Fun:Mathematical fallacies
Contents |
[edit] (Incorrect) proof that 1 = 2
assume:

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:

continuing:


Q.E.D.
A good example of why dividing by zero is a bad move.
[edit] (Incorrect) proof that 0 = -1 (or 1 = 2 if you prefer)

substitute "tan x":

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

but cos x * sec x = 1 so:

we substract both sides by ʃtan x dx:

then:

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



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 x∈ℝ,
. So, the last line should not be
, but
. In essence, the "proof" is claiming that
implies
.
[edit] Another proof
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
.
[edit] (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.
(Technically, this proof is correct, in the sense that the conclusion follows from the premise. It's just that the premise is wrong.)
[edit] Another (incorrect) proof that 1 = -1
And dividing by iπ:
This may be why assfly hates complex numbers.
[edit] References
- ↑ 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:
