Posted in algebra

One of the more surprising results a mathematician comes across in a university course is that the infinite sum $S = 1 + \frac{1}{4} + \frac{1}{9} + ... + \frac{1}{n^2} + ...$ comes out as $\frac{\pi^2}{6}$. If $\pi^2$s are going to crop up in sums like that, they should be

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Posted in ask uncle colin

Dear Uncle Colin, What is $\frac{1}{\infty}$? - Calculating A Number, Though Outside Reals Hi, CANTOR, and thanks for your message! The short answer: it's undefined. The longer answer: Infinity is not a number. It's not something you're allowed to divide by. The calculation doesn't make sense, and writing it down

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Posted in ninja maths

"So," said the Mathematical Ninja, "we meet again." "In fairness," said the student, "this is our regularly-scheduled appointment." The Mathematical Ninja was unable to deny this. Instead, it was time for a demand: "Tell me the square root of 22." "Gosh," said the student. "Between four-and-a-half and five, definitely. 4.7

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Posted in ask uncle colin

Dear Uncle Colin, I'm aware of Binet's formula for finding the $n$th Fibonacci number, $F_n = \frac{\phi^n - (-\phi)^{-n}}{\sqrt{5}}$, and wondered if there was an inverse version - to find $n$ given a Fibonacci number. -- Fibonacci Explicit Inverse, Getting Extremely Nervous But Am Understanding More Hi, FEIGENBAUM, and thank

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Posted in probability, ranting

In an early draft of my forthcoming book, The Maths Behind, which will be available wherever good books are sold from September, I believe, I took the following unprovoked dig at Australia: "... it crashed into the ocean about 1,600 miles to the west of Perth, Australia. There's nothing there.

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Posted in ask uncle colin, logarithms

Dear Uncle Colin, I'm a bit stumped by a logs question with a variable base: $\log_{\sqrt[3]{x+3}}(x^3 + 10x^2 + 31x + 30) = 9$. I know the basics of logarithms, but this is currently beyond me. -- Obtaining Underwhelming Grade, Having To Review Every Definition Hello, OUGHTRED, and thanks for

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Posted in podcasts

In this month's episode of Wrong, But Useful, we are joined by Special Guest Co-Host @jussumchick, who is Jo Sibley in real life. Colin's audio is unusually hissy in this one, which is why it's a little late; he apologises for both inconveniences. We discuss: Jo's background and work with

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Posted in probability

For all the grief I give @reflectivemaths on Wrong, But Useful, he does occasionally ask an interesting question. In episode 45, he wondered how many packs of Lego cards one would need to acquire, on average, to complete the set of 140?1 A simpler case Suppose, instead of 140 cards,

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Posted in ask uncle colin, quadratics

Dear Uncle Colin, In a recent exam, I was invited to solve $12x^2 - 59x + 72=0$ without a calculator. Is that a reasonable thing to ask? Very Irate EdExcel-Taught Examinee Hi, VIETE, and I don't blame you for being cross - in a non-calculator exam, I'm not sure that

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Posted in integration, puzzles

The brilliant @dragon_dodo sent me this puzzle: Evaluate $\int_0^1 \left(1-x^\frac{1}{7}\right)^3 - \left(1-x^\frac{1}{3}\right)^7 \d x$. I'm not going to give you the solution right now; that will come after I've rambled for a bit. After I'd solved the puzzle (see below), I wondered what each of the integrals actually evaluated to.

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