The Flying Colours Maths Blog: Latest posts

Ask Uncle Colin: A Fractional Equation

Dear Uncle Colin, I’m trying to solve $\frac{x}{x-1} = \frac{1}{x-1}$. I think the answer should be 1, but my teacher disagrees. What do you think? - First Results Are Contradicting Teachers’ - Is One Nonsense? Hi, FRACTION, and thanks for your message! It’s tempting, here, to multiply both sides by

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Binet’s formula and Haskell

This is an extended version of my entry in the Lockdown Mathoff at the Aperiodical Binet’s formula1 is a lovely way to generate the $n$th Fibonacci number, $F_n$. If $\phi = \frac{1}{2}\left(\sqrt{5} + 1\right)$, then $$F_n = \frac{ \phi^n - (-\phi)^{-n} }{\sqrt{5}}$$ Haskell and computation The main reason I’m writing

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Ask Uncle Colin: The Empty Product

Dear Uncle Colin, Why does $0! = 1$ and not 0? - Nothing Is Logical Hi, NIL, and thanks for your message! My best explanation for this - by which I mean, the one I can get some people to accept, goes like this: $4! = 4 \times 3 \times

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Dictionary of Mathematical Eponymy: The Quine-McCluskey Algorithm

There was a Fields Medallist named Dan Quillen, after whom are named several things in topics I’ve never head of. Other than Quillen, so far as I can tell, the only mathematical eponyms beginning with Q relate to Willard Van Ormine Quine. I know him from Godel, Escher, Bach, where

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Ask Uncle Colin: Rotating vectors

Dear Uncle Colin, I need to find a unit vector in the xy-plane that makes an angle of 45 degrees with the vector $3\bi + 4\bj$. How would you do that? - Don’t Enjoy Maths Of Integer Vectors Rotating Enough Hi, DEMOIVRE, and thanks for your message! I can think

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Some puzzles from Cav

A couple of puzzles that came my way via @srcav today: Cav’s solutions to this one are here; mine are below the line further down. Interesting angle puzzle — Cav (@srcav) July 8, 2019 And to this one, here Have a go yourself before you read on! I’ve

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Ask Uncle Colin: Two trig identities

Dear Uncle Colin These two trig questions are getting me frustrated! What do you recommend? Prove $\frac{\tan(2x) + \cot(x)}{\tan(2x) - \tan(x)} \equiv \cot^2(x)$ Prove $\frac{1 + \sin(2x)}{1+\cos(2x)} = \frac{1}{2}\left(1+\tan(x)\right)^2$ - I Don’t Like Equations Hi, IDLE, and thanks for your message! The great temptation here is to send you a

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Powers and remainders

Over on Reddit, a couple of “last digit” puzzles crossed my path, and I thought I’d share the tricks I used, as much for my reference as anything else. 1) Show that the last digit of $6^k$ is 6, for any positive integer $k$. There’s a standard way to prove

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Wrong, But Useful: Episode 78

In episode 78 of Wrong, But Useful, we're joined by @c0mplexnumber, who is Clarissa Grandi in real life. This month, we discuss: Clarissa's Artful maths books, available via Tarquin - the activity book and the teacher's guide Number of the podcast: $\phi$ (and 3D maths) @anniek_p's #mathartchallenge Aperiodical’s big math-off

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Ask Uncle Colin: An Additive Inverse

Dear Uncle Colin, I need to find $35^{-1} \pmod {234}$, but I’m not getting the right answer. Can you help me?1 - It’s Not Very Easy Resolving Such Expressions Hi, INVERSE, and thanks for your message! We’re looking for a number $x$ such that $35x = 234n + 1$, for

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