Posted in algebra

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, rather than

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Posted in ask uncle colin

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

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Posted in proof

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

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Posted in ask uncle colin

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

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Posted in dome

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 important) can

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Posted in ask uncle colin

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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Posted in 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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Posted in ask uncle colin

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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Posted in ninja maths

“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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Posted in podcasts

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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