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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?

reinout-g Jun 18, 2014