Author: Colin

Ask Uncle Colin: Dividing by halves

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 $\frac{x}{\br{\frac{1}{2}}} = y$

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On inverse-trig integration

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: $\int \frac{1}{\sqrt{1-x^2}} \dx = \arcsin(x) + C$ $\int \frac{-1}{\sqrt{1-x^2}}

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Ask Uncle Colin: A Strange Error

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

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“How many days old are you?”

“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.

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Wrong, but Useful: Episode 75

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 @SophieWarnes

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A “Components” Enigma

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. There’s a

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A few logarithmic tricks

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,

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Ask Uncle Colin: Matching sines

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

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The Dictionary of Mathematical Eponymy: The MacWilliams Identity

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, Jessie

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Ask Uncle Colin: Laying Bricks

Dear Uncle Colin, I have a question I can’t make sense of: “It takes 3 bricklayers four hours to lay 4200 bricks. How long would it take 2 bricklayers to lay 3150 bricks?” I can never figure out when to multiply and when to divide! Markscheme Obtusely Recommends Taking A

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