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November 4[edit]

Pi[edit]

Will there ever be an end of pi? JCI (talk) 01:59, 4 November 2008 (UTC)[reply]

Any terminating decimal is a rational number, and pi is irrational, so no. The decimal expansion of pi (3.1415....) never terminates. -GTBacchus^(talk) 02:07, 4 November 2008 (UTC)[reply]

That's really a somewhat silly way of asking the question. Usually it means "will there be an end to the decimal expansion of π, and it neglect the question of whether using a decimal representation as opposed to some other for π is somehow sacred. That there is no such end would be true even if π were rational and equal to 22/7. But then the expansion would end if it were in base 7 rather than base 10. That it will not end regardless of which base is chosen is not the essence of what it means to say π is an irrational number, but it is a corollary of the fact that π is an irrational number. See proof that π is irrational. Michael Hardy (talk) 03:12, 4 November 2008 (UTC)[reply]

Pi is only irrational when you're not working in base Pi. -mattbuck (Talk) 03:21, 4 November 2008 (UTC)[reply]

What? Irrationality is a property of numbers, not of number representations, and pi is always irrational. That doesn't mean it can't have a compact representation. Thus in base pi it is represented by ~~"1"~~ "10" and in ordinary mathematical discourse it is represented by "π", which certainly doesn't go on forever. Algebraist 10:23, 4 November 2008 (UTC)[reply]

Of course in base pi, pi is 10. 205.206.170.1 (talk) 22:17, 4 November 2008 (UTC)[reply]

That doesn't make it rational. A rational number is a ratio of two integers. Pi is not a ratio of two integers. Michael Hardy (talk) 23:28, 6 November 2008 (UTC)[reply]

No-one said it did. Algebraist 23:30, 6 November 2008 (UTC)[reply]

Yes, I think he meant that even if the never-ending expansion gives Pi a sort of unfinished or somehow random character, this phenomenon is just linked to the kind of representation, while as a real number Pi is determined and well defined, although irrational.--131.114.72.215 (talk) 11:37, 4 November 2008 (UTC)[reply]

I read this as Who will rid me of this troublesome number? :) In Contact (novel) by Carl Sagan there is the idea that at a suitable level in setting up the universe one could arrange the digits of pi, for instance to put pictures into it. Dmcq (talk) 14:04, 4 November 2008 (UTC)[reply]

Unfortunately, pi is a mathematical constant, not a physical one, so that's complete nonsense. It's a fun idea, though. --Tango (talk) 14:43, 4 November 2008 (UTC)[reply]

Well, depends. If you believe in a God who created mathematics, and even logic, then even though pi's decimal expansion might appear to us to be a priori necessary, it might be contingent for God. Of course this is a discussion that never really goes anywhere, because while in some sense you can't exclude that God might be sovereign even over logic, it doesn't leave us with much in the way of tools for figuring out "so, in that case, what?". --Trovatore (talk) 22:25, 4 November 2008 (UTC)[reply]

People have spent their lives debating whether or not God can change the value of pi... what a waste of a life... --Tango (talk) 22:34, 4 November 2008 (UTC)[reply]

I think it's a useful exercise to help clarify the concepts. I suppose my take on it would be that if God can change the value of π, then he can also change the value of 2. (Which, by the way, you can do, in older versions of Fortran :-). --Trovatore (talk) 04:30, 5 November 2008 (UTC)[reply]

"Not only I can change the value of 2, but I did it several times indeed and I am now quite bored to do it again and again to convince you atheistic mathematicians. The only things I can't change are your egg-heads!" (God, private communication)(talk) 09:44, 5 November 2008 (UTC)[reply]

Historically, pi is the ratio of the circumference to the diameter of a circle. Distances are still physical concepts, so I suppose God could create a version of the universe where the natural metric for distances was something other than Euclidean space, and hence give rise to a different version of the historically motivated pi. I'm not sure that you could do it in such a way that the effect would be noticeable and the universe would still be inhabitable by life as we know it though. Dragons flight (talk) 15:52, 4 November 2008 (UTC)[reply]

A significantly curved universe, perhaps? Pi wouldn't be constant then, though - the only way to find a meaningful constant would be to take the limit of pi as the radius goes to zero, which would give you the value of pi we know. And, if we're talking about positive curvature, to get a value sufficiently different to be noticed on every day scales would require a very small universe. I have no idea what would happen in the negative curvature case... --Tango (talk) 18:17, 5 November 2008 (UTC)[reply]

