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Fun:Mathematical fallacies

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(Incorrect) proof that 1 = 2[edit]



multiply both sides by A :


subtract B^2 from both sides:


factor both sides:


divide both sides by A-B :


as A and B are equal, substitute all As with Bs:





A good example of why dividing by zero is a bad move.

(Incorrect) proof that 0 = -1 (or 1 = 2 if you prefer)[edit]


substitute \tan(x) :


Integrate by parts, [1] assume u=\sec(x),dv=\sin(x)dx :


but \cos(x)\sec(x)=1 so:


we substract both sides by \int\tan(x)dx :

\int\tan(x)dx-\int\tan(x)dx =-1+\int\tan(x)dx-\int\tan(x)dx



(Incorrect) proof that 1 = -1[edit]

rewrite -1 two different ways:
take the square root of both sides:
using laws of square roots, rewrite both sides:
multiply both sides by \sqrt1\sqrt{-1} and reduce:
the square root of a number squared equals the number itself, so:

(Incorrect) proof that an elephant and a mosquito have the same mass[edit]

Let a = mass of elephant in kg
Let x = mass of mosquito in kg
Let y = their combined mass in kg


multiplying the two latter equations:


adding \left(\frac{y}{2}\right)^2 to both sides:


which can be rewritten:


from which derives:


and finally:


that is, mass of elephant = mass of mosquito.

The fallacy lies in the second to last step, when you take the square root of both sides. For all x\in\R , \sqrt{x^2}=|x| . So, the last line should not be a-\frac{y}{2}=x-\frac{y}{2} , but \left|a-\frac{y}{2}\right|=\left|x-\frac{y}{2}\right| . In essence, the "proof" is claiming that (-x)^2=(+x)^2 implies -x=+x.

Another proof[edit]

Consider the function f(x)=x , with domain the positive reals. Write

x=\underbrace{1+\cdots+1}_{x\text{ times}}

Then multiplying through by x we obtain

x^2=\underbrace{x+\cdots+x}_{x\text{ times}}

Differentiating yields

2x=\underbrace{1+\cdots+1}_{x\text{ times}}=x

Since by assumption x>0 we may divide through by x , whence 2=1 .

(Incorrect) proof that I am the Pope[edit]

This is a classic by the mathematician G. H. Hardy.

The Pope and I are two. [That is, two people.]

By the previous theorem, 2 = 1.

Therefore, the Pope and I are one.

(Technically, this proof is valid, in the sense that the conclusion follows from the premise. It's just that the premise is wrong.)

Another (incorrect) proof that 1 = -1[edit]

&\log(-1)=2\left(\frac{-\pi i}{2}\right)\\
&\pi i=-\pi i

And dividing by \pi i :


This may be why assfly hates complex numbers.

See also[edit]


  1. For a more complete discussion of this tactic, see Wikipedia. Here is a quick explanation of what is being done here:
    In the traditional calculus curriculum, this rule is often stated using indefinite integrals in the form
    \int f(x)g'(x)dx =f(x)g(x)-\int f'(x)g(x)dx
    or in an even shorter form, if we let u=f(x),v=g(x) and the differentials du=f'(x)dx,dv=g'(x)dx , then it is in the form in which it is most often seen:
    \int u\,dv=uv-\int v\,du