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Author Topic: Maths thread.  (Read 307167 times)

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skip wooznum

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Re: Maths thread.
« Reply #625 on: March 18, 2016, 02:24:41 am »
0

Is anyone familiar with the game Set? I was wondering, what is the most cards you could have out on the table where it would still be possible for there to be no sets?

If you Google this, the answer is 20. It looks like the actual proof you can't use 21 or more is a bit complicated, unfortunately.
im bad at google. Could you link it?
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Titandrake

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Kirian

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Re: Maths thread.
« Reply #627 on: March 18, 2016, 10:50:34 pm »
+2

I can't believe no one has yet posted the following limerick:



(I won't spoil the limerick quite yet for those who haven't seen it before.)
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Axxle

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Re: Maths thread.
« Reply #628 on: March 18, 2016, 11:03:15 pm »
0

a dozen a gross and a score
All added to three square root of four
You'll divide it by seven
And then add five elevens
??? I'm at a loss
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liopoil

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Re: Maths thread.
« Reply #629 on: March 18, 2016, 11:06:40 pm »
+5

Without reading Axxle's post or having seen it before:

A dozen, its square and a score
Plus three times the square root of four
divided by seven
plus five times eleven
is 9 squared and not a bit more


Okay, to be fair I do vaguely remember that last line from somewhere.

EDIT: Oh right, 144 is a gross. Also I'm unclear on how many syllables exactly the lines are supposed to have. 8-8-6-6-8 feels most natural to me. Is there supposed to be a specific meter?
« Last Edit: March 18, 2016, 11:15:21 pm by liopoil »
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liopoil

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Re: Maths thread.
« Reply #630 on: March 18, 2016, 11:35:19 pm »
0

Side note, I learned the following variant from people who play Set a lot: when you get to the end of the deck, leave the last card face down. You can figure out what the last card is from the face-up cards, and winning by taking a set with the facedown card is much more stylish.
This is awesome, but what if there is no set using that card :(
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Titandrake

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Re: Maths thread.
« Reply #631 on: March 19, 2016, 01:51:51 am »
+1

Side note, I learned the following variant from people who play Set a lot: when you get to the end of the deck, leave the last card face down. You can figure out what the last card is from the face-up cards, and winning by taking a set with the facedown card is much more stylish.
This is awesome, but what if there is no set using that card :(

Then you are sad, but you can feel good if you're the first to declare no remaining sets, because you need to figure out the last card to make sure.
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Re: Maths thread.
« Reply #632 on: March 19, 2016, 11:22:22 am »
+6

I think it goes like this:

                                 
                                         
         Moat         
                               
                                             

singletee

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Re: Maths thread.
« Reply #633 on: March 19, 2016, 02:56:01 pm »
+4

Without reading Axxle's post or having seen it before:

A dozen, its square and a score
Plus three times the square root of four
divided by seven
plus five times eleven
is 9 squared and not a bit more


Okay, to be fair I do vaguely remember that last line from somewhere.

EDIT: Oh right, 144 is a gross. Also I'm unclear on how many syllables exactly the lines are supposed to have. 8-8-6-6-8 feels most natural to me. Is there supposed to be a specific meter?

What matters is not so much syllables as stressed syllables.

oXooXooX
oXooXooX
oXooXo
oXooXo
oXooXooX

So what we have here is amphibrachic meter. We have two lines of trimeter (count the Xs), followed by two lines of dimeter (rare, I don't even know if that's the right term), and finally one more line of trimeter.

Now let's take another limerick (that I just wrote):

A writer desired a meter
So as to make poetry sweeter
He followed the rules
He had learned back in school
But he could not enamor the reader.

Let's look at the syllables here:

oXooXooXo
oXooXooXo
oXooX
ooXooX
ooXooXooXo

Notice lines 3 and 4 are missing a final unstressed syllable, and lines 4 and 5 start with an extra unstressed syllable. That's ok! The first and last foot don't have to match up exactly, as long as the stressed syllables do. There's a limit to how much you can fool around with things, and you just have to find out how much by trying it. In this case notice how the "missing" syllable is immediately followed by an extra syllable. It kind of reminds me of ionic bonding.

Axxle

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Re: Maths thread.
« Reply #634 on: March 22, 2016, 12:07:55 am »
0

I just mixed up diameter and circumference when trying to measure, and now I have these ridiculously long tassels hanging off my glass float net.  :-[
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Kirian

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Re: Maths thread.
« Reply #635 on: March 22, 2016, 12:47:55 am »
0

It shocks the hell out of me that you can use the prime numbers to calculate pi.  How the hell can those possibly be related!?!?
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sudgy

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Re: Maths thread.
« Reply #636 on: March 22, 2016, 01:10:00 am »
+2

It shocks the hell out of me that you can use the prime numbers to calculate pi.  How the hell can those possibly be related!?!?

Something I have learned from studying math: somehow, everything is related.
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   Quote from: sudgy on June 31, 2011, 11:47:46 pm

silverspawn

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Re: Maths thread.
« Reply #637 on: March 22, 2016, 01:19:44 am »
0

That's interesting. Do you use all prime numbers with diminishing relevance?

Titandrake

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Re: Maths thread.
« Reply #638 on: March 22, 2016, 01:32:00 am »
0

Kirian is probably referring to the result from https://en.wikipedia.org/wiki/Euler_product#Notable_constants



Edit: Looks like an explanation that generalizes requires more math than I know. An easier justification is at http://www.cut-the-knot.org/proofs/AfterEulerProduct.shtml. The idea is that you start with the Leibniz formula, and note that every odd number equal to 1 mod 4 must be the product of an even number of (not necessarily distinct) primes that are 3 mod 4.
« Last Edit: March 22, 2016, 01:38:06 am by Titandrake »
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Axxle

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Re: Maths thread.
« Reply #639 on: March 22, 2016, 01:37:26 am »
+1

I'm guessing Kirian saw this

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Kirian

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Re: Maths thread.
« Reply #640 on: March 22, 2016, 11:28:42 am »
0

I'm guessing Kirian saw this

Yep.

Kirian is probably referring to the result from https://en.wikipedia.org/wiki/Euler_product#Notable_constants



OK, so there are at least two ways of getting pi from the prime numbers.  The linked video uses Eqn 60 at this link:  http://mathworld.wolfram.com/PiFormulas.html

Actually, looking at them, they might be the same thing but written differently; it's really tough to tell.  Both equations are attributed to Euler.

Edit:  It appears the formula at MathWorld (and in the video near 2:00) gives the series 2/pi = (2/3)(6/5)(6/7)(10/11)..., which is both extremely similar to, yet completely different from, the series at that Wiki link.

--------

Of course, the fact that there is a formula that gives you the Nth hexadecimal digit of pi without calculating the previous digits is also a bit mind-blowing.
« Last Edit: March 22, 2016, 12:03:46 pm by Kirian »
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Watno

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Re: Maths thread.
« Reply #641 on: March 22, 2016, 11:40:19 am »
+2

I just mixed up diameter and circumference when trying to measure, and now I have these ridiculously long tassels hanging off my glass float net.  :-[

Recently, I was trying to figure out the size of a fondue pot with some friends (in order to find out wether the ecipe would fit in). Since we already had put some stuff in, we didn't want to fill it with water and measure how much water fit in, so instead we measured the height and the radius, and tried to calulate the volume. Our result didn't seem very realistic though, and it took us quite a while to figure out that the area of a circle wasn't 2*PI*r^2.
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liopoil

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Re: Maths thread.
« Reply #642 on: March 22, 2016, 04:43:51 pm »
0

My favorite is that, roughly speaking, the probability that two arbitrary integers have no common prime factors is 6/(pi^2). The proof involves first showing that it is 1/(zeta(2)) with fancy factoring, then it is the basel problem, which Euler solved with taylor polynomials of sine, which is where pi comes in. Unfortunately I forget all the details.
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Titandrake

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Re: Maths thread.
« Reply #643 on: March 22, 2016, 07:00:08 pm »
+4

Coincidentally, this discussion inspired a blog post where I talked about just that. You can read it at http://www.alexirpan.com/2016/03/22/primes-pi.html

(Shameless promotion is shameless.)
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Titandrake

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Re: Maths thread.
« Reply #644 on: March 31, 2016, 03:44:58 am »
0

Wow, how have I gone this long without using the fullpage LaTeX package?

It does what you think - if you add it as a required package, it will set all margins to 1 inch. My LaTeX-fu on math expressions is getting pretty solid, but the actual formatting is still pretty weak.
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liopoil

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Re: Maths thread.
« Reply #645 on: March 31, 2016, 07:25:35 am »
0

Wow, how have I gone this long without using the fullpage LaTeX package?

It does what you think - if you add it as a required package, it will set all margins to 1 inch. My LaTeX-fu on math expressions is getting pretty solid, but the actual formatting is still pretty weak.
Why is this so desirable? Isn't there a pretty simple way to set margins manually? Why 1-inch?
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Re: Maths thread.
« Reply #646 on: March 31, 2016, 10:14:01 am »
0

Wow, how have I gone this long without using the fullpage LaTeX package?

It does what you think - if you add it as a required package, it will set all margins to 1 inch. My LaTeX-fu on math expressions is getting pretty solid, but the actual formatting is still pretty weak.

I would use the geometry package over fullpage.  fullpage isn't really a recommended package.  (See http://kb.mit.edu/confluence/pages/viewpage.action?pageId=3907057.)