Another idea I've seen for getting rid of pi is to have a symbol for 2π instead, the symbol I saw was like π but had three feet say 'm', not sure what the called it. Then the circumference of a circle is mr and the area is ½mr² Dmcq (talk) 14:57, 4 November 2008 (UTC)[reply]

Sure, you can do that if you want, but what's the benefit? Your 'm' is still irrational. --Tango (talk) 15:39, 4 November 2008 (UTC)[reply]

They seemed to think it was a much more rational ;) No extra 2 in things like the normal distribution and the ½ is what one gets when getting the area integrating r. Dmcq (talk) 16:01, 4 November 2008 (UTC)[reply]

Well yes, it is true that pi is multiplied by 2 more often than not (other than areas, I'm struggling to think of an example where it isn't!). Your m could simply be defined as the circumference of the unit circle, or equivalently "the ratio of the circumference of a circle to its radius" which is more elegant since we usually work with radii rather than diameters. Oh well, it's far too late to change the convention now! --Tango (talk) 18:53, 4 November 2008 (UTC)[reply]

I have seen the argument that if we should demonstrate intelligence with radio signals to possible aliens, or look for intelligence, then binary digits of 2π might be more universally identifiable than π. PrimeHunter (talk) 19:03, 4 November 2008 (UTC)[reply]

That's a good one, they'd have to be pretty dim :) LOL Dmcq (talk) 19:16, 4 November 2008 (UTC)[reply]

I don't get the joke... That sounded like a pretty well thought out idea to me... --Tango (talk) 19:20, 4 November 2008 (UTC)[reply]

The binary expansion of 2π is somewhat similar to that of π. Algebraist 19:23, 4 November 2008 (UTC)[reply]

Oh... --Tango (talk) 19:33, 4 November 2008 (UTC)[reply]

My type of joke, non-obvious and very silly. As to the original question I think a poem by Martin Gardner covers it:

Pi goes on and on and on...

And e is just as cursed.

I wonder: Which is larger

When their digits are reversed?

Dmcq (talk) 09:26, 5 November 2008 (UTC)[reply]

"it is true that pi is multiplied by 2 more often than not (other than areas, I'm struggling to think of an example where it isn't!)" -Tango, above

The solution to the Basel problem is

{\frac {\pi ^{2}}{6}}

, and the solution to Buffon's needle problem is

{\frac {2}{\pi }}

, to name two examples. -GTBacchus^(talk) 15:23, 5 November 2008 (UTC)[reply]

In fact,

\pi

is by definition one half of

2\pi

, which is therefore a somehow more natural and primitive constant.--PMajer (talk) 17:48, 5 November 2008 (UTC)[reply]

Directed sets[edit]

Hi. This is not a homework question.

Is there some reasonable example of a directed set that is not a poset? -GTBacchus^(talk) 02:06, 4 November 2008 (UTC)[reply]

What do you mean by directed set? Do you mean in the sense of a net (topology)? In that situation, the antisymmetric relation property is not always assumed. So for example you could take the set to be {a,b} with both a <= b and b <= a. — Carl (CBM · talk) 03:38, 4 November 2008 (UTC)[reply]

The third example in directed set is very natural (it's what you use when you take the limit of a function) and is not a poset. Algebraist 10:20, 4 November 2008 (UTC)[reply]

Ah. I don't know how I missed that. Thank you. -GTBacchus^(talk) 15:16, 5 November 2008 (UTC)[reply]

Regular Polygon[edit]

Hello. A regular polygon of n sides is inscribed in a circle of radius r. The area of the polygon is $2r^{2}{\sqrt {2}}$ . How many sides does it have? Let C be the centre of the circle, A and B be adjacent vertices in which the circumference intersects the polygon, and h be the height of any $\triangle ABC$ .
If $C={\frac {360^{\circ }}{n}}$ , $A=B=90-{\frac {180}{n}}$ , and $c=r{\sqrt {2}}\,{\sqrt {1-\cos {\frac {360}{n}}}}$ , then $16=n^{2}\left(1-\cos {\frac {360}{n}}\right)\cos ^{2}{\frac {180}{n}}$ . How can I solve for n? Thanks in advance. --Mayfare (talk) 07:37, 4 November 2008 (UTC)[reply]

