Wikipedia:Reference desk/Archives/Mathematics/2006 August 1

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erosion prediction

At current erosion rates, what is the predicted time it would take for current land masses and continets to erode to see level?ĆΓїṢ–ḌĘḶőáĆḥ

That is like asking "what is your life expectancy if you stopped eating ?". The answer is (probably) not very long on a geological time scale, but erosion is not the only process that is going on - see orogeny, isostasy and plate tectonics for more details. Gandalf61 15:09, 1 August 2006 (UTC)[reply]

I would expect the land building processes and erosion processes to be more or less in equilibrium, meaning the average height above sea level would always be about the same. I might be wrong, but would want to see some evidence that I was before I would believe it. StuRat 04:08, 2 August 2006 (UTC)[reply]

BTW, this Q belongs on the Science Ref Desk. StuRat 04:10, 2 August 2006 (UTC)[reply]

Math Constants

Does grouping "e, π, phi, and Ω have any significance? What is the significance of the four basic constants? (e, pi, phi, and 2)--Tmchk 19:25, 1 August 2006 (UTC)[reply]

By Ω, do you mean the omega constant, or something else? I don't think grouping them has any significance. Also note that the golden ratio φ is not a fundamental constant: it's equal to

{1+{\sqrt {5}}} \over 2

. —Keenan Pepper 23:35, 1 August 2006 (UTC)[reply]

Also note that there are plenty of other fundamental constants unrelated to these: the Euler-Mascheroni constant γ, the Feigenbaum constants... see Category:Mathematical constants for many more. —Keenan Pepper 23:38, 1 August 2006 (UTC)[reply]

Is there a group of four constants that does have significance? Tmchk 00:18, 4 August 2006 (UTC)[reply]

What would that even mean? Melchoir 00:27, 4 August 2006 (UTC)[reply]

In a math competition I was asked, "What are the four _______ constants. The problem is I can't remember the adjective in the "blank". I was told the answear was e, π, Ψ, and ___. Can anyone fill in the blanks?Tmchk 01:35, 4 August 2006 (UTC)[reply]

Uh... "most popular", perhaps? What kind of math competition was this? Melchoir 21:44, 4 August 2006 (UTC)[reply]

~~Quizzle, in Mu Alpha Theta. I'm starting to think I'm just confused or mistaken.~~ Tmchk 23:00, 5 August 2006 (UTC)[reply]

I withdraw my questions as they are flawed and pointless. Tmchk 18:40, 7 August 2006 (UTC)[reply]

Is π natural?

Is there a "natural" reason for using π as a constant in mathematics, rather than one of its multiples, such as 2π or π/2?

("Natural" is opposed to "historical" here: a natural reason would be something like "taking π as our constant simplifies a bunch of important formulae"; a historical reason would be something like "Archimedes chose to study this value and everyone else has followed him in doing so".) Gdr 20:11, 1 August 2006 (UTC)[reply]

I think John Baez wrote somewhere (without evidence) that historically, pi arose from transporting heavy objects on rolling logs, an application where the diameter is more important than the radius. If you ask me, now that humanity has axles, complex analysis, and physics at our disposal, there's no excuse for not switching to 2pi, which is more "natural", and giving it its own letter. Unfortunately, when I ask mathematicians and physicists about this, only about half agree with me. The rest are, of course, lemmings the lot of 'em! Melchoir 20:40, 1 August 2006 (UTC)[reply]

I found a relevant e-mail by Baez at [1]: was this what you were referring to? Baez, like you, thinks 2π is more natural than π. What are the reasons for thinking this? Gdr 21:01, 1 August 2006 (UTC) P.S. I note that Baez's claims about Egyptian metrology are not confirmed by our article on Ancient Egyptian units of measurement, but maybe our article is not very trustworthy.[reply]

