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Concept
Proof that π is irrational
Summary
In the 1760s, Johann Heinrich Lambert was the first to prove that the number pi is irrational, meaning it cannot be expressed as a fraction , where and are both integers. 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 Nicolas Bourbaki. Another proof, which is a simplification of Lambert's proof, is due to Miklós Laczkovich. Many of these are proofs by contradiction.
In 1882, Ferdinand von Lindemann proved that is not just irrational, but transcendental as well.
In 1761, Lambert proved that is irrational by first showing that this continued fraction expansion holds:
Then Lambert proved that if is non-zero and rational, then this expression must be irrational. Since , it follows that is irrational, and thus is also irrational. A simplification of Lambert's proof is given below.
Written in 1873, this proof uses the characterization of as the smallest positive number whose half is a zero of the cosine function and it actually proves that is irrational. As in many proofs of irrationality, it is a proof by contradiction.
Consider the sequences of real functions and for defined by:
Using induction we can prove that
and therefore we have:
So
which is equivalent to
Using the definition of the sequence and employing induction we can show that
where and are polynomial functions with integer coefficients and the degree of is smaller than or equal to In particular,
Hermite also gave a closed expression for the function namely
He did not justify this assertion, but it can be proved easily. First of all, this assertion is equivalent to
Proceeding by induction, take
and, for the inductive step, consider any natural number If
then, using integration by parts and Leibniz's rule, one gets
If with and in , then, since the coefficients of are integers and its degree is smaller than or equal to is some integer In other words,
But this number is clearly greater than On the other hand, the limit of this quantity as goes to infinity is zero, and so, if is large enough, Thereby, a contradiction is reached.
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