By plugging in values for n. It has to be a not too large integer, and

n^{2}\left(1-\cos {\frac {360}{n}}\right)\cos ^{2}{\frac {180}{n}}

is monotonic, so a direct search is the easiest way. Dragons flight (talk) 08:12, 4 November 2008 (UTC)[reply]

For simplicity set r = 1. The equation for the double area is

4{\sqrt {2}}=n\sin {\frac {2\pi }{n}}

or

{\frac {\sin C}{C}}={\frac {2{\sqrt {2}}}{\pi }}

Such transcendental equations have no exact algebraic solution but can be solved numerically to some approximation. One shortcut is to truncate the Taylor series to get an approximate algebraic equation

1-{\frac {C^{2}}{6}}+{\frac {C^{4}}{120}}\approx {\frac {2{\sqrt {2}}}{\pi }}

so x = C² is the small solution to the quadratic equation

x^{2}-20x+120-{\frac {240{\sqrt {2}}}{\pi }}\approx 0

so

C^{2}\approx {10-{\sqrt {100-(120-{\frac {240{\sqrt {2}}}{\pi }}}})}

=2\left(5-{\sqrt {5\left({\frac {12{\sqrt {2}}}{\pi }}-1\right)}}\right)

and

n={\frac {2\pi }{C}}\approx {\frac {\pi {\sqrt {2}}}{\sqrt {5-{\sqrt {5\left({\frac {12{\sqrt {2}}}{\pi }}-1\right)}}}}}\approx 7.99809

So n = 8 is a good guess to check. Bo Jacoby (talk) 10:10, 4 November 2008 (UTC).[reply]

Nice. You could even truncate the Taylor series to two terms and get:

1-{\frac {C^{2}}{6}}\approx {\frac {2{\sqrt {2}}}{\pi }}

C^{2}\approx {\frac {6}{\pi }}\,(\pi -2{\sqrt {2}})\approx 0.598

so:

n={\frac {2\pi }{C}}\approx 8.125

CiaPan (talk) 12:00, 4 November 2008 (UTC)[reply]

Fraction Needed[edit]

What's the fraction for 2.571428 (decimal repeats afterwards)? 203.188.92.71 (talk) 12:34, 4 November 2008 (UTC)[reply]

We will not do your homework for you. Especially if you have given no indication you have attempted to solve the question, and especially especially if you haven't said what part of the decimal repeats. Algebraist 12:36, 4 November 2008 (UTC)[reply]

I've tried. It's part of a slope formula, but my calculator give decimals instead of fractions, but fractions seem to be prefered for slope. (I do, and the teacher also does.) However, I just figured out that I got the x and y wrong. The correct decimal is now 0.38. The 8 repeats. As far as I know, it's probably something over 9, possibly something divisive by 8 (I'm probably wrong.). All of the numbers I've tried don't work. Hope that helps. 203.188.92.71 (talk) 12:50, 4 November 2008 (UTC)[reply]

Presuming you have access to a calculator, assume that the questionner hasn't set a hard question and try the simple fractions. Enter .571428*2=, then .571428*3= etc until you get an answer that is almost a whole number. Then investigate what the fraction indicated would be - does it repeat appropriately? -- SGBailey (talk) 12:54, 4 November 2008 (UTC)[reply]

See my reply above (the 3rd one); it was the wrong decimal, the right one is in that reply. 203.188.92.71 (talk) 12:57, 4 November 2008 (UTC)[reply]

You have (at least) three obvious options: throw away your calculator and work out the fractional answer for yourself, get a better calculator that will give answers in the format you desire, or compute the sum of the (almost) geometric series

{\frac {3}{10}}+{\frac {8}{100}}+{\frac {8}{1000}}+{\frac {8}{10000}}+\cdots

. Algebraist 13:09, 4 November 2008 (UTC)[reply]

If you expect it is 'something over 9' why don't you just try to multiple 0.388888888×9 to see what the answer might be...? --CiaPan (talk) 07:50, 5 November 2008 (UTC)[reply]

You are being too hard on the poor kid. All the kid have to do is to read the article on Continued fraction and convert 0.38888888888... into continued fraction and convert it into an ordinary fraction. Simple. 122.107.228.237 (talk) 10:00, 5 November 2008 (UTC)[reply]

