Ask a Mathematician

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Ask a Mathematician

Postby CarrieVS » Tue Jan 12, 2016 9:10 pm

Since we're well on in the process of derailing the Bizarre and Inexplicable thread with things that, while bizarre, are not at all inexplicable, it seems like time to make a thread for maths problems you want answered or explained.

I'll try to explain things in intuitive terms, as well as mathematical proofs. I'm quite happy for other mathematically-minded TCSers to tag-team with me if anyone knows the answer to something.
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Re: Ask a Mathematician

Postby Learned Nand » Tue Jan 12, 2016 9:21 pm

Are there any non-trivial zeroes to the Riemann zeta function without a real part of 1/2?
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Re: Ask a Mathematician

Postby CarrieVS » Tue Jan 12, 2016 9:23 pm

The proof won't fit in this textarea.
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Re: Ask a Mathematician

Postby Windy » Wed Jan 13, 2016 4:24 am

Why is the surface area for a pyramid such a pain in the ass to calculate?

Also, real question:

So sin and cos are trigonometry things related to the angles of a triangle and the ratio of their sides. So how come I keep seeing them in graphs and things completely unrelated to triangles?
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Re: Ask a Mathematician

Postby CarrieVS » Wed Jan 13, 2016 11:23 am

Windy wrote:Why is the surface area for a pyramid such a pain in the ass to calculate?

Mainly because one of the key lengths you need - the distance from the tip to the centre of the base of each triangular face - isn't one of the standard measurements of a pyramid. Also partly because there's three different shapes of face, and there aren't even the same number of each.

Personally, I wouldn't bother with combining all the steps into one formula:
I'd calculate the 'height' of each triangular face (ie. how high it would be if it was tilted to be vertical): Sqrt(h^2 + (l/2)^2) for the 'side' triangle, by Pythagoras' theorem.
Meaning the area of that triangle is [that value] * 1/2 * w, by the usual triangle area formula
Reverse length and width and repeat to get the 'end' triangle.
Add those two areas together, and multiply by two because there's two of each kind of triangle.
Then add l * w for the rectangular base.


So sin and cos are trigonometry things related to the angles of a triangle and the ratio of their sides. So how come I keep seeing them in graphs and things completely unrelated to triangles?


Sines and cosines are properties of angles, whether that angle's in a triangle or not. They're usually defined in terms of right-angled triangles, but they have all sorts of other uses than simply measuring triangles. They're very prettily related to circles, for instance:
Image

That's a consequence of how triangles turn up in all sorts of places - anywhere you have a line at an angle, you have a triangle, if you draw in the other two sides. It's very often useful to describe things in terms of horizontal and vertical, and you need sines and cosines to go between description in terms of length and direction, and in terms of horizontal and vertical distance. So anything that uses vectors and coordinate systems is probably going to have trig functions turn up somewhere.

You can make any function that repeats itself - or any function if you only consider it on a finite domain - out of an infinite sum of sine waves, and that can be quite useful. In physics, you can model any periodic phenomenon using sines, and indeed many things can be modelled using just one sine wave, because it happens that that kind of motion turns up a lot in the real world.

They're also related to exponentials and complex numbers so you'll see them in that respect.

And they have quite interesting properties when you differentiate them, so you'll see them a lot in calculus.

I could go on. Sines and cosines get everywhere, they're not just for measuring triangles.
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Re: Ask a Mathematician

Postby Piter Lauchy » Wed Jan 13, 2016 6:28 pm

CarrieVS wrote:Sines and cosines get everywhere, [...]

"I don't like sines and cosines. They're coarse and rough and irritating..." - Anakin Sinecalculator
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Re: Ask a Mathematician

Postby CarrieVS » Wed Jan 13, 2016 6:29 pm

Piter Lauchy wrote:
CarrieVS wrote:Sines and cosines get everywhere, [...]

"I don't like sines and cosines. They're coarse and rough and irritating..."


Are not! They're as smooth as a function can very well be.
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Re: Ask a Mathematician

Postby A Combustible Lemon » Wed Jan 13, 2016 7:12 pm

*PEDANT MAN COMES SWINGING IN ON A ROPE*

The smoothest a single parameter function can be is if curvature f''(x)=0 which means f'(x)=k1 and f(x)=k1*x+k2. Which would make it a straight line.

*Swings out*
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Re: Ask a Mathematician

Postby CarrieVS » Wed Jan 13, 2016 7:52 pm

Really? I'd argue that's only as smooth, and smooth in a less interesting way.

I guess the exponential function might beat the harmonics, but you can differentiate sinx and cosx everywhere as many times as you please, and unlike polynomials, you can do so without ever getting a trivial answer.
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Re: Ask a Mathematician

Postby Windy » Thu Mar 17, 2016 4:42 pm

Why do mathematicians insist on writing things in the most confusing way possible?

e.g. Instead of writing 10.36 degrees, they write 10° 21' 36''
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Re: Ask a Mathematician

Postby Marcuse » Thu Mar 17, 2016 5:05 pm

Windy wrote:Why do mathematicians insist on writing things in the most confusing way possible?

e.g. Instead of writing 10.36 degrees, they write 10° 21' 36''


I would imagine it's a little bit like using complicated language; it communicates more.
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Re: Ask a Mathematician

Postby CarrieVS » Thu Mar 17, 2016 6:00 pm

Windy wrote:Why do mathematicians insist on writing things in the most confusing way possible?

e.g. Instead of writing 10.36 degrees, they write 10° 21' 36''


Why do you write 5' 7'' instead of 5.58 feet?

For that specific example, I'm going with 'we don't'.
I've never seen minutes and seconds used for angles instead of decimal fractions of degrees by a mathematician. I vaguely remember being told in school that you could write it that way, but we were never expected or encouraged to use it. You'll find some oddball who'll do it I'm sure, but you'd have to ask them why they do.

In general, we use shorthand such as symbols for units instead of writing everything out longhand because we're lazy. I guess it can be confusing, but if you were used to seeing sums written out like "five added to three is equal to eight" you'd probably find 3+5=8 just as confusing, and once you learned the shorthand you'd surely appreciate it.

It can make a page of advanced maths quite impenetrable to the layman, I agree, but so is a page of sheet music, for instance.
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Re: Ask a Mathematician

Postby NathanLoiselle » Thu Mar 17, 2016 7:08 pm

What's the square root of 37? My calculator's broke.
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Re: Ask a Mathematician

Postby jbobsully11 » Thu Mar 17, 2016 7:48 pm

NathanLoiselle wrote:What's the square root of 37? My calculator's broke.

6.08276253032982, to fifteen digits

You can calculate those in Microsoft Excel, y'know. Just type "=sqrt(37)" without the quotation marks and hit Enter. Google would also work.
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Re: Ask a Mathematician

Postby NathanLoiselle » Thu Mar 17, 2016 7:55 pm

jbobsully11 wrote:
NathanLoiselle wrote:What's the square root of 37? My calculator's broke.

6.08276253032982, to fifteen digits

You can calculate those in Microsoft Excel, y'know. Just type "=sqrt(37)" without the quotation marks and hit Enter. Google would also work.


Would you believe me if I said my google is broken too?
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