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Tag Archives: pi
Happy tau day!
Happy Tau Day! The Tau Manifesto by Michael Hartl is now available in print, if you’re in the market for a particularly nerdy coffee table book and conversation starter: I also wrote about Tau Day ten years ago; see that … Continue reading
The Basel problem
I wanted to follow up on something I mentioned in my previous post: I claimed that At the time I didn’t know how to prove this, but I did some quick research and today I’m going to explain it! It … Continue reading
Happy Tau Day!
Happy day! , of course, is the fundamental circle constant which represents the ratio of any circle’s circumference to its radius. (In the past people have also used the symbol “” to represent half of ; perhaps you’ve heard of … Continue reading
Book reviews: Math Jokes 4 Mathy Folks and Easy as Pi
Math Jokes 4 Mathy Folks A few months ago, Patrick Vennebush was kind enough to send me a review copy of his new book, Math Jokes 4 Mathy Folks. It’s a treasuretrove of mathrelated jokes with a huge range of … Continue reading
Irrationality of pi: the integral that wasn't
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 inequality, integral, irrational, Niven, pi
8 Comments
Irrationality of pi: the impossible integral
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 Fundamental Theorem of Calculus, integral, irrational, Ivan Niven, pi, proof
4 Comments
Irrationality of pi: curiouser and curiouser
I’ve been remiss in posting here lately, which I will attribute to Christmas and New Year travelling and general craziness, and then starting a new semester craziness… but things have settled down a bit, so here we go again! Since … Continue reading
Posted in famous numbers, proof
Tagged derivatives, irrationality, Ivan Niven, pi, proof
10 Comments
Irrationality of pi: derivatives of f
In my previous post in this series, we defined the function and showed that . Today we’ll show the surprising fact that, for every positive integer , although and are not necessarily zero, they are always integers. (The notation means … Continue reading
Posted in calculus, famous numbers, proof
Tagged derivatives, irrationality, Ivan Niven, pi, proof
10 Comments
Irrationality of pi: the unpossible function
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 irrational, Ivan Niven, pi, proof, symmetric
7 Comments
Irrationality of pi
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