Like this

Find the continued fraction for 3.245 (3 49/200)
$3\,$	$3.245\ (3\ 49/200)-3\,$	$=0.245\ (49/200)\,$	$1/0.245\ (200/49)\,$	$=4.082\ (4\ 4/49)\,$
$4\,$	$4.082\ (4\ 4/49)-4\,$	$=0.082\ (4/49)\,$	$1/0.082\ (49/4)\,$	$=12.250\ (12\ 1/4)\,$
$12\,$	$12.250\ (12\ 1/4)-12\,$	$=0.250\ (1/4)\,$	$1/0.250\ (4/1)\,$	$=4.000\,$
$4\,$	$4.000-4\,$	$=0.000\,$	STOP
continued fraction form for 3.245 (3 49/200) is [3; 4, 12, 4]
$3.245\ (3\ 49/200)=3+{\cfrac {1}{4+{\cfrac {1}{12+{\cfrac {1}{4}}}}}}$

122.107.228.237 (talk) 10:08, 5 November 2008 (UTC)[reply]

IMHO that's not a good idea. It requires (possibly) high-precision calculations to compute 1/0.388888... = 2.571428... The Algebraist's method with explicit geometric series is much, much better. --CiaPan (talk) 14:27, 5 November 2008 (UTC)[reply]

Yes, I'm a great fan of the continued fraction method for approximations, but if you know the period of a repeating decimal the approach suggested by the series is better. —Tamfang (talk) 17:14, 5 November 2008 (UTC)[reply]

I may be missing a comment where someone suggested this earlier, but if

x=0.388888\ldots

, then

10x=3.888888\ldots

, right? So, based on these, how can you figure out what

9x

is? And once you have

9x

, how can you solve for

x

? —Bkell (talk) 06:33, 6 November 2008 (UTC)[reply]

Notice that

10x=3.5+x\,\Rightarrow \,20x=7+2x\,

. Gandalf61 (talk) 10:07, 6 November 2008 (UTC)[reply]

By the way, the number 142857 is very well known to every old fox. I bet I can tell you immediately how much it is multiplied say by 1, 2, 3, 4, 5, 6 or even 7.--PMajer (talk) 13:24, 6 November 2008 (UTC)[reply]

I suppose it sounds horribly random, but you find those digits in the first decimals... Anyway, the articles are somewhat hard to understand with all the jargon, unless I'm mistaken. 203.188.92.71 (talk) 13:48, 6 November 2008 (UTC)[reply]

Not random at all, PMajer was dead serious: see 142857 (number). The geometric series and continued fraction articles are a bit complicated, yes, but certainly not needed here. As suggested, if you suspect it's something over 9, then why not try multiplying it by 9 to see what the numerator must be then? Or alternatively, learn how to let your calculator convert it to a fraction for you (if it's only slightly advanced, there's bound to be a way). -- Jao (talk) 14:21, 6 November 2008 (UTC)[reply]

Actually, the 9 one doesn't really work. Well then, anyway to do that on a graphic calculator - a TI-84? 203.188.92.71 (talk) 22:56, 6 November 2008 (UTC)[reply]

Multiplying by 9 does work, actually. There's a command to convert decimals into fractions on my TI-85, so I would be surprized if it hadn't made it to the TI-84. Why not try reading the manual? Algebraist 10:57, 7 November 2008 (UTC)[reply]

Yes, there is - after getting the number, press MATH, then use Frac, to convert to a fraction. --WikiSlasher (talk) 11:12, 7 November 2008 (UTC)[reply]

why the need[edit]

why the need for this long and complex proof?

Isn't it enough to say "If there were only finite primes, you could multiply them all together, add one, and get a finite number that, being relative prime to all of them, must have a prime factor not in 'all primes' -- an absurdity."? —Preceding unsigned comment added by 82.124.209.97 (talk) 17:36, 4 November 2008 (UTC)[reply]

That is essentially Euclid's proof, which is still the standard one. Furstenberg's proof is considered interesting in its own right, not because of the result it proves. Algebraist 18:13, 4 November 2008 (UTC)[reply]

Except that, contrary to what is often stated, sometimes by eminent number theorists, Euclid's proof was not phrased as a proof by contradiction. He just said if you start with any finite set of primes (not assumed to contain all primes), multiply them, add one, and factor the result, you get new primes. Also, there are only finite primes: every prime is finite. Presumably the poster meant "finitely many primes". Michael Hardy (talk) 23:23, 6 November 2008 (UTC)[reply]

The (topological) proof is also more interesting (to me). I am not really a fan of number theory so maybe a number theorist might think differently.

