Fun:Mathematical fallacies
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Contents
- 1 (Incorrect) proof that 1 = 2
- 2 (Incorrect) proof that 0 = -1 (or 1 = 2 if you prefer)
- 3 (Incorrect) proof that 1 = -1
- 4 (Incorrect) proof that an elephant and a mosquito have the same mass
- 5 Another proof
- 6 (Incorrect) proof that I am the Pope
- 7 Another (incorrect) proof that 1 = -1
- 8 See also
- 9 References
(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:
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]
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: