Posted in circles, geometry, trigonometry.

Here's a nice use of circle theorems: ever wondered why the sine rule works?

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Posted in ninja maths.

"Boom!" said the Mathematical Ninja, as the smoke cleared and the student jumped. Shell-shocked, he looked again at the whiteboard. "Wh-what just happened?" The Mathematical Ninja sighed. "OK, one more time. We're trying to estimate the size of the shapes you'd need to cover a sphere with $n$ patches, which

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Posted in fractions, reader questions.

A friend asks for REASONS: THAERTTHUGAL n. The smallest named fraction in the ancient Tamil language = 1/2323824530227200000000. — The QI Elves (@qikipedia) April 7, 2014 A who to the what now? A twelve-letter word, a thaerrhugaL, representing a number somewhere in the region of twenty-three ninety-ninths of a sextillionth.

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Posted in Introduction.

I don't quite remember who asked this, but at some point someone asked me to list ten books that have stayed with me. 1066 And All That - Sellar and Yeatman I've a beat-up copy of this I borrowed (what do you mean stole?) from my parents, and it's one

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Posted in podcasts.

In this month's episode of Wrong, But Useful, @reflectivemaths (Dave Gale) and I discuss: How Dave thinks Baby Bill should learn to count Decimal time McDonalds apple pies - Made 15:26. Ready 15:36 Discard 16:56 "School lunch could save you up to £437 a year. Only 1% of packed lunches

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Posted in algebra, binomial, further pure 2.

The Mathematical Ninja, some time ago, pointed out a curiosity about Pascal's Triangle and the Maclaurin1 (or Taylor2 ) series of a product: $\diffn{n}{(uv)}{x} = uv^{(n)} + n u'v^{(n-1)} + \frac{n(n-1)}{2} u'' v^{(n-2)} + ...$, where $v^{(n)}$ means the $n$th derivative of $v$ - which looks a lot like Pascal's

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Posted in logarithms.

A MathsJam classic question asks: Without using a calculator, which is bigger: $e^\pi$ or $\pi^e$? It's one of those questions that looks perfectly straightforward: you just take logs and then... oh, but is $\pi$ bigger than $e\ln(\pi)$? The Mathematical Ninja says "$\ln(\pi)$ is about 1.2, because $\pi$ is about 20%

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Posted in complex numbers, further pure 2.

A numerical curiosity today, all to do with $\i$th powers. Euler noticed, some centuries ago, that $13({2^\i + 2^{-\i}})$ is almost exactly $20$. As you would, of course. But why? And more to the point, how do you work out an $\i$th power? It's all to do with the exponential

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Posted in further pure 2.

A student asks: I know the method for finding the hyperbolic arcosine1 - but I get two roots out of my quadratic formula. Why is it just the positive one? A quick refresher, in case you don't know the method Hyperbolic functions are the BEST FUNCTIONS IN THE WHOLE WIDE

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Posted in complex numbers, further pure 2.

Just for a change, an FP3 topic. I've been struggling to tutor complex mappings properly (mainly because I've been too lazy to look them up), but have finally seen - I think - how to solve them with minimal headache. A typical question gives you a mapping from the (complex)

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