The estimable @colinthemathmo suggests a method for estimating the radius of the earth, which he credits to a sundial expert friend named Mike: Stand on a wall, perhaps two metres high, and wait for sunrise. When you see the sun just peak above the horizon, start the stopwatch, and jump

Read More →
Posted in ask uncle colin, inequalities

Dear Uncle Colin, I'm pretty good with quadratic inequalities and pretty good with absolute values, but when I get the two together, I get confused. For example, I struggled with the set of values satisfying $x^2 -\left| 5x-3\right| < 2 + x$. Can you help? - Nasty Absolute Value Inequalities

Read More →It's good to see @srcav back in the twitter and blogging fold - he's been missed! As part of his comeback, he shared this lovely geometry puzzle: Assuming the situation is symmetrical (which it needs to be to get a sensible solution), there are - as usual - several ways

Read More →
Posted in ask uncle colin, trigonometry

Dear Uncle Colin, When I solve $2\tan(2x)-2\cot(x)=0$ (for $0 \le x \le 2\pi$) by keeping everything in terms of $\tan$, I get four solutions; if I use sines and cosines, I get six (which Desmos agrees with). What am I missing? - Trigonometric Answers Not Generated - Expecting 'Nother Two

Read More →
Posted in podcasts

In this month's edition of Wrong, But Useful, @reflectivemaths and I are joined by special guest co-host @dragon_dodo, who is Dominika Vasilkova in real life. We discuss: What maths appeals to a physicist. Dominika's number of the podcast: $0.110001000000000000000001...$, Liouville's constant, which is $\sum_{n=1}^\infty 10^{-n!}$, the first constant to be

Read More →
Posted in algebra, big in finland, probability, puzzles

Zeke and Monty play a game. They repeatedly toss a coin until either the sequence tail-tail-head (TTH) or the sequence tail-head-head (THH) appears. If TTH shows up first, Zeke wins; if THH shows up first, Monty wins. What is the probability that Zeke wins? My first reaction to this question

Read More →
Posted in ask uncle colin, calculus

Dear Uncle Colin, I was asked to find the tangent to the curve $r=\frac{8}{\theta}$ at the point where $\theta = \frac{\pi}{2}$. I worked out $\dydx = \frac{ \frac{8 \left(\theta \cos(\theta)-\sin(\theta)\right)}{\theta^2}}{\frac{-8\left(\theta \sin(\theta)+\cos(\theta)\right)}{\theta^2} }$, which simplifies to $ -\frac{\theta \cos(\theta)-\sin(\theta)} {\theta \sin(\theta)-\cos(\theta)}$. Evaluated at $\theta = \frac{\pi}{2}$, that gives $\dydx=\frac{2}{\pi}$ and a

Read More →
Posted in ninja maths, trigonometry

As the student was wont to do, he idly muttered "So, that's $\cos(10º)$..." The calculator, as calculators are wont to do when the Mathematical Ninja is around, suddenly went up in smoke. "0.985," with a heavy implication of 'you don't need a calculator for that'. As the student was wont

Read More →
Posted in ask uncle colin, factorising

Dear Uncle Colin, If you know all of the factors of $n$, can you use that to find all of the factors of $n^2$? For example, I know that 6 has factors 1, 2, 3 and 6. Its square, 36, has the same factors, as well as 4, 9, 12,

Read More →
Posted in common errors, ninja maths

I'm a big advocate of error logs: notebooks in which students analyse their mistakes. I recommend a three-column approach: in the first, write the question, in the second, what went wrong, and in the last, how to do it correctly. Oddly, that's the format for this post, too. The question

Read More →