+2353 Funny fallacious 'proofs' (ordered by difficulty level);
'Proof' that a dog has nine legs;
A dog has nine legs;
No dog has five legs
A dog had four more legs than no dog
A dog had nine legs.
'Proof' that 2 = 1
'Proof' that 1 = 0
$$(n+1)^2 = n^2 + 2n + 1\\
(n+1)^2 -(2n+1) = n^2\\
(n+1)^2 - (2n+1) - n(2n+1) = n^2 - n(2n+1)\\
(n+1)^2 - (n+1)(2n+1) = n^2 - n(2n+1)\\
(n+1)^2 - (n+1)(2n+1)+\frac{(2n+1)^2}{4} = n^2-n(2n+1)++\frac{(2n+1)^2}{4}\\
((n+1)-\frac{2n+1}{2})^2 = (n-\frac{2n+1}{2})^2\\
(n+1)-\frac{2n+1}{2} = n - \frac{2n+1}{2}\\
n+1 = n\\
1 = 0\\$$
'Proof' that 1 is the largest number
'Proof' that 1 = -1
$$1 = \sqrt{1} = \sqrt{-1*-1} = \sqrt{-1}*\sqrt{-1} = i*i = i^2 = -1$$
'Proof' that all cars have the same color
Proof by induction
If there is one car, all cars have the same color.
Fix an arbitrary n. Suppose that for any set of n cars they have the same color. Consider any set of n+1 cars. Then car 1 through n have the same color. We also know that cars 2 through n+1 have the same color by the inductive hypothesis. Hence all cars must have the same color.
'Proof' that 0 = 1
Using integration by parts we have
$$\int\frac{1}{x} = x*\frac{1}{x} - \int x d\frac{1}{x}\\
\int\frac{1}{x} = 1 - \int x(-\frac{1}{x^2})dx\\
\int\frac{1}{x} = 1 + \int \frac{1}{x}dx\\
0 = 1$$
'Proof' that $$e^x = 1 \mbox{ }\forall x$$
$$e^x = exp(i2\pi*\frac{x}{i2\pi}) = exp(i2\pi)^\frac{x}{i2\pi} = 1^{\frac{x}{i2\pi}} = 1$$
Have I just disproven your world?
