$ be any enumeration of all Turing machines in which each machine appears an infinite number of times. Consider the language $D = \{i \, | \, M_i(i) \text{ does not accept within ...
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0answers
100 views

Where is this 1 = -1 proof wrong? [duplicate]

One of my friends had done a weird proof which showed that $1 = -1$, it goes something like this: $$ 1 \times 1 = -1 \times -1$$ $$\Rightarrow \frac{1}{-1} = \frac{-1}{1}$$ Taking square root on ...
2
votes
2answers
99 views

Fallacy applying Leibniz integral rule to problem of $x^2 = x+\ldots+x$ ($x$ times)

I was trying to provide the same answer my mathematics professor gave me when I asked to the problem raised in this thread. What I was told was that instead of summing what we were really doing is ...
2
votes
3answers
120 views

What's wrong with this demonstration? (1 = -1) [duplicate]

What's wrong with this demonstration?: $$A \iff 1 = 1^1$$ $$A \implies 1 = 1^\frac{2}{2}$$ $$A \implies 1 = (1^2)^\frac{1}{2}$$ $$A \implies 1 = ((-1)^2)^\frac{1}{2}$$ $$A \implies 1 = ...
7
votes
1answer
196 views

Infinite series

$$\log2=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-\frac{1}{6}+\cdots$$ $$\frac{\log2}{2}=\frac{1}{2}-\frac{1}{4}+\frac{1}{6}-\frac{1}{8}+\frac{1}{10}-\frac{1}{12}+\cdots$$ Adding these two ...
4
votes
1answer
167 views

Solve for $x$: $\frac{\sqrt{x}}{2} = -1$

I recently saw in the Thread on Mathematical Misconceptions a post which implied that the following equation has no known solutions: $$\frac{\sqrt{x}}{2} = -1$$ Why does this not have any solutions? ...
19
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9answers
3k views

Why is $i^3$ (the complex number “$i$”) equal to $-i$ instead of $i$? [duplicate]

$$i^3=iii=\sqrt{-1}\sqrt{-1}\sqrt{-1}=\sqrt{(-1)(-1)(-1)}=\sqrt{-1}=i $$ Please take a look at the equation above. What am I doing wrong to understand $i^3 = i$, not $-i$?
18
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9answers
2k views

$1/i=i$. I must be wrong but why? [duplicate]

$$\frac{1}{i} = \frac{1}{\sqrt{-1}} = \frac{\sqrt{1}}{\sqrt{-1}} = \sqrt{\frac{1}{-1}} = \sqrt{-1} = i$$ I know this is wrong, but why? I often see people making simplifications such as ...
5
votes
5answers
155 views

$\mathrm{card} ( \mathbb{Q})=\mathrm{card}( \mathbb{Q^c})$: Overcoming Wrong Intuition

This is a widespread intuitive argument, asserting that $\mathrm{card} ( \mathbb{Q})=\mathrm{card}( \mathbb{Q^c})$: Between any two rational numbers there's an irrational one and vice versa. So ...
1
vote
1answer
112 views

What is wrong with the following proof?

This originates from the problem $\#3.8$ of Rudin, the problem is as followed: If $ \sum a_{n} $ converges and if ${b_{n}}$ is monotonic and bounded, prove $\sum a_{n}b_{n}$converges. I know ...
0
votes
0answers
42 views

What's wrong with my method that gives $0.\bar9 = 1$? [duplicate]

This is what I learned : $x = 0.\bar3$ (I) $10x=3.\bar3$ (II) (II)-(I) : $9x = 3 => x = 1/3.$ And It's OK. But now, I think this similar calculation gives me wrong answer: $x = 0.\bar9$ ...
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votes
3answers
164 views

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 ...
0
votes
2answers
52 views

FAIL Logarithms

So if I understand $log_11^x=x$ and $log_11^y=y$ and because $1^x=1^y$ $\rightarrow$ $log_11^x=log_11^y$ therefore $x=y$ but here is an easy contradiction $1^7=1^8$ following my logic means $7=8$ ...
3
votes
0answers
79 views

Why is this proof false? (Why is $e^i \neq 1$?) [duplicate]

I found this on MathOverflow: $$e^i = (e^i)^{(2\pi/2\pi)} = (e^{2\pi i})^{1/2\pi} = 1^{1/2\pi} = > 1.$$ I first saw this one many years ago, written on the wall of a bathroom stall in ...
2
votes
1answer
212 views

Help debunk a proof that zero equals one (no division)?

