# August, 2016

## Ask Uncle Colin: How do I prove I’ve revised?

Dear Uncle Colin, I have to revise for an upcoming test — and my teacher is demanding proof! What’s the best way to convince my teacher I’ve revised? — Totally Evidence-based Study Time Hello, TEST – great question! Depending on the kind of test, you have several options that leave

## Fractions that generate Pythagorean Triples

An interesting tweet, some time ago, from @RJS2212: Two unit fractions where denominators differ by 2Add fractionsNumerator & denominator of the sum are two smaller numbers of Pythag triple — Robert J Smith (@RJS2212) May 30, 2016 And of course, you wonder two things: a) why does it work, and

## Ask Uncle Colin: An elastic speed limit

Dear Uncle Colin, I clumsily dropped a particle of mass $m$! Luckily, it’s attached to a light elastic string with a modulus of elasticity of $3mg$ and natural length $a$. The other end of the string is attached to the point where I dropped the weight from. When I say

## The Mathematical Ninja and the Tangents Near 1

“Forty-two degrees,” said the Mathematical Ninja, as smugly as possible while still using degrees. The student’s hand had barely twitched towards the calculator. “Go ahead, punk,” said the Mathematical Ninja. “Make my day.” “Righto,” said the student, and tapped in $\tan^{-1} \left( 0.9 \right)$, carefully closing the bracket. “41.987. That’s

Dear Uncle Colin, Inspired by a recent XKCD cartoon, I want to start measuring temperatures in radians celsius. How can I quickly convert between the two? Made Up Nonsense? Réaumur’s Octogesimal First up, MUNRO, that’s a really bad idea. I’ve said elsewhere that I don’t like degrees for measuring angles,

## On recurring decimals

It’s encouraging to see a few less-predictable questions coming up in the new GCSE and A-level specifications. @mathsjem highlighted an especially nice GCSE one: Question 26 from yesterday’s Edexcel Methods 2 GCSE paper. Helpful for revising recurring decimals. pic.twitter.com/78BghgBSvE — Jo Morgan (@mathsjem) June 17, 2016 This is unusual more

## Ask Uncle Colin: Rational Trigonometric Values

Dear Uncle Colin, You know how sometimes $\sin(2x)$ is rational and $\sin(5x)$ is rational and $\sin(7x)$ is rational, right? Would that necessarily mean that $\sin(12x)$ is rational? Asking for a friend. — Perhaps You THink All Geometry’s On Right Angled Stuff Hi, PYTHAGORAS, I believe it does! (In fact, I

## An alternative proof of the $\sin(2x)$ identity

Uncle Colin recently explained how he would prove the identity $\sin(2x) \equiv 2 \sin(x)\cos(x)$. Naturally, that isn’t the only proof. @traumath pointed me at an especially elegant one involving the unit circle. Suppose we have an isosceles triangle set up like this: The vertical ‘base’ of the triangle is $2\sin(\alpha)$

## Ask Uncle Colin: why does the normal distribution stop at $z=\pm 4$?

Dear Uncle Colin, In Statistics, we were shown a picture of the standardised normal distribution curve, and the base stops at +4 and -4. Why is it not $\pm 5$, $\pm 10$, or anything else? Is there something special about 4? — Got An Unanswered Statistics Struggle Dear GAUSS, The

## Repdigit endings to squares

Over at @onthisdayinmath, Pat highlights a @jamestanton question about squares: $2^2$ ends with 4 and $12^2$ ends with 44. Is there a square than ends 444? How about one that ends 4444? Pat’s answer (yes to the first — $38^2 = 1444$ is the smallest — and probably not to