I just happened upon this extra nifty little fact - the set of sums of two squares is closed under multiplication.
That is, facts like (102 + 92) × (12 + 12) = 192 + 12 occur for every
(a2 + b2) × (c2 + d2).
Take a look - simple result, I am surprised I hadn't seen it before...
See the Brahmagupta-Fibonacci identity at Wikipedia
Tuesday, June 23, 2009
Sunday, June 21, 2009
Trig identites and multiplication
We're used to the idea of using logarithms to convert multiplication problems into addition problems, because of the relation:
log(xy) = log(x) + log(y)
So if you have a table of logarithms, to multiply x and y you can look up the logarithms of each, add them, then find the number in your tables whose logarithm is that sum. Three table-lookups and an addition. [Well, in practice you do the last lookup in a table of antilogarithms, but that's only a minor benefit, it's not actually necessary to the calculation.]
However, there are other ways to make multiplication "simpler" than trying to directly multiply, such as this identity:
cos x cos y = [cos (x + y) + cos (x - y)]/2
This sort of relationship was actually used to do multiplication; it's a bit more complicated than logarithms, but it works well enough if you haven't invented logs yet, but do have cosine tables. It works like this.
You want to multiply two numbers, X and Y (and let's assume that they have already been scaled to lie in a range where this will work).
You find the numbers that they are the cosines of in your tables, X = cos x, Y = cos y. Add and difference those two numbers, giving (x+y) and (x-y). Look those numbers up in your cosine tables. Average them. That's the product. Four table lookups, an addition and a halving (and some rescaling back to the original problem, presumably, which probably just involves moving a decimal point), all much easier than general multiplication.
So far so good.
But here's what I am wondering. Humans came up with a nifty device to automate that multiplication via addition of logs, the slide rule. Is it possible to build a device that does mutliplication using a trig identity like the one above, perhaps some sort of "swivel rod"?
If we do it in two parts, I think maybe we can do it with a slide rule; one side does the additions and subtraction (and halving; halving is inverse doubling, which is just addition, so it can also be done there). The other does the conversions to and from normal scale to cosine scale - but you'd be forever transferring numbers across the different scales.
So I wonder if it can all be combined in a single, simpler system, whether it's a kind of slide rule or something more complex involving actual rotations to do the cosines (accuracy may be an issue though).
I don't think a device was ever made that did this, but I think that it might be possible.
If it were, I'd love to have one.
log(xy) = log(x) + log(y)
So if you have a table of logarithms, to multiply x and y you can look up the logarithms of each, add them, then find the number in your tables whose logarithm is that sum. Three table-lookups and an addition. [Well, in practice you do the last lookup in a table of antilogarithms, but that's only a minor benefit, it's not actually necessary to the calculation.]
However, there are other ways to make multiplication "simpler" than trying to directly multiply, such as this identity:
cos x cos y = [cos (x + y) + cos (x - y)]/2
This sort of relationship was actually used to do multiplication; it's a bit more complicated than logarithms, but it works well enough if you haven't invented logs yet, but do have cosine tables. It works like this.
You want to multiply two numbers, X and Y (and let's assume that they have already been scaled to lie in a range where this will work).
You find the numbers that they are the cosines of in your tables, X = cos x, Y = cos y. Add and difference those two numbers, giving (x+y) and (x-y). Look those numbers up in your cosine tables. Average them. That's the product. Four table lookups, an addition and a halving (and some rescaling back to the original problem, presumably, which probably just involves moving a decimal point), all much easier than general multiplication.
So far so good.
But here's what I am wondering. Humans came up with a nifty device to automate that multiplication via addition of logs, the slide rule. Is it possible to build a device that does mutliplication using a trig identity like the one above, perhaps some sort of "swivel rod"?
If we do it in two parts, I think maybe we can do it with a slide rule; one side does the additions and subtraction (and halving; halving is inverse doubling, which is just addition, so it can also be done there). The other does the conversions to and from normal scale to cosine scale - but you'd be forever transferring numbers across the different scales.
So I wonder if it can all be combined in a single, simpler system, whether it's a kind of slide rule or something more complex involving actual rotations to do the cosines (accuracy may be an issue though).
I don't think a device was ever made that did this, but I think that it might be possible.
If it were, I'd love to have one.
Monday, June 15, 2009
Annoying ways to ask a question...
I recently answered some mathematics questions on Yahoo Answers.
This reminded me of some of my pet peeves when people ask for mathematical help -
many of which are apparently attempts to minimize the scale of your contribution
* "Can I ask a question? It'll only take a moment."
-- yes, questions rarely take long to ask - but the answer won't be nearly as fast.
* "This is a simple problem"
-- it often isn't. If was so darn simple, you'd answer it yourself.
* "Why are you making this so complicated? It was a simple question!"
-- see question 1. Just because the question was brief doesn't mean it has a simple solution. Fermat's last theorem could be explained to a child and stated in a few lines!
Further annoyances more particular to YA include people just posting their entire homework with not the slightest indication that they've even attempted any of it, sneering responses when you don't just give them all the answers with fully worked solutions, and snarky reactions when you point out that merely typing their question as is into google would have got them the answer instantly.
This reminded me of some of my pet peeves when people ask for mathematical help -
many of which are apparently attempts to minimize the scale of your contribution
* "Can I ask a question? It'll only take a moment."
-- yes, questions rarely take long to ask - but the answer won't be nearly as fast.
* "This is a simple problem"
-- it often isn't. If was so darn simple, you'd answer it yourself.
* "Why are you making this so complicated? It was a simple question!"
-- see question 1. Just because the question was brief doesn't mean it has a simple solution. Fermat's last theorem could be explained to a child and stated in a few lines!
Further annoyances more particular to YA include people just posting their entire homework with not the slightest indication that they've even attempted any of it, sneering responses when you don't just give them all the answers with fully worked solutions, and snarky reactions when you point out that merely typing their question as is into google would have got them the answer instantly.
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