Posted in ask uncle colin.

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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Posted in ninja maths.

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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Posted in ask uncle colin.

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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Posted in factorising.

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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Posted in ask uncle colin, integration.

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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Posted in podcasts.

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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Posted in factorising.

There is a theorem that states: if a number can be written as the sum of two squares in two different ways, it is composite. Because of Twitter, I became interested in factorising $n=842,909$. Can this be written as the sum of two squares1? How - without cheating and using

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Posted in ask uncle colin.

Dear Uncle Colin, What's the correct method to find the general solution of $y''+4y'+4y=(3+x)e^{-2x}$? I've got the complementary function just fine (it's $y=Ae^{-2x}+Bxe^{-2x}$), but I'm going in circles with the particular integral. - Differential Equation's Solutions Are Really Gruesome; Ugly Exponential Scariness Hi, DESARGUES, and thanks for your message! I

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Posted in reviews.

The best science writers (as far as I'm concerned, at least) are the ones who make you feel like you're sitting down for a coffee with a smart friend, excited about a thing they know about, a thing they've found out, or a thing they've just put together. I have

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Posted in ask uncle colin.

Dear Uncle Colin, How do people do decimal calculations like $80 \times 0.15$ in their heads? It seems impossible. - Doesn't Everyone Seem Clever, Answering Readily These Evil Sums Hi, DESCARTES, and thank you for your message! There are several possible strategies for a question like this - how I

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