Irrationality of pi

Everyone knows that \pi—the ratio of any circle’s diameter to its circumference—is irrational, that is, cannot be written as a fraction a/b. This also means that \pi‘s decimal expansion goes on forever and never repeats …but have you ever seen a proof of this fact, or did you just take it on faith?

The irrationality of \pi was first proved (according to modern standards of rigor) in 1768 by Lambert, but his proof was rather complicated. A more elementary proof, using only basic calculus, was given in 1947 by Ivan Niven. You can read his original paper here, but it’s rather terse! Just as I did for Calkin and Wilf’s paper, Recounting the Rationals, I plan to write a series of posts explaining Niven’s proof in a bit more detail, with some accompanying intuition. I’ll assume a basic knowledge of calculus; if you don’t know calculus, just hang tight for a few posts!

Here’s the basic outline of the proof. We begin by supposing that \pi is rational: in particular, suppose \pi = a/b for some integers a and b. We’ll then use these values of a and b to define a special function f(x), about which we will show the following:

\int_0^\pi f(x) \sin(x) dx is an integer, AND

0 < \int_0^\pi f(x) \sin(x) dx < 1.

But this is absurd! There are no integers greater than zero and less than one. The inescapable conclusion will be that our initial assumption—that \pi = a/b—was false.

In my next post, we’ll define the special function f(x) and begin exploring some of its properties.

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14 Responses to Irrationality of pi

  1. Thank you!

    I have long wanted to understand this proof. I’ve looked it up online maybe a year ago, and then just a few days ago. I couldn’t follow the proofs I saw. (‘Couldn’t’ is probably not the right word. I wasn’t willing to put in the effort, when I wasn’t sure I’d make it out the other end.)

    I already like your intro. Yep, easy (for me) to see that showing those two things will show a contradiction, and prove our premise wrong.

  2. Brent says:

    Sue: Great! Please keep commenting as the series proceeds if there’s anything you find unclear.

  3. Dave says:

    Looking forward to it!

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  5. david says:


    I just found this site by accident.

    I can, for some reason, never remember the explanation for ‘sigma’ – as it was always explained in such complex language – but not on your site…. great stuff …. you had my attention……

    I saw “famous numbers” on the right, and clicked on it… to get this ‘PI’ page….

    How dare you create a new page yesterday that I have navigated too…. now I’ve got to come back to see the rest of the article (and probably waste more of my time on your site)

    My boss won’t be happy, (but it is my lunch break!)….

    but I will!

    Great site I look forward to the second instalment of ‘PI’


    ps I love and hate the net in equal proportions!

  6. Mike Manganello says:

    Very few things make me sit up and go, “Oh!” like this post did. I can’t wait to follow the rest of the problem!

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  12. Bonita says:

    Could you provide me with author and date information on “Irrationality of pi.” I and writing a paper on pi and would like to cite this…


  13. Brent says:

    Hi Bonita, you can find all the information here:

    Good luck with your paper!

  14. Pingback: Irrationality of pi: the unpossible function | The Math Less Traveled

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