# Let's discuss Mathematics

Discussion in 'Science & Technology' started by ParadigmShifter, Mar 16, 2009.

1. ### Robert FINMonty n' Roll

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Thank you. Awesome! The logic works now. Just didn't understand I had to factorize with the zero-points (in English??) method. I still got to think how to proof them that n(n-1)(n-2) is divisible by 3 but I think it will be an easier task, I'll ask if I have to. Thanks.

2. ### warpusIn pork I trust

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Base case is simple.

Assume true for n, prove for n+1 (induction step):

Also let x = (n+2)

We know that (n)(n-1)(n-2) is divisible by 3
Therefore (x+2)(x+1)(x) is also divisible by 3

Show that (x+3)(x+2)(x+1) is divisible by 3 to prove induction step

= (x)(x+2)(x+1) + (3)(x+2)(x+1)

We know that (x)(x+2)(x+1) is divisible by 3 as per our induction step assumption. (3)(x+2)(x+1) is also divisible by 3, obviously.

Add two numbers divisible by 3 together and the result will also be divisible by 3. (This would have to be proved in some way but I'm lazy)

Therefore (x+3)(x+2)(x+1) is divisible by 3
Therefore (n+1)(n)(n-1) is divisible by 3

Therefore the inductive step is proved.

Therefore n(n-1)(n-2) is divisible by 3

So of this notation is a bit off, you'll have to throw in k here and there and replace some of the n's I think, and expand on some of this, but you get the idea

3. ### KyriakosAlien spiral maker

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Finished reading a book account of the proof offered in the early 90s for a^x + b^x = c^x not true for x>2 and a,b,c integers (ie so-called 'Fermat's last theorem').
Book wasn't anything particularly good, (popularized account with general info, typically one chapter for each person of interest and his math used on this, and bio stuff). I liked the quest for unifying all math fields, cause i think it is obviously possible to do so (given all math is itself a subset in ways of human thought, and we use just that).
I do wonder what kind of proof Fermat had in mind, though. Maybe going with geometry (infinite collapse, or huge number of sides polygon leading to circle) and probability (since raised powers are directly tied to probability anyway; i suppose this is why x^0 = 1, cause there is only one case of empty set).

4. ### SamsonDeity

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I think I read that book. I have always assumed he was just wrong.

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5. ### KyriakosAlien spiral maker

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Me too. But i just assumed this for the book writer, not Fermat

6. ### FerocitusDeity

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I wrote a memetic algorithm about 20 years ago that found this arrangement
of hulls that minimises the wave-making resistance at one particular speed
(technically, Froude number).
It was surprising at first, but then very obvious after a little thought
and some mathematical squiggling. I love that aspect of AI/nonlinear search.

7. ### KyriakosAlien spiral maker

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^Not commenting cause i don't know what that is.
But i do love use of computers in graphic representation of math. It is something which allows for a (seemingly) very different source of input/impression, and thus may lead to furthering ideas on math itself (and not just consciously imo).

8. ### FerocitusDeity

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That one is simulating 4 ships travelling together and moving to the left
at increasing speed. The ships each make waves themselves, but the wave
patterns interfere. Sometimes the waves cancel at places behind the ship;
sometimes they reinforce and the waves at that point are larger. At one speed,
the waves almost disappear everywhere.
Waves created by ships means that energy is being wasted, in the sense that
it does not contribute to forward propulsion.

We found ways of calculating the wave pattern made by ships that were up to
100,000 times faster than was possible previously. That meant that I could
then create a population of artificial lifeforms that had the mathematical
behaviour of ships vis-a-vis waves. I then let that population evolve. By
that I mean I kept the best ships, replaced the worse ones with random
ships, and combined (or "mated") the best ships together. It's an extremely
crude approximation of Darwinian evolution. After 1 to 100 hours of computer
time, and many games of Civ while waiting, the population starts converging
on a pattern that makes the smallest waves, i.e. it has the lowest wave drag.
Without the mathematical work to devise fast accurate algorithms, it would have
been a ridiculous exercise. Waiting a day or a few is Ok; waiting 100,000 to
a million days is out of the question.
For me it then all became a game, with a high score to beat (in that case, the
lowest drag.)

