Proof that π is irrational
In the 18th century, Johann Heinrich Lambert proved that the number π (pi) is irrational: that is, it cannot be expressed as a fraction a/b, where a is an integer and b is a non-zero integer. In the 19th century, Charles Hermite found a proof that requires no prerequisite knowledge beyond basic calculus. Three simplifications of Hermite's proof are due to Mary Cartwright, Ivan Niven and Bourbaki. Another proof, which is a simplification of Lambert's proof, is due to Miklós Laczkovich.
In 1882, Ferdinand von Lindemann proved that π is not just irrational, but transcendental as well.
In 1761, Lambert proved that π is irrational by first showing that this continued fraction expansion holds:
Then Lambert proved that if x is non-zero and rational then this expression must be irrational. Since tan(π/4) = 1, it follows that π/4 is irrational and therefore that π is irrational. A simplification of Lambert's proof is given below.
This proof uses the characterization of π as the smallest positive number whose half is a zero of the cosine function and it actually proves that π2 is irrational. As in many proofs of irrationality, the argument proceeds by reductio ad absurdum.
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