Browsing category ask uncle colin

Ask Uncle Colin: What’s $sqrt{ 100! }$?

Dear Uncle Colin, How would I work out $\sqrt(\ln(100!))$ in my head? – Some Tricks I’d Really Like In Number Games Hi, STIRLING, and thanks for your message! I don’t know how you’d do it, but I know how the Mathematical Ninja would! Stirling’s Approximation1 says that $\ln(n!) \approx n

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Ask Uncle Colin: It’s Hip To Be Square

Dear Uncle Colin, I’m struggling to make any headway with this: find all integers $n$ such that $5 \times 2^n + 1$ is square. Any ideas? Lousy Expression Being Equalto Square Gives Undue Exasperation Hi, LEBESGUE, and thanks for your message! Every mathematician should have a Bag Of Tricks –

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Ask Uncle Colin: Computing $\sqrt{2}$

Dear Uncle Colin, If I didn’t have a calculator and wanted to know the decimal expansion of $\sqrt{2}$, how would I be best to go about it? Roots As Decimals – Irrational Constant At Length Hi, RADICAL, and thanks for your message! There are several options for finding $\sqrt{2}$ as

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Ask Uncle Colin: An Implicit problem

Dear Uncle Colin, I have to find the points $A$ and $B$ on the curve $x^2 + y^2 – xy =84$ where the gradient of the tangent is $\frac{1}{3}$. I find four possible points, but the mark scheme only lists two. Where have I gone wrong? I’ve Miscounted Points Like

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Ask Uncle Colin: Powers and Remainders

Dear Uncle Colin, I’m told that $2^a \equiv 9 \pmod{11}$. How do I find $a$? – Powers And Stuff, Calculating And Learning Hi, PASCAL, and thanks for your message! As so often, there are (at least) two reasonable ways to tackle this: a brute force way and an elegant way.

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Ask Uncle Colin: A dangling rope

A Hanging Rope Dear Uncle Colin, I’m designing a small cathedral and have an 80-metre long rope I want to hang between two vertical poles. The poles are both 50 metres high, and I want the lowest point on the rope to be 20 metres above the ground. How far

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Ask Uncle Colin: Integrating $\sec$ and $\cosec$

Dear Uncle Colin, I keep forgetting how to integrate $\sec(x)$ and $\cosec(x)$. Do you have any tips? – Literally Nothing Memorable Or Distinctive Hi, LNMOD, and thanks for your message! Integrating $\sec(x)$ and $\cosec(x)$ relies on a trick, and one the average mathematician probably wouldn’t come up with without a

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Ask Uncle Colin: A Strange Simultaneous Equation

Dear Uncle Colin, I have the simultaneous equations $3x^2 – 3y = 0$ and $3y^2 – 3x = 0$. I’ve worked out that $x^2 = y$ and $y^2 = x$, but then I’m stuck! – My Expertise Relatedto1 Simultaneous Equations? Not Nearly Enough! Hi, MERSENNE, and thanks for your message!

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Ask Uncle Colin: A Factorial Sum

Dear Uncle Colin, I have been given the series $\frac{1}{2} + \frac{1}{3} + \frac{1}{8} + \frac{1}{30} + \frac{1}{144} + …$, which appears to have a general term of $\frac{1}{k! + (k+1)!}$ – but I can’t see how to sum that! Any ideas? – Series Underpin Maths! Hi, SUM, and thanks

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Ask Uncle Colin: Some Ugly Trigonometry

Dear Uncle Colin, How do I verify the identity $\frac{\cos(\theta)}{1 – \sin(\theta)} \equiv \tan(\theta) + \sec(\theta)$ for $\cos(\theta) \ne 0$? – Struggles Expressing Cosines As Nice Tangents Hi, SECANT, and thanks for your message! The key questions for just about any trigonometry proof are "what’s ugly?" and "how can I

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