# Fun:Mathematical fallacies

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## Cool (but incorrect) proof that 1 = 2

assume:
$A = B$
multiply both sides by "A":
$AA = AB$
subtract $B^2$ from both sides:
$A^2 - B^2 = AB - B^2$
factor both sides:
$(A-B)(A+B) = B(A-B)$
divide both sides by (A-B):
$A + B = B$
as A and B are equal, substitute all "A"s with "B"s:
$B + B = B$
continuing:
$2B = B$
$2 = 1$
Q.E.D.

A good example of why dividing by zero is a bad move. Pedants would have us note that the above isn't a "proof" at all; the fact that you end up with 2 = 1 proves that the proof is wrong not that "2" actually equals "1".

## Cool (but incorrect) proof that 0 = -1 (or 1 = 2 if you prefer)

$\int\tan x\;dx = \int\tan x\;dx$
substitute "tan x":
$\int\tan x\;dx = \int\sin x\sec x\;dx$
Integrate by parts,[1] assume u = sec x and dv = sin x dx: $\int\tan x\;dx = -\sec x \cos x + \int\cos x\tan x\sec x\;dx$
but cos x * sec x = 1 so:
$\int\tan x\;dx = - 1 + \int\tan x\;dx$
we substract both sides by ʃtan x dx:
$\int\tan x\;dx - \int\tan x\;dx = - 1 + \int\tan x\;dx - \int\tan x\;dx$
then:
$0 = - 1$

## Cool (but incorrect) proof that 1 = -1

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

##  Cool (but incorrect) proof that an elephant and a mosquito have the same mass

Let a = mass of elephant in kg
Let x = mass of mosquito in kg
Let y = their combined mass in kg
Then:
$a + x = y$
$a = y - x$
$a - y = -x$
multiplying the two latter equations:
$a^2 - ay = x^2 - xy$
adding $(\frac{y}{2})^2$ to both sides:
$a^2 - ay + (\frac{y}{2})^2 = x^2 - xy + (\frac{y}{2})^2$
which can be rewritten:
$(a - \frac{y}{2})^2 = (x - \frac{y}{2})^2$
from which derives:
$a - \frac{y}{2} = x - \frac{y}{2}$
and finally:
$a = x$
that is, mass of elephant = mass of mosquito.

##  Another proof

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

$x = \underbrace{1 + \cdots + 1}_{x \; \textrm{times}}$.

Then multiplying through by $x$ we obtain

$x^2 = \underbrace{x + \cdots + x}_{x \; \textrm{ times}}$

Differentiating yields

$2x = \underbrace{1 + \cdots + 1}_{x \; \textrm{ times}} = x$.

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

##  Cool (but incorrect) proof that I am the Pope

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.

##  References

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 and dv = g′(x) dx, then it is in the form in which it is most often seen:
$\int u\,dv=uv-\int v\,du.$