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January 25

Extending the Steinmetz Solids to n dimensions...

If Vn is the (Hyper^n)volume in n dimensions of the intersections of all circles formed from the combinations of dj^2+dk^2=1 for any 1<j,k<n how does that volume Vn compare to the volume of the unit cube and the volume of the volume sphere in n dimension? (So in 4 dimensions, all points meeting w^2+x^2<=1, w^2+y^2<=1, w^2+z^2<=1, x^2+y^2<=1, x^2+z^2<=1 and y^2+z^2<=1, note this is *not* equal to the unit sphere as (sqrt(2),sqrt(2),sqrt(2),sqrt(2)) is on the edge of this volume and outside the unit sphere).Naraht (talk) 03:52, 25 January 2016 (UTC)

I get for the hypervolume for dimension n:

$2^{n}(n-1)\int _{0}^{\tfrac {\pi }{4}}\sin ^{n-2}\theta d\theta .$

Actually there are other ways to generalize to higher dimensions. For example in dimension 4 you could talk about the intersection of the 4 "hypercylinders" x²+y²+z²≤1, x²+y²+w²≤1, x²+z²+w²≤1, y²+z²+w²≤1. For any n and k, 1≤k≤n you could define the solid to be the intersection of the (n choose k) cylinders formed by taking coordinates k at a time and restricting to the k-ball. For k=1 this gives you the hypercube with side 2 and for k=n you get the n-ball, with the remaining sets a nested sequence between these two. --RDBury (talk) 09:36, 25 January 2016 (UTC)

Beautiful! Can the function be generalized with k as well?Naraht (talk) 19:22, 25 January 2016 (UTC)

I've been working on the k=3 case but it appears that the computation rapidly gets more difficult as k increases. For k=2 you end up integrating over a segment of a circle, which is where the limits 0 to π/4 come in. For k=3 you need to integrate over a spherical triangle which is tricky. For k=4 you need to integrate over a hyperspherical tetrahedron which is even worse. Of course there may be a simpler way that I'm missing; I'm trying to use spherical coordinates but maybe there's a different coordinate system that will work better. --RDBury (talk) 01:15, 26 January 2016 (UTC)

Want name of book

In one of Calvin Clawson's books on mathematics, chapter 12 deals with certain aspects of prime numbers, such as Goldbach's conjecture. Is the name of this book Mathematical Mysteries: The Beauty and Magic of Numbers?Bh12 (talk) 21:53, 25 January 2016 (UTC)

You can see the table of contents for that book at it's page on amazon. Chapter 12 is called "Goldbach's Conjecture", so it sounds like that's probably the book you're thinking of. Staecker (talk) 23:08, 25 January 2016 (UTC)

Resolved

January 28

What did people use before slide rule and calculators and computers were invented?

Yet another one from the Ohio troll.

What did people use before slide rule and calculators and computers were invented? An abacus? Their heads? Were people smarter back in the olden days because they might have had to do a lot of mental math while nowadays people just get lazy and use a calculator? 140.254.136.157 (talk) 18:23, 28 January 2016 (UTC)

There were look-up tables for things like trig functions. Of course, back then, few people really needed to do that type of math. As for whether people were "smarter" then, I'd say no, that just having to memorize more things doesn't make one smarter. We are coming into a period now where it really isn't necessary to remember many facts at all, as they are all available online, such as right here on Wikipedia. Not needing to do math in your head was just an earlier level of this movement from keeping all the world's knowledge in your head to storing it externally. The invention of writing was the first step in that direction. StuRat (talk) 18:46, 28 January 2016 (UTC)

Also see Calculator#Precursors to the electronic calculator. Loraof (talk) 18:50, 28 January 2016 (UTC)

Also Human computer. Loraof (talk) 18:54, 28 January 2016 (UTC)

And log tables→86.139.120.76 (talk) 20:09, 28 January 2016 (UTC)

Really, the question needs to be more specific. What did people use to do what sort of calculation? An abacus is great if you want to add and subtract numbers that may be several digits long, and it can help you do multiplication if you convert it to repeated additions: for example, to multiply 7,479 by 4,186, you could simply add 7,479,000 + 7,479,000 + 7,479,000 + 7,479,000 + 747,900 + 74,790 + 74,790 + 74,790 + 74,790 + 74,790 + 74,790 + 74,790 + 74,790 + 7,479 + 7,479 + 7,479 + 7,479 + 7,479 + 7,479, carefully keeping track of how many times you repeated each distinct term. Division would be of similar complexity, only you wouldn't know in advance how many steps there would be. And all of this would be possible even if you didn't have a positional notation like our 31,307,094 to express the answer in.

On the other hand, once people did have positional notation, they could also do calculations like multiplication and division using the long multiplication that I hope is still taught in schools today, and doing the additions mentally. And if they didn't want to work out the intermediate values directly (the ones you get by multiplying a multi-digit number by a single digit), they could use a set of Napier's bones for that purpose.

There is much more to say on this topic, but as I said, it really all depends on what sort of calculations you're talking about. --76.69.45.64 (talk) 23:32, 28 January 2016 (UTC)

The sort of calculations made by slide rule was: multiplication, division, square, square root, log, sin, cos, and tan. Before the slide rule these operations were done using paper and pencil and table lookup. Special formulas were designed to optimize trigonometric calculations. For example the angle A in a triangle having sides a, b and c was not computed by

$\cos A={\frac {b^{2}+c^{2}-a^{2}}{2bc}}$

but rather by

$\log \tan {\frac {A}{2}}={\frac {\log(s-b)+\log(s-c)-\log s-\log(s-a)}{2}}$ where $s={\frac {a+b+c}{2}}$

avoiding multiplications and using only 5 table lookups. Bo Jacoby (talk) 00:31, 29 January 2016 (UTC).

Recall also Mechanical calculators. --CiaPan (talk) 08:09, 31 January 2016 (UTC)

Can someone explain how the Romans did sums using their Roman numerals? I know they did it somehow. And what about the Greeks, J-e-w-s, etc., who used letters to denote numbers? 94.10.23.218 (talk) 13:52, 31 January 2016 (UTC)