Posted in ask uncle colin, logarithms.

Dear Uncle Colin, I have a problem with a limit! I need to figure out what $\left( \tan \left(x\right) \right)^x$ is as $x \rightarrow 0$. — Brilliant Explanation Required Now! Our Understanding’s Limited; L’Hôpital’s Inept Right, BERNOULLI, stop badmouthing L’Hôpital and let’s figure out this limit. It’s clearly an indeterminate

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Posted in podcasts.

This month’s episode is an interview with podcaster extraordinaire, @samuel_hansen, who is Samuel Hansen in real life. If you’re not listening to Relatively Prime, I don’t know what’s wrong with you. Go and listen to it.

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Posted in ninja maths.

“That @ColinTheMathmo chap had a blog post on Stirling’s approximation, too,” said the student, spotting a chance to move the lesson away from his disappointing mock exam results. “Used it to work out 52!” “I saw it,” said the Mathematical Ninja, polishing his weaponry smugly. “It… wasn’t bad, exactly…” “But

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Posted in ask uncle colin, number theory.

Dear Uncle Colin, I’ve been asked to find the last two digits of $19^{100}$. For what reason, I cannot tell. However, my calculator bums out before I get to $19^{10}$! — Many Other Digits, Unfindable Last Ones Hi, there, MODULO! What do you know, the clue to your problem is

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Posted in arithmetic.

There’s not much of a story to this post, except for a few curiosities the decimal system throws up (largely as a result of the binomial expansion). Some time ago, I looked at some Fibonacci witchcraft: $\frac{1}{999,998,999,999} = 0.000\,000\, 000\,001\, 000\,001\, 000\,002\, 000\,003\, 000\,005\, 000\,008\,…$, neatly enumerating the Fibonacci sequence

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Posted in ask uncle colin.

Dear Uncle Colin, I’ve been struggling to get my head around what happens if you chop infinity in two? Is half of infinity still infinity? Help! How Infinity Lies Beyond Every Reasonable Theory Hi, HILBERT1 The short answer is yes: halving infinity gives you infinity. (Once you get to infinity,

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Posted in ninja maths.

“I suppose,” said the Mathematical Ninja, “I can allow you to put $20!$ into a calculator. There’s absolutely no reason you should know that it turns out to be about $2.4 \times 10^{18}$.” The student tapped the numbers in, frowned, thought for a moment and said “OK, I’ll bite. How…?”

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Posted in ask uncle colin, revision.

Dear Uncle Colin I’m in year 9 and really annoyed: my little sister keeps beating me in maths tests! I’ve only ever beaten her once, and even then only by one point. It’s shameful! I always do the exercises in the book and work really hard at revising, but I

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Posted in integration, mechanics 1.

Most of the suvat equations are pretty easy to derive, as soon as you realise acceleration ($a$, assumed constant) is the derivative of velocity ($v$) with respect to time, and velocity is the derivative of position ($s$), also with respect to time. For example: $ a = \diff{v}{t}$ $ \int_0^t

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