Fun:Mathematical fallacies

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1+1=11


(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]

  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
    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: