Cool (but incorrect) proof that 1 = 2
multiply both sides by "A":
subtract from both sides:
factor both sides:
divide both sides by (A-B):
as A and B are equal, substitute all "A"s with "B"s:
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)
substitute "tan x":
Integrate by parts, assume u = sec x and dv = sin x dx:
but cos x * sec x = 1 so:
we substract both sides by ʃtan x dx:
 Cool (but incorrect) proof that 1 = -1
rewrite -1 two different ways:
take the square root of both sides:
using laws of square roots, rewrite both sides:
multiply both sides by and reduce:
the square root of a number squared equals the number itself, so:
 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
multiplying the two latter equations:
adding to both sides:
which can be rewritten:
from which derives:
that is, mass of elephant = mass of mosquito.
 Another proof
Consider the function , with domain the positive reals. Write
Then multiplying through by we obtain
Since by assumption we may divide through by , whence .
 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.
- ↑ 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
- 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: