Part of a convergent series on Mathematics
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1+1=11
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We are number one
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(Incorrect) proof that 1 = 2[edit]
assume:

multiply both sides by
:

subtract
from both sides:

factor both sides:

divide both sides by
:

as A and B are equal, substitute all
s with
s:

continuing:


Q.E.D.
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
:

Integrate by parts,
[1]
assume
:

but
so:

we substract both sides by
:

then:

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

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:
(Incorrect) proof that an elephant and a mosquito have the same mass[edit]
Let
= mass of elephant in kg
Let
= mass of mosquito in kg
Let
= their combined mass in kg
Then:

multiplying the two latter equations:

adding
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
,
. So, the last line should not be
, but
. In essence, the "proof" is claiming that
implies
.
Another proof[edit]
Consider the function
, with domain the positive reals. Write

Then multiplying through by
we obtain

Differentiating yields

Since by assumption
we may divide through by
, whence
.
(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]

And dividing by
:

This may be why assfly hates complex numbers.
See also[edit]
External links[edit]
References[edit]
- ↑ 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
and the differentials
, then it is in the form in which it is most often seen:
