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Author Topic: Mathematics Help Thread  (Read 187184 times)

MagmaMcFry

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Re: Mathematics Help Thread
« Reply #2490 on: July 04, 2017, 09:01:50 pm »

Quote
any polynomial degree 5 or above has no general solution
Correction: There exist some polynomials of degree 5 or above whose solutions have no closed form in terms of root expressions.

Also note that sin(pi/360) actually does have a closed form since sin(pi/5), sin(pi/8) and sin(pi/9) all have a closed form.
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Arx

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Re: Mathematics Help Thread
« Reply #2491 on: July 05, 2017, 01:37:14 am »

Yes, but then it begs the question of how we know that sin(pi/3) = (√3)/2.

Well, yes, but it's a fairly trivial geometric construction. Arguably that's dodging the question of how we know that that geometric construction is the right one, but at that point it's arguing about the definition of the trig functions and it's not particularly helpful.
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Reelya

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Re: Mathematics Help Thread
« Reply #2492 on: August 24, 2017, 07:20:25 pm »

https://www.theguardian.com/science/2017/aug/24/mathematical-secrets-of-ancient-tablet-unlocked-after-nearly-a-century-of-study

They've decoded a Babylonian clay tablet and it contains trigonometry calcuations that predate Pythagoras by over 1000 years. And apparently it's a whole different approach to doing trig. which might even have applications today.
« Last Edit: August 24, 2017, 07:31:54 pm by Reelya »
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frostshotgg

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Re: Mathematics Help Thread
« Reply #2493 on: August 25, 2017, 01:22:02 am »

I'm kinda stumped on where you're getting a sequence of length 54 to begin with. Shouldn't there be sequences of length 1, 3, 9, 27, then 81? Where does 54 come into play?
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MagmaMcFry

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Re: Mathematics Help Thread
« Reply #2494 on: August 26, 2017, 05:15:48 pm »

There's no method to construct your function, since it is not unique at all under only the conditions you've given us (the sequence is a permutation of the first 2*3^n natural numbers such that for all k the sequence is 2*3^k-periodic modulo 2*3^k). In fact, there are 341163456359156416512 sequences of length 54 exhibiting the exact same bahavior. So unless there are more details to that sequence, we can't answer your question.

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Parsely

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Re: Mathematics Help Thread
« Reply #2495 on: September 03, 2017, 02:11:59 pm »

I'm trying to solve this Trigonometry problem:

So basically we're solving for x, or theta.

I convert the cos(x) to x and solve it algebraically:

6x2-5x+1 = 0                    // easy enough to factor it
(3x-1)(2x-1)                    // then set them equal to 0 and solve for x

---

3x-1 = 0
3x = 1
x = 1/3                          // x = cos(x)

2x-1 = 0
2x = 1
x = 1/2

---

cos(x) = 1/2
x = cos-1(1/2)
x = 1.05 radians                // Questions asks for solutions rounded to hundredths

cos(x) = 1/3
x = cos-1(1/3)
x = 1.23 radians

x = theta = 1.05, 1.23 radians

These are positive and they're on my interval so I figure that's it, but I only got partial credit so either my answer is incomplete or one of those numbers is wrong. What am I overlooking?
« Last Edit: September 03, 2017, 02:13:42 pm by GUNINANRUNIN »
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MagmaMcFry

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Re: Mathematics Help Thread
« Reply #2496 on: September 03, 2017, 03:25:08 pm »

There's more than one x with cos(x) = 1/2.

Also for reference it's not a good idea to give two different variables (x and cos(x)) the same name, because it could confuse you if you're not careful, but more importantly because it is confusing and hard to read for anyone reading your math (and strictly speaking it's ambiguous and ambiguous is not math). You could call cos(x) y instead, then you have 6y² - 5y + 1 = 0 which you can solve for y, then solve y = cos(x) for x.
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Parsely

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Re: Mathematics Help Thread
« Reply #2497 on: September 03, 2017, 08:44:09 pm »

There's more than one x with cos(x) = 1/2.

Also for reference it's not a good idea to give two different variables (x and cos(x)) the same name, because it could confuse you if you're not careful, but more importantly because it is confusing and hard to read for anyone reading your math (and strictly speaking it's ambiguous and ambiguous is not math). You could call cos(x) y instead, then you have 6y² - 5y + 1 = 0 which you can solve for y, then solve y = cos(x) for x.
Oh, thank you, I think I understand. Though I'm not actually sure how I was to know that these answers would be in the 1st and 4th quadrant.

I'll keep that in mind, and thanks again.
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TheDarkStar

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Re: Mathematics Help Thread
« Reply #2498 on: September 03, 2017, 09:30:32 pm »

You can end up with extraneous solutions to trig equations. Make sure to check them by substituting them into the original equation to make sure that they're actually solutions.
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askovdk

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Re: Mathematics Help Thread
« Reply #2499 on: September 04, 2017, 07:46:29 am »

An old mathematical curiosity has come back to haunt me, as I can’t complete a rigid proof.
I probably have it from a Martin Gardner (Scientific American and more), and the statement is:

Write the two fractions*:
   0           1
   1           0

(* The last element is not a fraction, and will be removed later, but it’s needed to make the algorithm going.)
Between these write the fraction, where the numerator is the sum of the numerators on each side, and the denominator is the sum of the denominators on each side.
   0      1      1
   1      1      0
Do it again
   0    1    1   2    1
   1    2    1   1    0
And again
   0  1  1  2  1  3  2  3  1
   1  3  2  3  1  2  1  1  0
And again
   0 1 1 2 1 3 2 3 1 4 3 5 2 5 3 4 1
   1 4 3 5 2 5 3 4 1 3 2 3 1 2 1 1 0
And again … and again …
Claim (ignoring the last 1/0 (and the first depending on your attitude towards 0 being positive)):
Eventually every positive rational number will be found once in its non-reducible form.

So beside wishing to share this beautiful nugget of math, my questions to the forum are:
•   Does anybody know if this has any official name, that I can Google for?
•   Can anyone prove the claim?
It’s easy to show that each generation will be a strictly increasing list of number. I have a strange but valid proof that all elements are irreducible, but the fact that we get them all still eludes me.

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da_nang

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Re: Mathematics Help Thread
« Reply #2500 on: September 04, 2017, 08:41:02 am »

A similar sequence is the Farey sequence.

In fact, perhaps you're looking for the Stern-Brocot sequence?
« Last Edit: September 04, 2017, 08:46:43 am by da_nang »
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Helgoland

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Re: Mathematics Help Thread
« Reply #2501 on: September 04, 2017, 11:32:30 am »

Looking at the example you wrote out (thanks for that), a fairly quick proof should be doable by looking at the differences between numerators and denominators. Every difference - positive and negative - occurs with every possible denominator.
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bloop_bleep

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Re: Mathematics Help Thread
« Reply #2502 on: September 04, 2017, 12:57:56 pm »

There's a math thread?
WOOT WOOT
Also, I've found a proof to askovdk's problem, I'm writing it up now.
EDIT: Found unexpected contingency. Will have to work on it some more.
EDIT 2: Contingency solved. Writing it up again.
« Last Edit: September 04, 2017, 01:38:12 pm by bloop_bleep »
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WealthyRadish

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Re: Mathematics Help Thread
« Reply #2503 on: September 04, 2017, 03:34:50 pm »

I'm writing a function to simulate the Ricardian Law of Rent for part of an EU4 mod (something for fun), and think I've reached an acceptable solution but am not positive that all the steps I took are sound. The units of the final rent function seem to check out and it exhibits the properties that I would expect, but I still have doubts if it's conceptually correct.

In the mod, there is a function labor(x) which represents the amount of labor necessary to produce a unit of goods in some industry as a function of x, land. Less productive land requires more labor to produce the same unit of goods, and it's assumed that the most productive land will be used first. So for example the excellently-situated 0.1th unit of land may require 15 people to produce a bushel of turnips in a month, but to make that same bushel of turnips on the crummy 7.8th unit of land may require 50 people, and that's what this function represents. I'm using simple polynomials to model the labor function composed of one x variable and two constants 'a' and 'c', e.g. labor(x) = a*x + c, labor(x) = a*x^2 + c, or more generally as labor(x) = a*x^n + c.

The function I want to arrive at is the total amount of rent (in money) that would be appropriated under these conditions given the total amount of land in production and the labor function. The Law of Rent suggests that the amount a landlord can charge in rent for producing on their land is limited by the difference between the amount a laborer can produce on that land compared to the amount the same laborer could produce if they were working on the best rent-free land available (or take everything beyond subsistence if no land is available or they are forcibly prevented from moving). I took the following steps to try and reach a function of that rent value from the labor function:

1) The difference in the amount of labor to produce one unit of goods on a unit of land 'x' compared to the final unit of land in production would be given by labor(x) - labor(T), where 'T' is the total amount of land in production. The units of this would be in labor.

2) The difference in the amount of money a laborer could produce on a unit of land compared to the last unit of land would be P*s/labor(x) - P*s/labor(T), where 'P' is price (money per goods) and 's' is general productivity of all units of land (goods per land). The units of this function would be in money per land per labor. This function could be factored to P*s(1/labor(x) - 1/labor(T))

3) Since the above is the difference per laborer, to get the difference per unit of land I think I need to multiply the function again by labor(x). I'm not positive if this step actually makes sense conceptually, and this is where most of my doubts come from, but the result after some simplification is r(x) = P*s(1 - labor(x)/labor(T)), in units of money per land.

4) From there by taking the integral from 0 to T with respect to x of r(x), I should get a function R(T) that I think represents the total rent appropriated in units of money. Substituting back in the labor function -- labor(x) = a*x^n + c -- the integral results in a nice and clean function R(T) = P*s (n*a*T^(n+1) / ( (n+1)(a * T^n + c) ) in units of money. This function is attractive because it exhibits all the behaviors I would expect, can be computed with simple arithmetic, and appears to be in the correct units, but I just don't know if it's logically consistent with the Law of Rent and actually represents the amount of rent that would be appropriated under those conditions.

So what I'd like to ask is, if anyone actually made it to the bottom of this post:

1) Did I goof the math
2) Does the logic seem consistent with what I'm trying to model?
« Last Edit: September 04, 2017, 03:55:04 pm by UrbanGiraffe »
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askovdk

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Re: Mathematics Help Thread
« Reply #2504 on: September 05, 2017, 04:20:22 am »

A similar sequence is the Farey sequence.

In fact, perhaps you're looking for the Stern-Brocot sequence?

Thank you very much!  :D
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