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, My friend claims that $\frac { 2 - \frac{2 \sin(x)}{\cos(x)}}{\sin(x) - \cos(x)} \equiv -2\sec(x)$. I think she’s crazy. What do you think? -- I Don’t Even Need Trigonometry, I Teach Yoga Hi, IDENTITY – even yoga teachers need trigonometry, though! Well, …
So much wasted time. I spent much of my first two years at university cursing the names of Gauss and Jordan, railing at my lecturer (who grim-facedly assured me there were no more useful uses of a student’s thinking time than ham-fistedly rearranging these things), and thinking “there MUST be a …
Dear Uncle Colin, I’m struggling with A-level. I used to love maths when I did [one board] at GCSE and now I’m doing [another board] at A-level, I don’t enjoy it any more – when I see a question, I can’t even tell what it is they’re asking. My teachers are no help …
It’s billed as the calculator that won’t think until you do: if you give it something to evaluate, it will refuse to give you an answer until you give it an acceptable approximation. On the surface, that’s a great idea. If I had a coffee for every time I’ve rolled my eyes at …
Dear Uncle Colin, I have a disagreement with my teacher about the integral $\int_{-1}^{1} x^{-1} \dx$. I understand you have to split the integral into two parts, which I’m happy with. They split it from -1 to $a$, letting $a \rightarrow 0_-$ and from $b$ to 1, letting $b \rightarrow 0_+$; …
When I was researching the recent Mathematical Ninja piece for Relatively Prime, I stumbled on something at MathWorld1 I’d never noticed before: if $t = \tan(x)$, then: $\tan(2x) = \frac{2t}{1 - t^2}$ $\tan(3x) = \frac{3t - t^3}{1 - 3t^2}$ $\tan(4x) = \frac{4t - 4t^3}{1 - 6t^2 + t^4}$ …
Dear Uncle Colin, Up here in Scotland, we’ve got an election tomorrow. It’s not as simple as the stupid first past the post elections you have down there in England, but even with our superior Scottish intelligence, some people are still struggling to understand how the system works. Do …
Some time ago, someone asked Uncle Colin what the last two digits of $19^{1000}$ were. That caused few problems. However, Mark came up with a follow-up question: how would you estimate $19^{1000}$? I like this question, and set myself some rules: No calculators (obviously) Only rough memorised …
Dear Uncle Colin, I’ve come across a seemingly simple question I can’t tackle: solve $x^2 + 2x \ge 2$. I tried factorising to get $x(x+2) \ge 2$, which has the roots 0 and -2, but the book says the answer is $x < -1-\sqrt{3}$ or $x > -1 + \sqrt{3}$. Where have I gone wrong? -- …
A STEP question (1999 STEP II, Q4) asks: By considering the expansions in powers of $x$ of both sides of the identity $(1+x)^n (1+x)^n \equiv (1+x)^{2n}$ show that: $\sum_{s=0}^{n} \left( \nCr{n}{s} \right)^2 = \left( \nCr{2n}{n} \right)$, where $\nCr{n}{s} = \frac{n!}{s!(n-s)!}$. By considering …
Dear Uncle Colin, I have a triangle with sides 4.35cm, 8cm and 12cm; the angle opposite the 4.35cm side is 10º1 and need to find the largest angle. I know how to work this out in two ways: I can use the cosine rule with the three sides, which gives me 151.3º – I believe that to be correct …
“One equals two” growled the mass of zombies in the distance. “One equals two.” The first put down the shotgun. “I’ve got this one,” he said, picking up the megaphone. “If you’re sure,” said the second. “I’M SURE.” The …
Dear Uncle Colin, I’m supposed to solve $(1+i)^N = 16$ for $N$, and I don’t know where to start! -- Don’t Even Mention Other Imaginary Variations – Reality’s Enough Hello, DEMOIVRE, there are a couple of ways to attack this. The simplest way (I think) is to convert the …
Somewhere deep in the recesses of my email folder lurks a puzzle that looks simple enough, but that several of my so-inclined friends haven’t found easy: A circle of radius $r$, has centre $C\ (0,r)$. A tangent to the circle touches the axes at $A\ (9,0)$ and $B\ (0, 2r+3)$. Find $r$. Now, I …
Dear Uncle Colin, I’m trying to find a definite integral: $\int_0^\pi \sin(kx) \sin(mx) \dx$, where $m$ and $k$ are positive integers and the answer needs to be simplified as far as possible. I’ve wound up with $\left[\frac{ (k+m) \sin( (k-m) \pi) - (k-m)\sin( (k+m)\pi) …
