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Tag Archives: Ivan Niven
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