# The Flying Colours Maths Blog: Latest posts

## Ask Uncle Colin: another vile limit

Dear Uncle Colin, Apparently, you can use L'Hôpital's rule to find the limit of $\left(\tan(x)\right)^x$ as $x$ goes to 0 - but I can't see how! - Fractions Required, Example Given Excepted Hi, FREGE, and thanks for your question! As it stands, you can't use L'Hôpital - but you can

## Another of Alison’s Ace Puzzles, Revisited

This is a guest post from @ImMisterAl, who prefers to remain anonymous in real life. It refers to the problem in this post: a semi-circle is inscribed in a 3-4-5 triangle as shown; find $X$. As with any mathematical problem, my first thought was to sort out exactly what I

## Ask Uncle Colin: Are normals… normal?

Dear Uncle Colin, I don't understand why the normal gradient is the negative reciprocal of the tangent gradient. What's the logic there? -- Pythagoras Is Blinding You To What's Obvious Hi, PIBYTWO, and thanks for your message! My favourite way to think about perpendicular gradients is to imagine a line

## From Euclid to Cantor

One of my favourite quotes is from Stefan Banach: "A good mathematician sees analogies between theorems. A great mathematician sees analogies between analogies." This post is clearly in the former camp. I'm fairly sure it's a trivial thing, but it's not something I'd noticed before. One of the first serious

Dear Uncle Colin, When I have an angle in the second quadrant, I can find it just fine using $\cos^{-1}$ - but using $\sin^{-1}$ or $\tan^{-1}$ gives me an angle in the fourth quadrant. I don't understand why this is! -- I Need Verbose Explanations; Radians Seem Excellent Hi, INVERSE,

## A surprising overlap

Every so often, my muggle side and mathematical side conflict, and this clip from @marksettle shows one of them. My toddler's train track is freaking me out right now. What is going on here?! pic.twitter.com/9o8bVWF5KO — marc blank-settle (@MarcSettle) April 6, 2016 My muggle side says "wait, what, how can

## Ask Uncle Colin: Perpendicular vectors

Dear Uncle Colin, I'm struggling a bit with my C4 vectors. Most of it is fine, except when I have to find a point $P$ on a given line such that $\vec{AP}$ is perpendicular to the line, for some known $A$. How do I figure that out? -- Any Vector

## Wrong, But Useful: Episode 41

This month on Wrong, But Useful, @reflectivemaths and @icecolbeveridge are joined by @evelynjlamb, who is Evelyn Lamb in real life. She writes the Roots Of Unity column for Scientific American. We discuss: How Evelyn got into maths, into writing and into France Evelyn picks the numbers of the podcast: 339,613

## Another of Alison’s Ace Puzzles

A nice puzzle this week, via NRICH's magnificent @ajk44: a semicircle is inscribed in a 3-4-5 triangle as shown. Find $X$. I think it's a nice puzzle because Alison's way of doing it was entirely different to mine, but thankfully got the same answer. You might like to try it

## Ask Uncle Colin: Bridges, Donkeys and Triangles

Dear Uncle Colin, I'm struggling to understand why, if you know a triangle has two sides the same, the base angles must be the same. Can you explain? -- I'm Struggling Over Some Coherent Explanation Leveraging Equal Sides Hi, ISOSCELES, and thanks for your message! There are several good proofs