Proof by contradiction (aka a LaTeX test)

What is a proof by contradiction? A proof by contradiction is if ¬PF is true.

One assumes that a proposition P is False, and uses that to derive until a contradiction is reached, which can’t be True.

A popular example: Let’s prove that 2 is irrational. An irrational number is something that cannot be expanded into a fraction. (A common misconception is that Pi is 227 and therefore rational; no it is not exactly 227.

See http://mathworld.wolfram.com/PiFormulas.html

Assume that 2 is rational; such that we can represent it as ab, where ab is a fraction in lowest terms.

2=ab 2=a2b2 2b2=a2 2a 4a2 42b2 2b2 2b If both a and b are even, they are not in lowest terms, as both can be divided by 2 for further simplification. Hence we have a contradiction.

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