Browsing category ask uncle colin

Ask Uncle Colin: about summation notation

Dear Uncle Colin, I'm having trouble getting my head around sum notation! I can't tell whether $\sum_{n=0}^{6}{5}$ means $0\times5 + 1\times 5 + ... + 6\times 5$ or $0 + 1 + 2 + ... + 6$ or $5 + 5 + 5 ... + 5$. Wolfram|Alpha just gives me

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Ask Uncle Colin: A hellish trigonometric identity

Dear Uncle Colin, @CmonMattTHINK unearthed the challenge to prove that: $\tan\left( \frac 3{11}\pi \right) + 4 \sin\left( \frac 2{11}\pi \right) = \sqrt {11}$. Wolfram Alpha says it's true, but I can barely get started on the proof and I'm worried no-one will like me. Grr, Really Obnoxious Trigonometry Has Evidently

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Ask Uncle Colin: Cambridge!

Dear Uncle Colin, Do you have any advice about Cambridge interviews? I have one coming up. How About Wanting Kings (If Not, Girton) Hello, HAWKING! My Cambridge interviews were a couple of decades ago, so I don't know how current my advice is. A few things: be ready to demonstrate

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Ask Uncle Colin: An emergency limit!

Dear Uncle Colin Help! An emergency has arisen and my L'Hôpital's Rule is out of order -- but I still need to find $\lim_{x \to 0_+} \frac{\ln(x)}{\cot(x)}$! This is not a drill! -- Help! Inexplicable Limitations Bring Everyone Ridiculous Trouble Don't panic, HILBERT! Don't panic. Now, for the benefit of

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Ask Uncle Colin: Missing Numbers

Dear Uncle Colin, My teacher recently challenged me to tackle the missing numbers problem below and I don't know where to start! It's driving me to disgust with the whole number system. Crazy Old Numbers, Wacky And Yucky #Edtech and gimmicks won't help students solve this, but considering the nature

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Ask Uncle Colin: Random Points on a Sphere

Dear Uncle Colin, In one of Randall Munroe's What If? articles he says that the maths of finding a random point on a sphere is a bit hairy. Can't you just pick a random latitude and longitude? -- Surely Places Have Equal Random Expectations You would think so, wouldn't you,

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Ask Uncle Colin: Multiplying negatives

Dear Uncle Colin, I'm finding it hard to understand why, when you multiply two negative numbers together, you get a positive number. I accept that it's true, but I was brought up to believe that two wrongs don't make a right. -- Positive Equals A Negative Otherwise? There is a

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Ask Uncle Colin: Is my friend crazy?

Dear Uncle Colin, A friend of mine told me that $1 + 2 + 4 + 8 + ... = -1$. Is he crazy, or is there something going on here? -- Somehow Enumerating Ridiculous Infinitely Extended Sum Dear SERIES, There are a couple of 'proofs' of this non-fact that

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Ask Uncle Colin: A logarithmic coincidence?

Dear Uncle Colin, I noticed that $2^{\frac{1}{1,000,000}} = 1.000 000 693 147 2$ or so, pretty much exactly $\left(1 + \frac{1}{1,000,000} \ln(2)\right)$. Is that a coincidence? Nice Interesting Numbers; Jarring Acronym Dear NINJA, The easiest way to see that it's not a coincidence is to check out $3^{\frac{1}{1,000,000}} $, which

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Ask Uncle Colin: an integral that’s giving me a headache

Dear Uncle Colin, I've been trying to work out $I = \int_0^{\frac \pi 4} x \frac{\sin(x)}{\cos^3(x)} \d x$ for hours. It's the fifth time this week I've been up until the small hours working on integration and it's affecting my work and home life. I'm worried I'm becoming a calcoholic.

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