Posted in ask uncle colin.

Dear Uncle Colin, I’ve figured out that $x^{x^{x^{\dots}}} = 2$ when $x = \sqrt{2}$, but I’m struggling to make sense of the function - it seems to have a vertical gradient when $x = e^{\frac{1}{e}}$, but it doesn’t seem to have what I think of as an asymptote there. What

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Posted in ninja maths.

“We’ve been through this a hundred times, sensei. I say something like ‘$10^{1.35}$. Hm, let me get my calculator’ and you torture me in some unspeakable way an blurt out the answer…” “22.4” “… thank you, especially for refraining from the torture bit.” “You’re welcome.” “Then, of course, you tell

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Posted in ask uncle colin.

Dear Uncle Colin, On a recent revision course, my tutor couldn’t integrate $\int_0^{\piby2} \frac{\sin(2\theta)}{1+\cos(\theta)} \d \theta$. Can you? - Reasonable Expectation For University’s Nameless Don? Hi, REFUND! Far be it from me to disparage a fellow professional’s integration skills, especially in the heat of the moment; I frequently mistake $2\times

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Posted in dome.

I’m a big fan of the doodle. My lecture notes, even my schoolbooks, are covered with geometric patterns and impossible shapes and simple cartoons. Today’s entry in the Dictionary of Mathematical Eponymy started jumped out of Stanislaw Ulam’s notes while he was listening (according to Martin Gardner) to a ‘long

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Posted in ask uncle colin.

Dear Uncle Colin I wonder: at what height is the volume of a cone above that height equal to the volume below? What about the surface area? Are there any cones where it’s the same height? Finally Researching Uniformly Splitting Things Up, Mate Hi, FRUSTUM, and thanks for your message!

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Posted in fractions.

I’m on a bit of a continued fractions jaunt at the moment, as they’re my current “oo! That’s a tool I haven’t played with enough, and that I think might be interesting!” One of their applications (surprisingly to me) is to answer the question “A survey firm says ‘82.43% of

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Posted in ask uncle colin.

Dear Uncle Colin, Can you explain why $\frac{a+b}{c + d}$ is between $\frac{a}{c}$ and $\frac{b}{d}$? - Grinding Out Solid Proof Explaining Rationals Hi, GOSPER, and thanks for your message! As per usual, there are several methods to show this. I’m going to assume (since it hasn’t been stated) that $\frac{a}{c}$

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Posted in fractions.

Lately I’ve been playing around with continued fractions - I blame credit @evelynjlamb for pointing me at this post by @johndcook. I’ve done most of my learnin’ from here and from various Wikipedia pages, and I thought I’d revisit some of it for the benefit of others. In fact, I’ll

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Posted in ask uncle colin.

Dear Uncle Colin, I need to solve $5^{2x+2} +16\cdot 15^x - 9^{x+1} = 0$ but I’ve hit a dead end! Can you help? - Puzzling Out Wild Exponent Relations Hi, POWER, and thanks for your message! From the working you sent, it looks like you’ve picked a good strategy: you’ve

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Posted in geometry.

A nice observation from Futility Closet: Draw two circles of any size and bracket them with tangents, as shown. The chords in blue will always be equal. I’m hardly going to let that pass by without a proof, now, am I? Spoilers below the line. My proof Definitions Let the

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