Wikipedia:Reference desk/Archives/Mathematics/2008 August 9

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August 9

NonStandard Number Theory

If a nonstandard integer is divisible by infinitely many finite primes, must it also be divisible by infinitely many infinite primes? Black Carrot (talk) 02:06, 9 August 2008 (UTC)

Yes; this is a consequence of the overspill principle in arithmetic. I don't think there is a Wikipedia article on that subject. — Carl (CBM · talk) 02:12, 9 August 2008 (UTC)

I didn't know that name for it. In rough terms, though, here's the reason. First note that if there were a nonstandard integer divisible by infinitely many standard primes but only finitely many nonstandard ones, you could divide out by all the nonstandard ones, and you'd have one divisible by infinitely many standard primes but no nonstandard ones.

But then you'd have a way of defining, within the nonstandard model, which of its integers were finite (they'd be just the ones less than some prime divisor of your special number). This is definability with a parameter, of course, but that doesn't matter; it still means the model could tell which integers were finite and which weren't, and that's impossible. If the model knew some of its integers were infinite, it would realize that they weren't really integers. --Trovatore (talk) 02:29, 9 August 2008 (UTC)

Say the number n is divisible by infinitely many standard primes. Rather than dividing out the nonstandard primes (which is a good idea), you could also use the idea that there can't be a least nonstandard prime dividing n, because you could use that number and n to define the standard numbers. Thus there will be an infinite decreasing sequence of nonstandard primes dividing n. Moreover, given any nonstandard number m whatsoever, there must be a nonstandard prime p < m dividing n, or again you could use n and m to define the standard naturals. The general statement of that last fact is called "overspill" in the context of models of Peano arithmetic. — Carl (CBM · talk) 12:12, 9 August 2008 (UTC)

Another proof of $0.999\dots$

Is this is a good proof for 0.999?

${\begin{aligned}0.999\dots +0.999\dots &{}=\quad ?\\0.999\dots +0.999\dots &{}=0.888\dots +{\mbox{carry}}\\0.999\dots +0.999\dots &{}=0.888\dots +1.111\dots \\0.999\dots +0.999\dots &{}=1.999\dots \\0.999\dots +0.999\dots &{}=1.000\dots +0.999\dots \\0.999\dots &{}=1.000\dots \\\end{aligned}}$

Ohanian (talk) 09:10, 9 August 2008 (UTC)

It's only a proof if you accept that the arithmetical operations work that way. The idea of a "carry" is troubling when there is no rightmost digit. Irrational numbers can't be added with the right-to-left algorithm learned in school. The way the addition is calculated depends on how you actually define .999... - if you define it as a supremum of the set {.9.,99,.999,...}, then the proof would go like this:

${\begin{aligned}0.999\dots &{}=\sup\{.9,.99,.999,\ldots \}\\0.999\dots +0.999\dots &{}=\sup\{1.8,1.98,1.998,\ldots \}\\0.999\dots +0.999\dots &{}=1.999\dots \\0.999\dots +0.999\dots &{}=1.000\dots +0.999\dots \\0.999\dots &{}=1.000\dots \\\end{aligned}}$

—Preceding unsigned comment added by CBM (talk • contribs) 12:02, 9 August 2008

If you're asking whether it's good enough for inclusion in the article, then no. Unless you can find reliable sources saying this works, it isn't verifiable. Paragon12321 (talk) 16:30, 9 August 2008 (UTC)

Iain Evans constant

What is it? Kittybrewster ☎ 11:12, 9 August 2008 (UTC)

I deleted the mention since it gets no Google hits and isn't mentioned on his page. Apparently its value was 47. It could be a facetious reference to a number he made up for rhetorical purposes, like Joseph McCarthy and his 57 known Communists. -- BenRG (talk) 11:45, 9 August 2008 (UTC)

Momentum

I'm in the middle of reading a book on Particle Physics and I have just come to a chapter concerning momentum. It says that p=mv does not hold for velocities that are a significant fraction of the speed of light and that the formula for relativistic momentum is $p={\frac {mv}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}$. However it is not totally clear on the applications of this formula; is it useful in situations where an object's velocity is not a significant fraction of c?

Also does p=mv ever give the true momentum of an object? I'm asking for a straight yes or no and not whether or not it is an extremely good approximation to the momentum. Thanks. 92.4.189.176 (talk) 19:30, 9 August 2008 (UTC)

The formula for relativistic momentum holds at all speeds, both those which are a significant fraction of the speed of light and those which are not. The formula p=mv gives the exact answer if v = 0, but at no other speeds. (It is an extremely good approximation for low v, but only exactly correct at v = 0).

To understand that, compare the true formula $p_{true}={\frac {mv}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}$ against the approximation $p_{approx}=mv$. The two formula agree exactly when

$mv={\frac {mv}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}$

$1={\sqrt {1-{\frac {v^{2}}{c^{2}}}}}$

$0={\frac {v^{2}}{c^{2}}}$

which is satisfied if and only if v = 0. Eric. 82.139.83.173 (talk) 20:23, 9 August 2008 (UTC)

The variable m may refer either to rest mass or to relativistic mass. If it refers to relativistic mass then p = mv exactly. The modern convention is to use rest mass exclusively, but it's still common to see relativistic mass in textbooks. If m refers to rest mass then $p={\frac {mv}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}$ exactly, and since this is the mathematics desk I'll point out that you can Taylor expand the right hand side to obtain

$p=mv\left(1+{\frac {1}{2}}\left({\frac {v}{c}}\right)^{2}+{\frac {3}{8}}\left({\frac {v}{c}}\right)^{4}+{\frac {5}{16}}\left({\frac {v}{c}}\right)^{6}+\cdots \right)$

which should give you an idea of how accurate p=mv is when v/c is small. -- BenRG (talk) 20:40, 9 August 2008 (UTC)

Labour-theory value of human life

In Canada, what is the value of a newly minted adult if calculated based on the value of materials and labour spent to produce one and the risks incurred along the way? NeonMerlin 23:46, 9 August 2008 (UTC)

Isn't that just another way of asking "What's the cost of bringing up a child in Canada?"? It's not really a maths questions, you'll just have to look up the statistics. Try google. --Tango (talk) 00:14, 10 August 2008 (UTC)

A quick search gives me $154,000 for goods and services, but that doesn't include the parents' own labour, or government services paid for through taxes. NeonMerlin 00:24, 10 August 2008 (UTC)

Your question implies a fallacy. The value of something is not the cost of what goes into the production of the thing, but rather what people are willing to give up to obtain the thing. For example, you could spend four hours making an apple pie and I could spend four hours making a mud pie. That doesn't mean that the value of the mud pie is the same as the value of the apple pie. Wikiant (talk) 00:40, 10 August 2008 (UTC)

I agree it's not a very theory, but that is what the Labor theory of value says. --Tango (talk) 04:19, 10 August 2008 (UTC)