Topology Expert (talk)

No need. The idea politicians always have of encouraging mathematics because it is useful makes me hiss and grit my teeth and want to kick them hard in the ankles. This proof is the mathematical equivalent of going along to a music recital. Dmcq (talk) 09:44, 5 November 2008 (UTC)[reply]

Only mathematicians appreciate mathematics for its inherent beauty. Everyone else appreciates it because it's useful, and that's why they are willing to pay mathematicians to do what we love. Don't complain! --Tango (talk) 16:00, 5 November 2008 (UTC)[reply]

Different proofs in mathematics are interesting (from latin: interesse) because they show hidden connections between different areas. Sometimes, using new techiques to produce a new proof of an old result is just a way to test the power of a theory. The point is that maths (like all science) is not a definitive and fixed matter, and facing a new proof our question is: what's behind? what's next? what if I used it in an open problem? If it was just a matter of virtuosity, then I could agree with Dmcq (although there are aesthetic reasons for all that, for those who appreciate)--PMajer (talk) 16:10, 5 November 2008 (UTC)[reply]

Characteristic polynomial of product of two matrices[edit]

Does someone have a proof (or a reference) for the following claim (in Characteristic polynomial):

"If A is an m*n-matrix and B an n*m-matrix (with coefficients in the same field or ring), then AB and BA have the same characteristic polynomial, up to a power X^(n-m) of the indeterminate"?

For m=n, a proof is given (AB and BA are similar for A or B invertible), but I don't see how it can be generalised to non-square matrices. --Roentgenium111 (talk) 20:49, 4 November 2008 (UTC)[reply]

I have never thought about this before, but you can do it with just a calculation. Suppose n > m, A is n x m and B is m x n. Let i = n - m. Let

0^{a,b}

represent an a x b matrix of 0s. Compute:

{\begin{bmatrix}A&0^{n,i}\\\end{bmatrix}}{\begin{bmatrix}B\\0^{i,n}\end{bmatrix}}=AB,

where both matrices on the left are square; and

{\begin{bmatrix}B\\0^{i,n}\end{bmatrix}}{\begin{bmatrix}A&0^{n,i}\\\end{bmatrix}}={\begin{bmatrix}BA&0^{m,i}\\0^{i,m}&0^{i,i}\end{bmatrix}}

Then use the fact that the characteristic polynomials of NM and MN are always the same when M and N are square matrices of the same size. My guess is you can also prove this in some more algebraic way, and I'd be interested if anyone has a proof of that. — Carl (CBM · talk) 23:22, 4 November 2008 (UTC)[reply]

Actually, there is an easy way to kill two birds with one stone. If A,B are rectangular matrices and x is nonzero, then I−BA/x is invertible if and only if I−AB/x is. Explicitly,

\left(I-{\frac {BA}{x}}\right)^{-1}=I+{\frac {1}{x}}B\left(I-{\frac {AB}{x}}\right)^{-1}A

.

Cheers, siℓℓy rabbit (talk) 00:01, 5 November 2008 (UTC)[reply]

And a mnemonic for this identity is Neumann series. Notice also that for any $x\neq 0$ , $\textstyle A$ and $\textstyle B$ give by restriction the isomorphisms $\textstyle A:ker(BA-x)\to ker(AB-x)$ and $\textstyle B:ker(AB-x)\to ker(BA-x)$ (they are inverse of each other up to a nonzero costant). Thus any $x\neq 0$ is an eigenvalue of $\textstyle BA$ if and only if it is an eigenvalue of $AB$ , the geometric multiplicities being the same. The algebraic multeplicities are also the same: do the same with $\textstyle ker(BA-x)^{n}$ and $\textstyle ker(AB-x)^{n}$ . In the invertible case, one can also observe that $\textstyle MN$ and $\textstyle NM$ are conjugated : $\textstyle MN=N^{-1}(NM)N$ , so their spectral theories are the same. --PMajer (talk) 12:44, 5 November 2008 (UTC)[reply]

Thank you all for your answers. Actually, I figured out Carl's way in the meantime as well. --Roentgenium111 (talk) 13:16, 5 November 2008 (UTC)[reply]