Tag Archives: irrational

Book review: The Irrationals

Posted on October 25, 2012 by Brent

The Irrationals: A Story of the Numbers You Can’t Count On Julian Havil Princeton University Press sends me lots of cool books to review! Here’s one. Remember the irrational numbers, which can’t be expressed as a ratio of integers ? … Continue reading →

Posted in books, number theory, review | Tagged , , , | 4 Comments

Irrationality of pi: the integral that wasn't

Posted on February 11, 2010 by Brent

And now for the punchline! Today we’ll show that, for large enough values of , completing the proof of the irrationality of . First, let’s show that is positive when . We know that is positive for . But I … Continue reading →

Posted in algebra, calculus, convergence, famous numbers, proof, trig | Tagged , , , , | 8 Comments

Irrationality of pi: the impossible integral

Posted on February 6, 2010 by Brent

We’re getting close! Last time, we defined a new function and showed that and are both integers, and that . So, consider the following: The first step uses the product rule for differentiation (recalling that and ); the last step … Continue reading →

Posted in famous numbers, proof | Tagged , , , , , | 4 Comments

Irrationality of pi: the unpossible function

Posted on December 12, 2009 by Brent

Recall from my last post what we are trying to accomplish: by assuming that is a rational number, we are going to define an unpossible function! So, without further ado: Suppose , where and are positive integers. Define the function … Continue reading →

Posted in calculus, famous numbers, proof | Tagged , , , , | 7 Comments

Irrationality of pi

Posted on December 7, 2009 by Brent

Everyone knows that —the ratio of any circle’s diameter to its circumference—is irrational, that is, cannot be written as a fraction . This also means that ‘s decimal expansion goes on forever and never repeats …but have you ever seen … Continue reading →

Posted in calculus, famous numbers, proof | Tagged , , , | 14 Comments