The Flying Colours Maths Blog: Latest posts

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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Ask Uncle Colin: A quadratic inequality

Dear Uncle Colin, I’ve come across a seemingly simple question I can’t tackle: solve $x^2 + 2x \ge 2$. I tried factorising to get $x(x+2) \ge 2$, which has the roots 0 and -2, but the book says the answer is $x < -1-\sqrt{3}$ or $x > -1 + \sqrt{3}$.

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A STEP expansion

A STEP question (1999 STEP II, Q4) asks: By considering the expansions in powers of $x$ of both sides of the identity $(1+x)^n (1+x)^n \equiv (1+x)^{2n}$ show that: $\sum_{s=0}^{n} \left( \nCr{n}{s} \right)^2 = \left( \nCr{2n}{n} \right)$, where $\nCr{n}{s} = \frac{n!}{s!(n-s)!}$. By considering similar identities, or otherwise, show also that: (i)

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Ask Uncle Colin: Two Answers For One Triangle

Dear Uncle Colin, I have a triangle with sides 4.35cm, 8cm and 12cm; the angle opposite the 4.35cm side is 10º1 and need to find the largest angle. I know how to work this out in two ways: I can use the cosine rule with the three sides, which gives

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