Anyone with a keen sense of aesthetics can see that π was properly chosen, to produce the most beautiful equation in mathematics, e^iπ+1 = 0. ;-) --KSmrq^T 21:26, 1 August 2006 (UTC)[reply]

But e^2πi = 1 is not much worse. Grue 21:46, 1 August 2006 (UTC)[reply]

You may wink, but I'm gonna rant all the same. That whole "+1=0" thing is just an excuse to avoid a minus sign and needlessly complicate an already inelegant formula. Whose idea was it to start writing the identity that way, anyway? Hell, I can add negative one and one all fancy-like, too:

{\frac {72}{\pi ^{2}}}\sum _{a,b=1}^{\infty }{\frac {a}{b^{2}}}+{\frac {\mu _{0}\epsilon _{0}}{c^{2}}}={\frac {1}{\infty ^{2}}}

Look, ma, I unified trigonometry, calculus, electricity, relativity, and the first three Latin letters, using addition, division by squares, and infinity! Too bad what I wrote is useless! Grrrr..... Melchoir 22:49, 1 August 2006 (UTC)[reply]

Or you can view it as the case n = 2 of the more general identity Σ_{k = 0..n-1}e^2kπi/n = 0. --Lambiam ^Talk 02:13, 2 August 2006 (UTC)[reply]

Okay, that's actually a good one. I'd forgotten about it; you may want to add it to Euler's identity (But notice that it's naturally exp(2pi i/2), not exp(pi i)...) Melchoir 02:24, 2 August 2006 (UTC)[reply]

So done. --Lambiam ^Talk 03:39, 2 August 2006 (UTC)[reply]

To Gdr: 2pi is the length of the unit circle. It shows up everywhere. Anyway, give me an application for pi and I'll (probably) show you how it's better with 2pi. Melchoir 22:52, 1 August 2006 (UTC)[reply]

Well, the radian is the fundamental unit of angle. So, is a half-circle or full-circle the more fundamental angle? EdC 23:24, 1 August 2006 (UTC)[reply]

Right, you can't say "half-circle" without the "half"; pi is just half of something else. Melchoir 23:27, 1 August 2006 (UTC)[reply]

I call for immediate reform! Let's choose a new Greek letter to stand for 2π and start using it instead of π. How about κ for κυκλος? —Keenan Pepper 23:31, 1 August 2006 (UTC)[reply]

Hmm... I've always sucked at writing kappa by hand. Maybe something easier? Melchoir 23:59, 1 August 2006 (UTC)[reply]

π is better than 2π because derivatives of trig functions in radians come out nicer. The derivative of $sinrad(x)\,$ (sine in radians) is obviously $cosrad(x)\,$ (cosine in radians). However, this doesn't work for degrees or "diameters". The sine function for degrees, $sindeg(x)=sinrad(x\pi )\,$ and the sine function for "diameters", $sindiam(x)=sinrad(x/2)\,$ . Thus, $sindeg'(x)={\frac {\pi }{180}}sindeg(x)$ and $sindiam'(x)={\frac {1}{2}}sindiam(x)$ . It's here at The significance of radians. (I actually learned this from the reference desk) JianLi 03:11, 2 August 2006 (UTC)[reply]

That's irrelevant to the present discussion. The trigonometric functions work the same whether we use π or 2π as the basic constant. —Keenan Pepper 03:59, 2 August 2006 (UTC)[reply]

Hmm...ok I take it back. I mistakenly conflated 3.14 with radius and 6.28 with diameter :p In fact, if pi = 6.28, then 360 degrees would equal π radians :)JianLi 04:21, 2 August 2006 (UTC)[reply]

What about the four basic constants? They could not be linked as nicely if pi was any other value. (Phi^x + 2^x + e^x + pi^x - RTANG = 0, x) | x =.445704478798 [2]--Tmchk 15:24, 2 August 2006 (UTC)[reply]

That's some impressive nonsense. Fredrik Johansson 15:35, 2 August 2006 (UTC)[reply]