The Mathematical Ninja and $\sin(15º)$ | Colin

The Mathematical Ninja sniffed. “$4\sin(15º)$? Degrees? In my classroom?”

“Uh uh sorry, sensei, I mean $4\sin\br{\piby{12}}$, obviously, I was just reading from the textmmmff.”

“Don’t eat it all at once. Now, $4\sin\br{\piby{12}}$ is an interesting one. You know all about Ailes’ Rectangle, of course, so you know that $\sin\br{\piby{12}}=\frac{\sqrt{6}-\sqrt{2}}{4}$, which makes the whole thing $\sqrt{6}-\sqrt{2}$. Now, obviously, that’s the correct, exact answer. But…”

“Hmmmf. Thank you.”

“… the textbook wants it to three decimal places, for some unfathomable reason. Don’t touch…

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Ask Uncle Colin: A mental quotient | Colin

Dear Uncle Colin,

I have to work out $851 \div 37$ without pen, paper or calculator. How would you do it?

– Simple Mental Arithmetic Looks Easy

Hi, SMALE, and thanks for your message!

I have three ways to tackle it.
Brute force and estimation

It’s fairly obvious that the answer is between 20 and 30.

Knowing that $20 \times 37 = 740$ means you’re 111 short, which is $3 \times 37$ – making a total of $23 \times 37$.
Number fact alert!

One of my favourite things about 37 is that $3 \times 37 = 111$, a very pleasing number.

In particular, multiplying everything by 3 gives…

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Wrong, But Useful: Episode 52 | Colin

In this month’s episode of Wrong, But Useful, we are joined by maths communication superhero @stecks, who is Katie Steckles in real life.

We discuss:
Katie’s numerous and various activities as a maths freelancer, including maths busking and being a mathematician-in-residence at the Stephen Lawrence Gallery at the University of Greenwich.
Number of the podcast: 145; it’s a factorion, like 1, 2 and 40585 (and no others).
A discussion between @jussumchick, @colinthemathmo, @mrmattock, and dozens more on the difference between an expression, equation, formula and identity.
The 1089 trick…

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On $0 \div 0$ | Colin

A question that frequently comes up in the insalubrious sort of place a mathematician might hang around is, what is that value of $0^0$. We generally sigh and answer that the same way every time.

It was nice, then, to see someone ask a more fundamental one: what is $0 \div 0$?

The short answer is, it’s not defined, even though $0 \div a = 0$ pretty much everywhere. But why?

There are probably dozens of explanations for this. My favourite is to look at what division means.

$a \div b$ asks the question “what do you multiply by $b$ to get $a$?” So, $6 \div 2 = 3$ because $3 \times 2 =…

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Ask Uncle Colin: A surprising order | Colin

Dear Uncle Colin,

How come $0.3^{0.3} > 0.4^{0.4}$?

– Puzzling Over It, Some Surprisingly Ordered Numbers

Hi, POISSON, and thank you for your message!

It is a bit surprising, isn’t it? You would expect $x^x$ to increase everywhere, at first glance.
Why it doesn’t

We can see that this isn’t the case if we differentiate $y=x^x$ – or rather, $\ln(y)=x\ln(x)$, which is much more tractable. We get $\frac{1}{y} \dydx = \ln(x) + 1$, so $\dydx = x^x \br{\ln(x)+1}$.

That clearly gives a turning point when $\ln(x)=-1$, which it does when $x=e^{-1}$ – bang in between your two values…

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The Maths Behind… Cakes | Colin

“Cooking,” said my friend Liz in a recent Facebook post, “is one of the activities where maths is most useful in my everyday life.” She added this picture:

I’ve got several reasons for wanting to share this.
1. It’s pretty much a model answer

Imagine you’re in a GCSE exam, and the paper asks:

Liz has a recipe that calls for a cake tin with a 7-inch diameter, but only has an 8-inch diameter tin available. How should she adjust the recipe, assuming all cake tins are the same depth?

This is a perfectly plausible question, incidentally, although I’d be surprised to see inches…

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Ask Uncle Colin: Solve this! | Colin

Dear Uncle Colin,

I need to find the largest solution to $e^x + \sin(x)=0$ and I don’t really know where to start. Any ideas?

– Some Options Look Virtually Equal

Hi, SOLVE, and thanks for your message!

That is something we in the trade call ‘not a very nice equation to solve’. It’s quite unusual for mixtures of exponentials and trig functions to have simple roots, and this, I’m afraid, is one of those cases.
Finding bounds

We can put some bounds on it, though: we know that when $x=0$, $e^x=1$ and $\sin(x)=0$, and a sketch shows that there can be no positive solutions….

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