May, 2016

A curious identity

There's something neat about an identity or result that seems completely unexpected, and this one is an especially nice one: $$ e^{2\pi \sin \left( i \ln(\phi)\right) }= -1$$ (where $\phi$ is the golden ratio.) It's one of those that just begs, "prove me!" So, here goes! I'd start with the

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Ask Uncle Colin: Trigonometric craziness

Dear Uncle Colin, My friend claims that $\frac { 2 - \frac{2 \sin(x)}{\cos(x)}}{\sin(x) - \cos(x)} \equiv -2\sec(x)$. I think she's crazy. What do you think? -- I Don't Even Need Trigonometry, I Teach Yoga Hi, IDENTITY -- even yoga teachers need trigonometry, though! Well, there's one way to find out

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How to invert a $3 \times 3$ matrix

So much wasted time. I spent much of my first two years at university cursing the names of Gauss and Jordan, railing at my lecturer (who grim-facedly assured me there were no more useful uses of a student’s thinking time than ham-fistedly rearranging these things), and thinking “there MUST be

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Ask Uncle Colin: I’ve lost my mojo

Dear Uncle Colin, I'm struggling with A-level. I used to love maths when I did [one board] at GCSE and now I'm doing [another board] at A-level, I don't enjoy it any more -- when I see a question, I can't even tell what it is they're asking. My teachers

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Review: The QAMA Calculator

It's billed as the calculator that won't think until you do: if you give it something to evaluate, it will refuse to give you an answer until you give it an acceptable approximation. On the surface, that's a great idea. If I had a coffee for every time I've rolled

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Ask Uncle Colin: An Undefined Integral

Dear Uncle Colin, I have a disagreement with my teacher about the integral $\int_{-1}^{1} x^{-1} \dx$. I understand you have to split the integral into two parts, which I'm happy with. They split it from -1 to $a$, letting $a \rightarrow 0_-$ and from $b$ to 1, letting $b \rightarrow

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A lovely trigonometric identity

When I was researching the recent Mathematical Ninja piece for Relatively Prime, I stumbled on something at MathWorld1 I'd never noticed before: if $t = \tan(x)$, then: $\tan(2x) = \frac{2t}{1 - t^2}$ $\tan(3x) = \frac{3t - t^3}{1 - 3t^2}$ $\tan(4x) = \frac{4t - 4t^3}{1 - 6t^2 + t^4}$ $\tan(5x) =

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Ask Uncle Colin: Don’t be daunted by D’Hondt?

Dear Uncle Colin, Up here in Scotland, we've got an election tomorrow. It's not as simple as the stupid first past the post elections you have down there in England, but even with our superior Scottish intelligence, some people are still struggling to understand how the system works. Do you

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A frankly ludicrous bit of paper arithmetic

Some time ago, someone asked Uncle Colin what the last two digits of $19^{1000}$ were. That caused few problems. However, Mark came up with a follow-up question: how would you estimate $19^{1000}$? I like this question, and set myself some rules: No calculators (obviously) Only rough memorised numbers ($e \approx

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I teach in my home in Abbotsbury Road, Weymouth.

It's a 15-minute walk from Weymouth station, and it's on bus routes 3, 8 and X53. On-road parking is available nearby.

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