Edit: My default is something like

Code: [Select]
\usepackage{geometry}
\newgeometry{margin=0.75in}

or more compactly

Code: [Select]
\usepackage[margin=0.75in]{geometry}

The advantage here is that you can change geometry in different places; for example, a cover page or final page. [/code]
« Last Edit: March 31, 2016, 10:18:45 am by Witherweaver »
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Titandrake

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Re: Maths thread.
« Reply #647 on: March 31, 2016, 02:05:23 pm »
0

I was freaking out because I had never bothered to learn how to change margin size. Thought it would be really annoying for some reason, when it wasn't.
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Re: Maths thread.
« Reply #648 on: April 05, 2016, 04:04:05 pm »
+3

Came across this after some related Quora discussion:

http://www.maa.org/external_archive/devlin/devlin_06_08.html

I have big issues with this rant.  Mainly, it misses the main point that the definitions are, by and large, arbitrary.  In some cases, some definitions simplify things, but really the definition is part of the math, and it can be personal preference on what to take as your axiom and what to take as your conclusions.

The basic claim is that multiplication is not defined through addition.  But it can be.  Roughly: suppose you know what binary  x+y  means for any real numbers x and y.  Association lets you define n-ary '+'.  Then you define

m*x = x*m = sum(i=1..m, x) = x + x + ... + x (m times)

for any integer m and real number x. (Note that commutivity is by definition.)

Now, given some integer m!=0, we may define  a real number m^{-1} = (1/m) by the real number that satisfies

m*(1/m) = (1/m) * m = 1. 

Now, suppose p and q are rational numbers, p=n/m, q=k/l, where n,m,k,l are integers (m, l nonzero).  Then we define

p*q = (n*k)*(1/(m*l)).

This is perfectly defined as n*k is an integer and 1/(m*l) is a real number. 

Now, what is x*y for any real numbers x and y?  Well, we take a sequence of rational numbers x_n -> x and y_n -> y.  (This could actually be done relative to any topology; the normal topology on the real numbers coincides with the usual Euclidean distance.  You could argue that we need to know how to multiply real numbers to measure convergence, but you could do it on the level of topology.  Or you could simply take the Euclidean norm as given, and it's through that norm that we define multiplication.  Though maybe if you take any topology and require multiplication to be continuous wrt that topology you would get the same thing.. I'm not really sure.)    Then  x*y is defined to be the limit of the sequence x_n*y_n.  A small amount of work shows it's well defined. 

So we started with addition of reals and some notion of a topology and ended up with multiplication of reals.  There is nothing to say that this definition of multiplication is not "really" multiplication---it surely is.  Moreover, in statements like:

Quote
And telling them that multiplication is repeated addition definitely requires undoing later.

How much later? As soon as the child progresses from whole-number multiplication to multiplication by fractions (or arbitrary real numbers). At that point, you have to tell a different story.

"Oh, so multiplication of fractions is a DIFFERENT kind of multiplication, is it?" a bright kid will say, wondering how many more times you are going to switch the rules.

The author is missing a big issue.  There ARE different kinds of multiplication.  Just because an algebra A is a subset of some B and you have a binary operation to make B an algebra doesn't mean A with the induced operation of B is the same thing as B.  In fact, A with induced operation may not even be an algebra.  In other words, even if I take a "multiplication" with the real numbers as axiomatic, that doesn't necessarily mean you should have the same notion of multiplication on the integers. Sure, it turns out the natural definitions coincide, but even so (Z,*), (Q,*) and (R,*) are entirely different objects.  Moreover, if you consider, for example, the set X = {0,1,2,3,4,5}, this is certainly a subset of real numbers.  You can even induce R's multiplication on X... except you get into trouble.  Because * does not take X into X; it takes X into some larger set.  In fact, the thing we may be interested in is not R's * but a different kind of * that maps X into X, some special *_X.  Maybe x(*X)y = (x*y) mod 6.  In fact, there are unlimited numbers of multiplications.

Though, notably, in my example multiplication by fractions is not 'changing any rules'.  It is operating on the same rule that multiplication by an integer is. 

Also, this:

Quote
Why not say that there are (at least) two basic things you can do to numbers: you can add them and you can multiply them. (I am discounting subtraction and division here, since they are simply the inverses to addition and multiplication, and thus not "basic" operations. This does not mean that teaching them is not difficult; it is.) Adding and multiplying are just things you do to numbers - they come with the package. We include them because there are lots of useful things we can do when we can add and multiply numbers. For example, adding numbers tells you how many things (or parts of things) you have when you combine collections. Multiplication is useful if you want to know the result of scaling some quantity.

is arbitrary.  You could just as easily say that there are two things you can do to numbers: you can subtract them and you can divide them.  I discount addition and multiplication , since these are just inverses.  It may be more natural to define addition and multiplication as axiomatic (though as I said, you don't 'need' to define multiplication; you can define topology or metrics instead), probably because of how we count, but it doesn't *have* to be done this way; it's a choice.
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Re: Maths thread.
« Reply #649 on: April 05, 2016, 05:19:30 pm »
0

It shocks the hell out of me that you can use the prime numbers to calculate pi.  How the hell can those possibly be related!?!?

You can also use, say, the odd numbers (ok, also 4)
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