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.
An interesting “creative” integral pointed my way by the marvellous @DavidKButlerUoA: Find $\int {\frac{1}{(1+x^2)^2}} \dx$ There are “proper” ways to do this - in his tweet, David shows a clever way to do it by parts, and suggests a trig substitution as an alternative. I want to show a third way: …
Dear Uncle Colin, I’m trying to invert $y = \arcsin\br{x^2 + 4x - 5}$, but I get stuck at $x^2 + 4x = \sin(y) + 5$. Any pointers? - All Routine Calculations Should Invert Naturally Hi, ARCSIN, and thanks for your message! Just before I dive into this, I have an itch I need to deal with: the domain …
On a recent1 episode of Wrong, But Useful, Dave noted that 33 of the first 100 Fibonacci numbers were even, 333 of the first 1000, and so on. My reaction wasn’t quite as it should have been: I said something like “well, yes, obviously”. While there’s a not-too-difficult reason behind it (which I’ll …
Dear Uncle Colin, I need to work out $\lim_{x\to\infty} \left\{ \left[ \frac{ 2^{1/x} + 8^{1/x} }{2} \right]^x \right\}$. The textbook says it’s four, but my engineering friends all get different answers. How would you work it out? - Giving Engineers Real Maths Analysis Is Naive Hi, GERMAIN, and …
Dear Uncle Colin, I’m meant to be sitting my A-levels this summer and there’s an awful disease going around. I don’t know if school will be open next week or even if the exams will go ahead. I don’t know what that means for university. I’m scared. What do I do? School Closures And Related Exam …
“The square root of two… I don’t even know how to say this. The square root of two to the square root of threeth power?” “$\sqrt{2}^{\sqrt{3}}$?” said the Mathematical Ninja. “I wouldn’t bother saying it, I’d just write it down.” “But what does it mean? I mean, I can just about get on board with …
In this month’s Pi Day Special episode, we’re joined by @aap03102, who is Chris Smith in real life. We discuss: Negative primes Surdoku (look in issue $2^5$) Chris’s dark past Chalkdust book of the year Book (or writing) ideas (I’m trying to compile a big list of maths things I - …
Dear Uncle Colin, I’ve been asked to show how to do $23241 \div 31$ in base 5 by long division - I can barely do it in base 10! Help! - Lots Of Number Games Hi, LONG, and thanks for your message! There are several steps to working a long division problem, but it comes down to taking away as large a …
OK, listen up, there seems to be some confusion about this. I shall make the point several times here, but first I shall make it in bold: Cylinders are not prisms. In fact, it’s the other way around. A cylinder is a 3D shape that has parallel end-faces and a cross-section that’s “the same” all the …
What’s this? Didn’t we have a WBU last week? We did indeed. Only we cut out a section of our conversation with @dr_teeluck about APR to keep the episode length (somewhat) reasonable. Here it is as a special bonus episode for your enjoyment.
Dear Uncle Colin, What do you call the ‘point’ of the absolute value graph - for instance, $(0,0)$ on the graph of $y=|x|$? It can’t be a minimum because the gradient is undefined! Proof Or It’s Not True Hi, POINT, and thanks for your message! I hate to go ‘well, actually’, but I’m afraid I have to …
Today’s entry in the Dictionary of Mathematical Eponymy is, by some distance, the entry that’s been most useful to me since I learned about it. (The Elo rating is probably in second place.) It’s also a unique entry in that I have next to no information about its originator. What is Osborn’s Rule? …
In this episode of Wrong, But Useful, we’re joined by @dr_teeluck, who is Dr Vijay Teeluck in real life. We discuss: sunrise (posts by @colinthemathmo and by me got the same answer, someone else disagrees, nice thing to explore) favourite numbers a lovely post by @samhartburn about tetrahedra …
Dear Uncle Colin, I got stuck trying to prove that $(\sec(x) - \tan(x))^2 \equiv \frac{1-\sin(x)}{1+\sin(x)}$. Can you help? - Bringing Over Is No Good! Bringing Over Is No Good! Hi, BOINGBOING, and thank you for your message! Here’s how I’d tackle it, starting from the left hand side: $\sec(x) - …
On Twitter, @Trianglemanscd posed a pertinent problem: Stand back everyone! I have compasses and a straight-edge and I’m not afraid to use them; the Geogebra demonstration below shows one way to do it, eschewing things like ‘strings’ and ‘protractors’ in favour of proper geometry. [iframe …
