The Flying Colours Maths blog has been running posts twice weekly since 2012, covering maths from the basics to… well, the most advanced stuff I have a clue about.
Here they all are, sorted by date. Some day, other ways to filter them will be possible.
Dear Uncle Colin, I’m working on a (3D) computer graphics course, and my notes have some equations in that I don’t understand. I have a point light at (4D) position $\vec L$, an object translated by a 4D matrix $\bb M$ and a (4D) point on the surface at $\vec p$, as measured from the …
This is a guest post, courtesy of @Dragon_Dodo and @FennekLyra. Thanks, Dominika and Eva! It was an unusually quiet morning at the Maths Police HQ. ‘If a mathematician is a device for turning coffee into theorems, what do you reckon the conversion rate is?’ mused agent Dodo over her morning cup of …
Dear Uncle Colin, I have an inequality that involves two pairs of modulus signs, and I can’t make head nor tail of it. It’s $\left| \left| 3 - x \right| - 2x\right| \le 9$. Please help! -- Absolutely Bewildered, Driven Absolutely Barmy Well, ABDAB, I’d normally recommend sketching …
One of the key measures of how much I like a book is how quickly I get through it. Turing’s Cathedral went back to the library today after having being renewed the maximum of three times; I did finish it, but only just. I can’t put my finger on what it was that didn’t do it for me. …
Dear Uncle Colin, I’ve carelessly interlocked a couple of regular pentagons together like this and need to find the area of the overlap because of reasons. For still other reasons, I don’t want to use trigonometry. How do I negotiate my way through this minefield? Trigonometry’s …
A week or two back, it was announced that computers had verified that $2^{74,207,281}-1$ was a 22-and-a-bit million digit prime number. Cue headlines on the BBC and silliness on Twitter and slight derailment of maths lessons across the country. But me? It leaves me cold, and here’s why: …
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 …
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.
“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 …
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 in your name, who …
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\, …
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 …
“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 …
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 never seem to get more …
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 a \d{T} …
Dear Uncle Colin, I wanted to work out $3^{41}\mod 13$: Wolfram|Alpha says it’s 9, but MATLAB says it’s 8. They can’t both be right! What gives? MATLAB Obviously Doesn’t Understand Logical Operations Hi, MODULO! First up, when computers disagree, the best thing to do is check …
“Hm,” I thought, “that’s odd.” I don’t often work in degrees, but the student’s syllabus insisted. And $\sin(50º)$ came up. It’s 0.7660, to four decimal places. But… I know that $\sin\left(\frac 13 \pi\right)$, er, sorry, $\sin(60º)$ is 0.8660 …
In this month’s festive WBU, Colin and Dave discuss… Bill’s second birthday and train set Colin’s latest book is due out any day. First number of the podcast: 998 (days since our first recording) Colin gloats about Spoof My Proof and Dave chews on sour grapes Second number …
Dear Uncle Colin, I’m having trouble getting my head around sum notation! I can’t tell whether $\sum_{n=0}^{6}{5}$ means $0\times5 + 1\times 5 + … + 6\times 5$ or $0 + 1 + 2 + … + 6$ or $5 + 5 + 5 … + 5$. Wolfram|Alpha just gives me the answer, not an explanation! …
A question, some time ago, from my favourite Egyptologist on Twitter: There’s a very simple answer to this question: it doesn’t just depend on who you’re asking, it may also depend on what mood they’re in at the time. Of course, there are extremists on both sides of the …
Dear Uncle Colin, @CmonMattTHINK unearthed the challenge to prove that: $\tan\left( \frac 3{11}\pi \right) + 4 \sin\left( \frac 2{11}\pi \right) = \sqrt {11}$. Wolfram Alpha says it’s true, but I can barely get started on the proof and I’m worried no-one will like me. Grr, Really …
I’ve always had a soft spot for the Countdown numbers game, a challenge pitched just perfectly right for my mental capacities. It would take a very good numbers game to displace Countdown in my heart. Mathador is a very good numbers game1 with similarities to Countdown: you’re given a …
The following puzzle/trick came up on Futility Closet, one of my favourite sites for recreations. Here’s how they describe it: Arrange cards with values ace through 9 in a row, in counting order, with the ace on the left. Take up a card from one end of the row — left or right, your choice. Do …
Dear Uncle Colin, Do you have any advice about Cambridge interviews? I have one coming up. How About Wanting Kings (If Not, Girton) Hello, HAWKING! My Cambridge interviews were a couple of decades ago, so I don’t know how current my advice is. A few things: be ready to demonstrate that …
In terms of tone and style, The Joy of $x$ is an absolute delight – Strogatz has a knack for finding the right analogy and the right anecdote that is the envy of maths writers everywhere. It’s an enjoyable read, and I’d recommend it to anyone who thinks “I’d love to …
“Where are they all coming from?” asked the first. “Facebook, mainly,” said the second. “Usually with a slick video showing how to multiply 321 by 13 using only a few diagonal lines.” “Gah,” said the second. A few hollow voices chanted “4 1 7 …
Dear Uncle Colin Help! An emergency has arisen and my L’Hôpital’s Rule is out of order – but I still need to find $\lim_{x \to 0_+} \frac{\ln(x)}{\cot(x)}$! This is not a drill! -- Help! Inexplicable Limitations Bring Everyone Ridiculous Trouble Don’t panic, HILBERT! …
It was an incredible idea: gather together as many really smart people as possible and set them to work on discovering the essential nature of things. At least, it was an incredible idea around the time of Pythagoras, or the time of the House of Wisdom, or any time anyone else had it. The unusual …
Oh, the days – weeks, even – of my university life I spent working out the determinants of matrices. The 3×3 version was the main culprit, of course, usually needing to be split down into three smaller determinants, and usually requiring a sign change in one or two that I’d almost …
Dear Uncle Colin, My teacher recently challenged me to tackle the missing numbers problem below and I don’t know where to start! It’s driving me to disgust with the whole number system. Crazy Old Numbers, Wacky And Yucky Dear CONWAY, That feeling of disgust is a completely normal part of …