# Fun:Mathematical fallacies

## (Incorrect) proof that 1 = 2

assume:

$A=B$ multiply both sides by $A$ :

$AA=AB$ subtract $B^{2}$ from both sides:

$A^{2}-B^{2}=AB-B^{2}$ factor both sides:

$(A-B)(A+B)=B(A-B)$ divide both sides by $A-B$ :

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

$B+B=B$ continuing:

$2B=B$ $2=1$ A good example of why dividing by zero is a bad move.

## (Incorrect) proof that 0 = -1 (or 1 = 2 if you prefer)

$\int \tan(x)dx=\int \tan(x)dx$ substitute $\tan(x)$ :

$\int \tan(x)dx=\int \sin(x)\sec(x)dx$ Integrate by parts,  assume $u=\sec(x),dv=\sin(x)dx$ :

$\int \tan(x)dx=-\sec(x)\cos(x)+\int \cos(x)\tan(x)\sec(x)dx$ but $\cos(x)\sec(x)=1$ so:

$\int \tan(x)dx=-1+\int \tan(x)dx$ we substract both sides by $\int \tan(x)dx$ :

$\int \tan(x)dx-\int \tan(x)dx=-1+\int \tan(x)dx-\int \tan(x)dx$ then:

$0=-1$ ## (Incorrect) proof that 1 = -1

assume:
$\mathbf {-1} =\mathbf {-1}$ rewrite -1 two different ways:
${\frac {1}{-1}}={\frac {-1}{1}}$ take the square root of both sides:
${\sqrt {\frac {1}{-1}}}={\sqrt {\frac {-1}{1}}}$ using laws of square roots, rewrite both sides:
${\frac {\sqrt {1}}{\sqrt {-1}}}={\frac {\sqrt {-1}}{\sqrt {1}}}$ multiply both sides by ${\sqrt {1}}{\sqrt {-1}}$ and reduce:
${\sqrt {1}}{\sqrt {1}}={\sqrt {-1}}{\sqrt {-1}}$ the square root of a number squared equals the number itself, so:
$\mathbf {1} =\mathbf {-1}$ ## (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:

{\begin{aligned}&a+x=y\\&a=y-x\\&a-y=-x\end{aligned}} multiplying the two latter equations:

$\displaystyle a^2-ay=x^2-xy$

adding $\displaystyle \left(\frac{y}{2}\right)^2$ to both sides:

$\displaystyle a^2-ay+\left(\frac{y}{2}\right)^2=x^2-xy+(\frac{y}{2})^2$

which can be rewritten:

$\displaystyle \left(a-\frac{y}{2}\right)^2=\left(x-\frac{y}{2}\right)^2$

from which derives:

$\displaystyle a-\frac{y}{2}=x-\frac{y}{2}$

and finally:

$\displaystyle a=x$

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 $\displaystyle x\in\R$ , $\displaystyle \sqrt{x^2}=|x|$ . So, the last line should not be $\displaystyle a-\frac{y}{2}=x-\frac{y}{2}$ , but $\displaystyle \left|a-\frac{y}{2}\right|=\left|x-\frac{y}{2}\right|$ . In essence, the "proof" is claiming that $\displaystyle (-x)^2=(+x)^2$ implies $\displaystyle -x=+x$ .

## Another proof

Consider the function $\displaystyle f(x)=x$ , with domain the positive reals. Write

$\displaystyle x=\underbrace{1+\cdots+1}_{x\text{ times}}$

Then multiplying through by $\displaystyle x$ we obtain

$\displaystyle x^2=\underbrace{x+\cdots+x}_{x\text{ times}}$

Differentiating yields

$2x=\underbrace {1+\cdots +1} _{x{\text{ times}}}=x$ Since by assumption $x>0$ we may divide through by $x$ , whence $2=1$ .

## (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 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

{\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}} And dividing by $\pi i$ :

$1=-1$ This may be why assfly hates complex numbers.