Dear Uncle Colin, I have twelve coins, one of which is counterfeit. I don’t know if it’s heavier or lighter than the others, but I’m allowed three goes on a balance scale to determine which coin it is and whether it’s light or heavy. Can you help? Some Coins Are Light, Evidently Hi, SCALE, and …
Some time ago, I was surprised to see the following question in a predicted GCSE paper: Solve for $x$: $ \frac{2x}{3x+2} \leq \frac{3}{4x+1}$ Give your answers to two decimal places (3 marks) Why surprised? Surprised because the techniques you need to solve it correctly are Further Maths A-level, …
Dear Uncle Colin, The question says “Under what circumstances does $|\mathbf{a} + \mathbf{b} | = | \mathbf{a} | + | \mathbf{b} |$ and I don’t have a clue. Can you help? Sines? Cosines? Angle Leaves A Residue Hi, SCALAR, and thanks for your message! I have several ways to approach this, but I …
A question that comes up a lot in class is, “how do you get good at proofs?” (It’s usually framed as “I don’t like proofs”, but we’re not having any of that negativity here, thank you very much.) I don’t have a silver bullet for that. I do have some advice. Keep a notebook In your special proofs …
Dear Uncle Colin I have the set-up below - how do I find the cumulative density function for $d^2$? - Circles Don’t Fit Hi, CDF, and thanks for your message! Before I start, I’m going to make two simplifying assumptions that don’t change anything1 - that the radius of the outer circle $R=1$, and …
We’ve just reached the halfway point of the Dictionary of Mathematical Eponymy project, and it’s time for a fairly famous one (and again, one I’ve been meaning to understand better). What is Noether’s Theorem? Emmy Noether has several theorems named for her, but the first (and probably most …
Dear Uncle Colin, Why is dividing by a half the same as doubling? - How Arithmetic Leverages Fractions Hi, HALF, and thanks for your message! I’m going to give a couple of reasons: first, the algebra of it, then the logic. Algebra Let’s set this up as $\frac{x}{\br{\frac{1}{2}}} = y$ - you’re asking …
When faced with something like $\int \frac{1}{\sqrt{1+x^2}} \dx$, my first instinct has usually been to panic, and then to try trig (or hyperbolic) substitutions more or less at random. But is there a better way? There are six such integrals altogether: $\int \frac{1}{\sqrt{1-x^2}} \dx = \arcsin(x) …
Dear Uncle Colin, If $(-2)^3 = -8$, why doesn’t $\log_{-2}(-8) = 3$? Exponential Rules: Rubbish Or Reasonable? Hi, ERROR, and thanks for your message! As is so often the case, there are two (or possibly three) answers to this. A problem of definition The way logarithms are usually1 defined as …
“How old are you?” asked young Fred. (This is not, technically, an ‘Ask Uncle Colin’. It’s an ‘Ask Daddy’.) “I’m 42, buddy.” “42 days?” “No, 42 years!” “Oh. But how many days is that?” It is not quite 7:15am on New Year’s Day. I have not yet had my coffee. $42 \times 365$, plus however many leap …
In episode 75 of Wrong, but Useful, we talk with @astrastefania, who is Stefania Delprete in real life. We discuss: Dreams about Maths and doing Maths in lucid dreams Goodreads maths reading challenge Newsletters I like: Stuff Evelyn Wants You To Read by @evelynjlamb and and Fair Warning by …
Dear Uncle Colin, Help! I’m working on a top-secret project somewhere in Buckinghamshire. I can tell you that it involves… components. The… machine has three slots for components, and there are five different components available (one of each) - so there are 60 ways to arrange the components. …
I love a good logarithm. Logarithms are a standby for when I want to work something out ninja-style, and there’s something very satisfying about taking something horrible in the powers, bringing it down to the working line, and finding that it wasn’t so horrible after all. I’m an old hand, though. …
Dear Uncle Colin, How do I solve $\sin(3x) = \sin(5x)$ for $0 \le x \lt 360$? Seems I Need Extra Smarts Hi, SINES, and thanks for your message! This involves a formula I always have to look up, or work out from scratch. Today I am in a working out from scratch sort of mood. The formula we need The …
After the Second World War, there was a boom in the study of transmitting encoded data. In likelihood, I imagine the boom started earlier, and the boom was more about the declassified publication of papers on this topic than about a sudden increase in productivity. This month’s mathematical hero, …