Unlike the more common variant of proof that 0=1, this does not use division. So, the reasoning goes like this: \begin{align} 0 &= 0 + 0 + 0 + \ldots && \text{not too controversial} \\ ...
8
votes
2answers
297 views

Why is $\lim\limits_{n\to\infty } 1=0$ incorrect?

$$ \lim_{n\to\infty } 1 =\lim_{n\to\infty }\frac{n}{n} =\lim_{n\to\infty }\frac{\overbrace{1+1+\ldots+1}^{n \text{ times}}}{n} =\lim_{n\to\infty }\frac{1}{n} + \lim_{n\to\infty }\frac{1}{n} + \ldots ...
-12
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1answer
198 views

Is this proof for the Goldbach Conjecture? [closed]

Let's prove the conjecture by induction. Claim: For every even number $n≥4$, there exist primes p and q such that p+q=n. Base case: $n=4$. Let $p=q=2$. Induction step: Say that we know that the ...
1
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1answer
78 views

Am I correctly identifying the fallacy in this induction “proof?”

The prompt states: Let us accept as true that a person can always walk an extra mile. Does the Principle of Induction then prove that a person can walk forever? Where is the fallacy? No. ...
1
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2answers
70 views

Proof that the derivative of a linear function is $0$.

We have a function $f(x)=x$ which is defined and is continuous on the set $S$ of all real numbers. The derivative at point $x$, $$f'(x)=\lim_{h\to 0} \frac{f(x+h)-f(x)}h$$ Using the theorems of ...
0
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2answers
97 views

Why is this proof wrong?

I am taking a course on logical equations and I found this exercise while reading about proofs and how to prove a given sentence and what kind of mistakes usually occur when you are trying to prove ...
0
votes
1answer
96 views

Fake Proof for 2-3=0 [closed]

I get that this doesn't make any sense but I'm not sure what exactly is wrong with it: Let's just let # = infinity |1+2+3...| = |-1-2-3...| so # = #/2 and 2# = # Since 2# = #, 3# = # and 2# = 3# Then, ...
9
votes
4answers
206 views

Which step is wrong in this proof

Proof: Consider the quadratic equation $x^2+x+1=0$. Then, we can see that $x^2=−x−1$. Assuming that $x$ is not zero (which it clearly isn't, from the equation) we can divide by $x$ to give ...
-1
votes
1answer
85 views

Two equal to One? Is it correct?

Some people ask me that two, Two equal to One ...
11
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2answers
252 views

Indefinite integral. Where is the mistake?

The problem was to compute $I=\int x^2\sin^{-1}(x)\ dx$ (where $\sin^{-1}(x)$ is the inverse function of $\sin(x)$). The answer of my students: firstly, we put $u=\sin^{-1}(x)$, so $x=\sin(u)$ and ...
2
votes
3answers
129 views

What's wrong with that proof?

What wrong with this proof? $(-1)=(-1)^{\frac{2}{2}}=(-1)^{2\times \frac{1}{2}}=\sqrt{1}=1$ then $1=-1$
1
vote
0answers
55 views

A weird infinity problem. [duplicate]

A weird infinity problem. I saw this on youtube but could not understand it: Let us add 1 + 2 + 4 + 8 + 16 + ... up to infinity x=(1+2+4+8+...) = 1(1+2+4+8+...) = (2-1)(1+2+4+8+...) = ...
3
votes
2answers
86 views

What am I doing wrong?

I am trying to prove the integral test for series, but got a strange result. Assume that $f$ is decreasing and positive. Because the series can be imagined as the area-sum of $1$-wide rectangles of ...
3
votes
2answers
366 views

why 64 is equal to 65 here?

how is this possible? I know there is some trick, should someone please explain?!