“No, no, wait!” said the student. “Look!” “8.000 000 072 9,” said the Mathematical Ninja. “Isn’t that $\frac{987,654,321}{123,456,789}$? What do you think this is, some sort of a game?” “It has all the hallmarks of…” …
In this month’s podcast, @reflectivemaths and I discuss: Colin’s book being available to buy Number of the podcast: Catalan’s constant, which is about 0.915 965 (defined as $\frac{1}{1} - \frac{1}{9} + \frac{1}{25} - \frac{1}{49} + … + \frac{1}{(2n+1)^2} - \frac{1}{(2n+3)^2} …
Dear Uncle Colin, Why can’t I work out $\int \left( \ln(x) \right)^2 \dx$ using the reverse chain rule? -- Previously Acceptable, Reasonable Technique Stumbles Hello, PARTS, There are two answers to this: the first is, you can’t use the reverse chain rule – which I learned as …
This post is based on work by Mark Ritchings; I know of no finer1 maths tutor in Bury. A few weeks ago, I pointed in the vague direction of a few decimal curiosities – fractions that spit out lovely patterns in their decimal expansions. Having found one that generated the squares, I asked the …
Dear Uncle Colin, I recently came across a problem in which I had to integrate $\cos^3(x)$. Somewhere in my mind, I recall that the thing to do is to make it into something involving $\cos(3x)$, but I couldn’t put the details together. Could you help? -- Not A Very Inspired Expression …
Is This Prime? is probably the most infuriating, addictive, revolting and unbearably simple games that has ever disgraced my computer screen. I love it, hate it, am glad of its existence and wish it had never been written. It’s pretty tough to think of a simpler premise: you’re given an …
Dear Uncle Colin, I was doing a STEP paper and it asked me to calculate $\int_0^1 x^3 \arctan\left(\frac{1-x}{1+x}\right) \dx$, given that $\int_0^1 \frac{x^4}{1+x^2} \dx = \frac{\pi}{4} - \frac{2}{3}$. Nut-uh. College Asked Me Back: Rocked Interview. Daren’t Get Excited Hello, CAMBRIDGE! Is …
This article has also been published as part of the Relatively Prime zine. The student yelped, and found his wrists and ankles strapped to the wheel before the lesson had even started. “Good morning,” said the Mathematical Ninja. “I think you and I need to have a little… …
Dear Uncle Colin, I’m supposed to use the change of variable $z = \sin(x)$ to turn $\cos(x) \diffn{2}{y}{x} + \sin(x) \dydx - 2y \cos^3(x)= 2\cos^5(x)$ into $\diffn{2}{y}{z} - 2y = 2\left(1-z^2\right)$. Yeah but no but. No idea. Lacking, Obviously, Something Trivial Hello, LOST, Right, yes. …
Once in a while, a student puts me on the spot; it’s not always deliberate. In this case, changing a variable in a second-order differential equation, he blithely said “Well, $\diffn 2yx = \frac{1}{\diffn 2xy}$…” “Whoa whoa whoa…” “… …
In Episode $n$ of Wrong, But Useful (where $n$ is a semiprime followed by two semiprimes and a square), @reflectivemaths and I discuss: Colin’s trip to Center Parcs and whether it is sufficiently central The number of the podcast, which is the tetrahedral angle, …
Dear Uncle Colin, In Basic Maths For Dummies, you mention a method for multiplying numbers from 6 × 6 to 10 × 10 on your fingers. It’s almost magical! Why does it work? -- Does It Guarantee Interesting Times Sums? Hello, DIGITS, I’m glad you’re finding the book helpful! The method …
Big disclaimer here: I’m not a mental health professional. I’m someone who has suffered from anxiety and depression. If you are experiencing regular panic attacks, or struggling with depression, please seek medical help. I endured many years of misery because I was too stubborn to go to …
Dear Uncle Colin1, Supposing $\pi$ were to start repeating – how long would it take to confirm that it actually was? - Recurring Over Uncanny Numbers of Decimals Dear ROUND2 , Great question! $\pi$, as I’m sure you know, is a number close to 3.1416 that represents the ratio between a …
The Mathematical Ninja placed his quadruple espresso on the table; @dragon_dodo looked up from her laptop and smiled. “You’re taking it easy on the caffeine this morning, I see?” The Mathematical Ninja nodded. “Yeah, this is only my third cup. What are you working on?” …