Category Archives: famous numbers

Mersenne primes and the Lucas-Lehmer test

Mersenne numbers, named after Marin Mersenne, are numbers of the form . The first few Mersenne numbers are therefore , , , , , and so on. Mersenne numbers come up all the time in computer science (for example, is … Continue reading

Posted in arithmetic, computation, famous numbers, iteration, modular arithmetic, number theory, primes | Tagged , , , , | 3 Comments

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

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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 , , , , | 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

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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 , , , , | 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 , , , , | 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 , , , , | 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

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

Carnival of Math #48, and Monday Math Madness #25

The 48th Carnival of Mathematics is posted at Concrete Nonsense. My favorite posts include Foxmath’s post about a strange iterated sequence involving pi and this amazing picture of a fractal cabbage. Also near and dear to my heart is Mark … Continue reading

Posted in challenges, counting, famous numbers, fractals, links | Tagged , , , , | 1 Comment

Predicting pi: pretty graphs and convergents

Recall the challenge I posed in a previous post: given the sequence of integers , what can you learn about (assuming you didn’t know anything about it before)? The answer, as explained in another post, is that you can learn … Continue reading

Posted in convergence, famous numbers, pattern, sequences | Tagged , , , | 5 Comments