The Flying Colours Maths Blog: Latest posts

Ask Uncle Colin: How did they get $\ln(50)$?

Dear Uncle Colin, I get $-\frac{\ln(0.02)}{0.03}$ as my answer to a question. They have $\frac{100\ln(50)}{3}$. Numerically, they seem to be the same, but they look completely different. What gives? — Polishing Off Weird Exponents, Really Stuck Dear POWERS, What you need here are the log laws (to show that $-\ln(0.02)=\ln(50)$,

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The Bigger Fraction

Some while back, Ben Orlin of the brilliant Maths With Bad Drawings blog posted a puzzle he’d set for some eleven-year-olds: Which is larger, $\frac{3997}{4001}$ or $\frac{4996}{5001}$? Hint: they differ by less than 0.000 000 05. He goes on to explain how he solved it (by considering the difference between

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Ask Uncle Colin: An absolute quadratic inequality

Dear Uncle Colin, I have to solve the inequality $x^2 – \left|5x-3\right| \lt 2+x$. I rearranged to make it $x^2 – x – 2 \lt \left|5x-3\right|$ , but the final answer is eluding me. — Put Right Inequality Muddle Hello, PRIM! You’re off to a good start; the next thing

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On the square root of a third

While I’m no Mathematical Ninja, it does amuse me to come up with mental approximations to numbers, largely to convince my students I know what I’m doing. One number I’ve not looked at much1 is $\sqrt{\frac{1}{3}}$, which comes up fairly frequently, as it’s $\tan\left(\frac{\pi}{6}\right)$2 . Ninja-chops taught me all about

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Ask Uncle Colin: Am I Smart Enough?

Dear Uncle Colin, My maths mock went terribly, and I got a U. Since then I’ve done some real revision and got a good grade on a paper I did off my own bat. However, I’m a long way behind on the new material and I feel like it’s too

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The Mathematical Ninja and the twenty-sixths

The Mathematical Ninja played an implausible trick shot, not only removing himself from a cleverly-plotted snooker, but potting a red his student had presumed safe and setting himself up on the black. Again. “One!” he said, brightly, and put some chalk on the end of his cue. The student sighed.

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Wrong, But Useful: Episode 35

In this month’s Wrong, But Useful, Dave and Colin discuss: Colin gets his plug for Cracking Mathematics in early Colin is upset by a missing apostrophe Dave teases us with the number of the podcast and asks about the kinds of things it’s reasonable to expect students to know, and

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Ask Uncle Colin: Simultaneous Equations

Dear Uncle Colin, Could you please tell me how to solve simultaneous equations? I have a rough idea, but I get confused about it. — Stuck In Mathematical Examinations/Qualifications Hello, SIMEQ! Here’s how I attack linear simultaneous equations, such as: $5x + 6y = -34$ (A) $7x + 2y =

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A curious identity

There’s something neat about an identity or result that seems completely unexpected, and this one is an especially nice one: $$ e^{2\pi \sin \left( i \ln(\phi)\right) }= -1$$ (where $\phi$ is the golden ratio.) It’s one of those that just begs, “prove me!” So, here goes! I’d start with the

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Ask Uncle Colin: Trigonometric craziness

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, there’s one way to find out

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