We are creating the world's most trusted encyclopedia and knowledge base.
Once you join us and log in, you'll be able to edit this page instantly!

Proof that π is irrational

From Citizendium, the Citizens' Compendium

Jump to: navigation, search
Image:Statusbar2.png
Main Article
Talk
Definition [?]
Related Articles  [?]
Bibliography  [?]
External Links  [?]
 
This is a draft article, under development. These unapproved articles are subject to a disclaimer.

Although the mathematical constant known as π (pi) has been studied since ancient times, it was not until the 18th century that it was proved to be an irrational number.

In the 20th century, proofs were found that require no prerequisite knowledge beyond integral calculus. One of those is due to Ivan Niven, another to Mary Cartwright (reproduced in Jeffreys).

Niven's proof

Like all proofs of irrationality, the argument proceeds by reductio ad absurdum. Suppose π is rational, i.e. π = a / b for some integers a and b, which may be taken without loss of generality to be positive.

Given any positive integer n we can define functions f and F as follows:

f(x) = {x^n(a - bx)^n \over n!},\,
F(x) = f(x) + \cdots + (-1)^j f^{(2j)}(x) + \cdots + (-1)^n f^{(2n)}(x). \,

Then f is a polynomial function each of whose coefficients is 1/n! times an integer. It satisfies the identity

f(x) = f(\pi - x)\,

and the inequality

0 \le f(x) \le {\pi^n a^n \over n!} \mbox{ for } 0 \le x \le \pi.\qquad\qquad(*)

Observe that for 0 ≤ j < n, we have

f(j)(0) = f(j)(π) = 0.

For j ≥ n, f (j)(0) and f (j)(π) are integers. Consequently F(0) and F(π) are integers. Next, observe that

F + F'' = f,\,

and hence

(F'\cdot\sin - F\cdot\cos)' =f\cdot\sin.

It follows that

\int_0^\pi f(x)\sin(x)\,dx

is a positive integer. But by the inequality (*), the integral approaches 0 as n approaches infinity, and that is impossible for a sequence of positive integers.

References

  • Ivan Niven, "A Simple Proof that π is Irrational", Bull.Amer.Math.Soc. v. 53, p. 509, (1947)
  • Harold Jeffreys, Scientific Inference, 3rd edition, Cambridge University Press, 1973.
Retrieved from "http://en.citizendium.org/wiki/Proof_that_%CF%80_is_irrational"
Views
Personal tools