(Incorrect) proof that 1 = 2[edit]
assume:
![{\displaystyle A=B}](https://wikimedia.org/api/rest_v1/media/math/render/svg/045cafe35b1e9c9ac889481fd7178d6f59a77fdb)
multiply both sides by
:
![{\displaystyle AA=AB}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2fb5d75ea289eaf257cdd4017b5a8beff2d88a41)
subtract
from both sides:
![{\displaystyle A^{2}-B^{2}=AB-B^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/65c147b9a6e05391aa5c9ddb479b0437e3ffbef3)
factor both sides:
![{\displaystyle (A-B)(A+B)=B(A-B)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bf54b3b1dddfd97f8b937e0a784de99c1834a808)
divide both sides by
:
![{\displaystyle A+B=B}](https://wikimedia.org/api/rest_v1/media/math/render/svg/10f78898ab8483a00ac44a175dc4d757fd87d83c)
as A and B are equal, substitute all
s with
s:
![{\displaystyle B+B=B}](https://wikimedia.org/api/rest_v1/media/math/render/svg/88157d159148d8e7984b2efc1d4ecaab5961f930)
continuing:
![{\displaystyle 2B=B}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9d4da3d9b892a2f12e3498fee6c864732d3dfb10)
![{\displaystyle 2=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/66cf677955dbda7dbb2f214252c9b9833669022b)
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]
![{\displaystyle \int \tan(x)dx=\int \tan(x)dx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2099c469b577bf858de412c1397242f8f10c06e9)
substitute
:
![{\displaystyle \int \tan(x)dx=\int \sin(x)\sec(x)dx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ce6ed8b3f3c271c5250331ca83cb33e263042b4e)
Integrate by parts,
[1]
assume
:
![{\displaystyle \int \tan(x)dx=-\sec(x)\cos(x)+\int \cos(x)\tan(x)\sec(x)dx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6d6de994b6a5a8ed866997d13b4f1452475dd020)
but
so:
![{\displaystyle \int \tan(x)dx=-1+\int \tan(x)dx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/241c83b3c11992bc4e56e6b21d9fb962bd654e09)
we substract both sides by
:
![{\displaystyle \int \tan(x)dx-\int \tan(x)dx=-1+\int \tan(x)dx-\int \tan(x)dx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/420ef5d0b0bac56c83056763c4a0f2ba944d73d1)
then:
![{\displaystyle 0=-1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5ddd90de97b76a81d92f562ac32fae0f7d9ade64)
(Incorrect) proof that 1 = -1[edit]
assume:
![{\displaystyle \mathbf {-1} =\mathbf {-1} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/01dc5c9e8ecdaafe49d924958baf4092267d669d)
rewrite -1 two different ways:
![{\displaystyle {\frac {1}{-1}}={\frac {-1}{1}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0de4b3862d55ed5c0e41e0d2c00df7a775f54729)
take the square root of both sides:
![{\displaystyle {\sqrt {\frac {1}{-1}}}={\sqrt {\frac {-1}{1}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b85d05197e604aae438961ddb25c98b4697539c)
using laws of square roots, rewrite both sides:
![{\displaystyle {\frac {\sqrt {1}}{\sqrt {-1}}}={\frac {\sqrt {-1}}{\sqrt {1}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/db277c3af5801596722e14548e84a3efcfbbdb17)
multiply both sides by
and reduce:
![{\displaystyle {\sqrt {1}}{\sqrt {1}}={\sqrt {-1}}{\sqrt {-1}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/baf75d49d496acea34155a39fd797b3ba63bb0bc)
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:
![{\displaystyle {\begin{aligned}&a+x=y\\&a=y-x\\&a-y=-x\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f0e803bb76247dcd88c7d85d33c1e722d10f079b)
multiplying the two latter equations:
![{\displaystyle a^{2}-ay=x^{2}-xy}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9385214b7f653fbfa2231288e0ff7ecf03d1b3c6)
adding
to both sides:
![{\displaystyle a^{2}-ay+\left({\frac {y}{2}}\right)^{2}=x^{2}-xy+({\frac {y}{2}})^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c17498635503831336038f07249b3e6aadd36542)
which can be rewritten:
![{\displaystyle \left(a-{\frac {y}{2}}\right)^{2}=\left(x-{\frac {y}{2}}\right)^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2ef1f47d290e7daaf2241b3deb52a8128e5a1e17)
from which derives:
![{\displaystyle a-{\frac {y}{2}}=x-{\frac {y}{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cad894dcbd01ef68db2126d80299cc4f246d5dec)
and finally:
![{\displaystyle a=x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/60c2905b9231255762355e8d012c24310eeeb667)
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
![{\displaystyle x=\underbrace {1+\cdots +1} _{x{\text{ times}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3582134c1aa96bd011f7e8c4e2cd9d9a3cf35bdb)
Then multiplying through by
we obtain
![{\displaystyle x^{2}=\underbrace {x+\cdots +x} _{x{\text{ times}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cb728c1d8c01ca3f0f4061a4096200950ec396e2)
Differentiating yields
![{\displaystyle 2x=\underbrace {1+\cdots +1} _{x{\text{ times}}}=x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f96fbda4327893048c0619d87983fc266d1f4e93)
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]
![{\displaystyle {\begin{aligned}&\log \left((-i)^{2}\right)=\log \left((-i)^{2}\right)\\&\log \left((-i)^{2}\right)=2\log(-i)\\&\log(-1)=2\left({\frac {-\pi i}{2}}\right)\\&\pi i=-\pi i\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/909e624f44fd5d725bcd0de64baa74cba3f7fcbb)
And dividing by
:
![{\displaystyle 1=-1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7fe2156ea89dbb501ff119c337fb4819b1fca5ca)
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
![{\displaystyle \int f(x)g'(x)dx=f(x)g(x)-\int f'(x)g(x)dx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2889eaf9084ef92b29de58414b1005c3905dad75)
- 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:
![{\displaystyle \int u\,dv=uv-\int v\,du}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bddb2871f48408d699da1d94af2076a15008989a)