Posted in ask uncle colin, complex numbers, quadratics

Dear Uncle Colin, I'm told that $z=i$ is a solution to the complex quadratic $z^2 + wz + (1+i)=0$, and need to find $w$. I've tried the quadratic formula and completing the square, but neither of those seem to work! How do I solve it? – Don't Even Start Contemplating

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Posted in algebra

It turns out I was wrong: there is something worse than spurious pseudocontext. It's pseudocontext so creepy it made me throw up a little bit: This is from 1779: a time when puzzles were written in poetry, solutions were assumed to be integers and answers could be a bit creepy…

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Posted in algebra, ask uncle colin, fractions

Dear Uncle Colin, I recently had to decompose $\frac{3+4p}{9p^2 – 16}$ into partial fractions, and ended up with $\frac{\frac{25}{8}}{p-\frac{4}{3}} + \frac{\frac{7}{8}}{p-\frac{4}{3}}$. Apparently, that's wrong, but I don't see why! — Drat! Everything Came Out Messy. Perhaps Other Solution Essential. Hi, there, DECOMPOSE, and thanks for your message – and your

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Posted in podcasts

In this month's episode of Wrong, But Useful, @reflectivemaths1 and I are joined by consultant and lapsed mathematician @freezingsheep2. We discuss: Mel's career trajectory into 'maths-enabled type things that are not actually maths', although she gets to wave her hands a lot. What you can do with a maths degree,

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Posted in reviews

There is a danger, when your book comes plastered in praise from people like Art Benjamin and Ron Graham, that reviewers will hold it to a higher standard than a book that doesn't. That would be unfair, and I'll try to avoid that. What it does well This is a

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Posted in ask uncle colin, trigonometry

Dear Uncle Colin, In an answer sheet, they've made a leap from $\arctan\left(\frac{\cos(x)+\sin(x)}{\cos(x)-\sin(x)}\right)$ to $x + \frac{\pi}{4}$ and I don't understand where it's come from. Can you help? — Awful Ratio Converted To A Number Hello, ARCTAN, and thank you for your message! There's a principle I want to introduce

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Posted in calculus

Last week, I wrote about the volume and outer surface area of a spherical cap using different methods, both of which gave the volume as $V = \frac{\pi}{3}R^3 (1-\cos(\alpha))^2(2-\cos(\alpha))$ and the surface area as $A_o = 2\pi R^2 (1-\cos(\alpha))$. All very nice; however, one of my most beloved heuristics fails

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Posted in ask uncle colin, mechanics 1

Dear Uncle Colin, One of my students recently attempted the following question: "At time $t=0$ particle is projected upwards with a speed of 10.5m/s from a point 10m above the ground. It hits the ground with a speed of 17.5m/s at time $T$. Find $T$." They used the equation $s

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Posted in calculus, integration

What is the volume above a plane, and inside a sphere of radius $r$, such that the radius of the circle where the two intersect is $R \sin(\alpha)$? What is this spherical sector's curved surface area? I've lost the precise wording of the question that drove a small cabal of

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Posted in ask uncle colin, geometry

Dear Uncle Colin, I’m trying to sew a traditional football in the form of a truncated icosahedron. If I want a radius of 15cm, how big do the polygons need to be? — Plugging In Euler Characteristic’s Excessive Hello, PIECE, and thank you for your message! Getting an exact answer

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