I also like creating patterns just for their own intrinsic beauty.
The green one is just a collection of singularities (like tiny black holes and
white holes) travelling just under the surface of an imaginary ocean.
Grav-Mass is Richard Stallman's idea of celebrating Isaac Newton's birthday
on the 25th of December.

The grey picture is a simple wave pattern created by one singularity travelling
towards the bottom left.

It has some features that are similar to those in the bottom graphic, which
is the pattern made by wind-driven clouds as they pass over a small island in
the Indian Ocean.

Of course, the real world is far more complicated than we could ever hope to
simulate, but logic and mathematics (and physics) sometimes make you feel as
if you are getting a glimpse into the structure of the universe and how it works.
At another level, it's delusional, and that feeling of satisfaction is probably
the same as when somebody with OCD lines up hundreds of matches on a table in
nice straight lines.

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9. ### Robert FINMonty n' Roll

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EDIT: I did the exam already so I don't need the answer anymore. They didn't ask it in the exam luckily DD

I have the final exam of maths tomorrow. (high school)

A question:

How to count the radius when using integration to count volume? The formula is like: pi * (the integral of r^2 from starting point to ending point). And if the object spins around y or x axis, then the r is going to be just the function that is going to spin. But it changes somehow when the object spins around something like x=2 or y=-4 etc. In one example I read, there the object spins around x=3 so then they said that r=3-x and inserted that to the formula. How did they count the r, I don't understand the logic because they didn't explain.

Again, sorry for bad English math terms. And thank you in advance.

Last edited: Mar 22, 2017
10. ### Danai GuriraChieftain

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I have questions about my maths homework too as I am taking a summer courses and learning calculus. I am having problem with the derivatives of trigonometric functions. Here is the question:
y = sin^4 (x)
y = sin(x^4)
Are the derivatives of these two the same?
I need some help plz.

11. ### MichkovEmperor

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No they are not the same. How did you come up with the notion that the two derivatives would be the same?

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@Danal Gurira -- Hint: Study the examples in the link you provided and use the chain rule.
@Michkov -- The question of whether or not the two derivatives are equal is the problem to be solved. Danal is not asserting that it's true.

13. ### MichkovEmperor

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Just trying to understand hangup is in the thought process. I could say apply the chain rule and see what happens, but that assumes she knows the chain rule and how to use it.

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Solution in spoilers in case you really can't solve it

Spoiler :
(Sin (x^4) )' = 4*(x^3)*cos (x^4) (it's (f (g (x)))' = g'(x)*f'(g (x)) with f = sin and g = x^4)
(Sin (x)^4)' = cos (x)*4*(sin(x)^3) (this time f = x^4 and g = sin)

So the derivatives of the two functions are different

Last edited: Jul 8, 2017

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Please do not solve other member's homework for them. It is much more useful to provide hints. Thanks.

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I'd say that it depends on at what stage the person needing help is. Sometimes you're at "I'm having a small problem with this can someone drop me a hint" and sometimes you're at "I'm at a complete loss I need to see it done properly once to get the basic principle". But you're right I should have at least spoilered it. (now that I have you may want to delete the quoted part of your message)

17. ### Lohrenswald老任森林

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to have f(x) -> 0 for x -> positive infinity, A has to be 0, right?
or is there something I'm missing?

18. ### OlleusDeity

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If a squared->0 faster than x->infinity you can also have A = B I believe

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If a>0 A=0. If a=0 A=B.

20. ### Lohrenswald老任森林

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alright so, a chain mail or pyramid scheme kind of deal:

we begin with one person, who "recruits" number a amount of people. Each of these a people are tasked with recruiting a more people. Also index the steps as n (wheather that first original guy's recruitment should be n=0 or n=1 I don't know (yet lol)). total number of people recruited is N.

So, here's my lazy question lol: what does the function N(n) look like? It can't be as simple as a^n, because that doesn't factor in the people recruited before step n

also, the obvious question is how many steps it takes to recruit all of the earth's population