# The Flying Colours Maths Blog: Latest posts

## A student asks: How do you simplify surds?

A student asks: How could I simplify a sum like $(\sqrt 3+\sqrt 2)(\sqrt 3-\sqrt 2)$? Great question! The trick is to treat it like it's an algebraic bracket, like this: $(x + y)(x - y) = x^2 + yx - xy - y^2$ But then you've got $+yx -xy$ in

## A trigonometric trick: Secrets of the Mathematical Ninja

"Have you seen this trick?" asked the student. "If you know all three sides of a right-angled triangle, you can estimate the other angles - $A \simeq \frac{86a}{\frac b2 + c}$!" The Mathematical Ninja thought for a moment, and casually threw a set-square into the wall, millimetres from the student's

## Euclid the Game: level 20

Note: since I wrote this post, level 20 has moved to level 23. It may move again in the future, I suppose. Rather than keep updating, it's the one with the tangent to two circles. I LOVE Euclid, the game - it's a brilliant, interactive way to get students (and

## A proof of the sine rule

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

## A Ninja Masterclass

"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

## Tamil Fractions

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.

## Ten books that stayed with me

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

## Wrong, But Useful: Episode 16

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

## Why the Maclaurin series gives you Pascal’s Triangle

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

## Which is larger?

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%