The Flying Colours Maths Blog: Latest posts

Ask Uncle Colin: Some number theory

Dear Uncle Colin, I need to show that $\sqrt{7}$ is in $\mathbb{Q}[\sqrt{2}+\sqrt{3}+\sqrt{7}]$ and I don’t really know where to start. We Haven’t Approached Tackling Such Questions Hi, WHATSQ, and thanks for your message! I am absolutely not a number theorist, although I must admit to getting a bit curious about

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A moment of neatness

Working through an FP2 question on telescoping sums (one of my favourite topics – although FP2 is full of those), we determined that $r^2 = \frac{\br{2r+1}^3-\br{2r-1}^3-2}{24}$. Adding these up for $r=1$ to $r=n$ gave the fairly neat result that $24\sum_{r=1}^{n} r^2 = \br{2n+1}^3 – 1 – 2n$. Now, there are

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Ask Uncle Colin: Powers

Dear Uncle Colin, I’m ok with my basic power laws, but I don’t understand why $x^0$ is always 1, and I get mixed up when it’s a fraction or a negative power. Can you help? Running Out Of Time Hi, ROOT, and thanks for your message! If it’s any consolation,

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Ten great books to give an interested mathematician

Oh no! Your favourite mathematician has a birthday/Christmas/other present-giving occasion coming up and you don’t know what book to get them! They’ve already got Cracking Mathematics and The Maths Behind, obviously… so what can you give them this year? Fear not, dear reader. I am at hand to list some

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Ask Uncle Colin: Some Rigour Required

Integration by substitution, rigorously Dear Uncle Colin, Can you explain why integration by substitution works? I get that you’re not allowed to ‘cancel’ the $dx$s, but can’t see how it works otherwise. – Reasonable Interpretation Got Our Understanding Ridiculed Hi, RIGOUR, and thanks for your message. First up, confession time:

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The Mathematical Ninja and the Cube Root of 4

The student swam away, thinking almost as hard as he was swimming. The cube root of four? The square root was easy enough, he could do that in his sleep. But the cube root? OK. Breathe. It’s between 1 and 2, obviously. What’s 1.5 cubed? The Mathematical Ninja isn’t going

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Ask Uncle Colin: A Ridiculous Restriction

Dear Uncle Colin, In a recent test, I was asked to differentiate $\frac{x^2+4}{\sqrt{x^2+4}}$. Obviously, my first thought was to simplify it to $\br{x^2+4}^{-\frac{1}{2}}$, but I’m not allowed to do that: only to use the quotient rule and the fact that $\diff {\sqrt{f(x)}}{x} = \frac{f'(x)}{2f(x)}$. When Evaluating, Inappropriate Rules Demanded Hi,

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Factorising a large number, part II: Gaussian integers

You know how things escalate on Twitter sometimes? Somebody makes an off-hand comment wondering whether a number is prime and suddenly you’re neck deep in number theory? This is the story of how you might factorise 842,909 on paper. In fact, it’s the second part of the story; we join

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Ask Uncle Colin: a nasty integral

Dear Uncle Colin, How would you integrate $e^x \sin(x)$ (with respect to $x$, obviously)? – Difficult Integral, Just Kan’t See The Right Answer Hi, DIJKSTRA, and thanks for your message! As seems to be the way recently, there are several ways to approach this. My favourite way One of the

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Wrong, But Useful: Episode 60

In this month’s episode of Wrong, But Useful, we’re joined by Special Guest Cohost @macaronique, who is Angela Brett in real life. Angela recites her poem They Might Not Be Giants We discuss joy in teaching and learning, and crosswords Number of the podcast: 44, the number of derangements of

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