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Category Archives: induction
PIE: proof by algebra
In my previous post I stated a very formal, general form of the Principle of InclusionExclusion, or PIE.1 In this post I am going to outline one proof of PIE. I’m not going to give a completely formal proof, because … Continue reading
Book review: Roads to Infinity
What is infinity? What is proof? These are two of the biggest questions mathematicians have grappled with over the years. In this wellwritten and fascinating book, John Stillwell takes us on a tour through some of the answers to these … Continue reading
Posted in arithmetic, books, computation, induction, infinity, logic, proof, review
Tagged infinity, John Stillwell, proof, roads
Comments Off on Book review: Roads to Infinity
Cassini’s identity
My previous post asked you to take any Fibonacci number, square it, and also multiply the two adjacent Fibonacci numbers, and see if a pattern emerged. Here’s a table I made for the first 6 Fibonacci numbers: (Hmm, the numbers … Continue reading
The hyperbinary sequence and the CalkinWilf tree
And now, the amazing conclusion to this series of posts on Neil Calkin and Herbert Wilf’s paper, Recounting the Rationals, and the answers to all the questions about the hyperbinary sequence. Hold on to your hats! The CalkinWilf Tree First, … Continue reading
Posted in arithmetic, computation, induction, iteration, number theory, pattern, proof, recursion, sequences, solutions
Tagged algorithm, binary, CalkinWilf, Euclidean, Haskell, hyperbinary, tree
6 Comments
More hyperbinary fun
When I originally posed Challenge #12, a certain Dave posted a series of comments with some explorations and partial solutions to part II (the hyperbinary sequence). Although I gave the “solution” in my last post, no solution to any problem … Continue reading
Posted in challenges, induction, pattern, proof, recursion, sequences, solutions
Tagged hyperbinary, induction
12 Comments
Challenge #12 solution, part II
Yes, that’s right, that Challenge #12, posted one year, five months, and a day ago. You see, I have this nasty habit of starting things and not finishing them… well, better late than never! Question two of the aforementioned challenge … Continue reading
Posted in challenges, counting, induction, pattern, sequences, solutions
Tagged binary, hyperbinary
15 Comments
Recounting the Rationals, part IV
Continuing a series about the CalkinWilf tree (see those links for some background), today I’d like to show why all the rationals in the tree must be in lowest terms. Let’s start off with a little number theory! What do … Continue reading
Posted in induction, number theory, pattern, proof
11 Comments
Golden powers
So, we know from a previous challenge that . That’s a pretty interesting property, which is shared only by its cousin, . I wonder whether other powers of have special properties too? Let’s see: Interesting! What about ? And ? … Continue reading
Posted in famous numbers, fibonacci, golden ratio, induction, proof
6 Comments