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 $23 \times 37$.

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.