Wikipedia:Reference desk/Archives/Mathematics/2014 October 27

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October 27

dividing an exponential curve

(Note, I moved this question and StuRat's comment here from another desk where I mistakenly posted.)

I’m looking for a formula to find the following values: Given a section of an exponential curve on a cartesian coordinate plane that starts at zero and ends at 23.0, how can I find the y values that correspond to equally spaced points along the x axis? That is, for instance, given points (0, 0), (3.83, y), (7.66, y), (11.49, y)... etc, how can I solve for y? (Note: Although this sounds like homework, it is not. Rather this is part of precomposition for a work of contemporary classical music.) Thanks for your help! I wouldn't know how to start this calculation... --104.172.61.168 (talk) 02:38, 27 October 2014 (UTC)

The Math Desk would be a better place to ask. I don't quite understand the Q, though. Where to the values you listed come from ? StuRat (talk) 02:43, 27 October 2014 (UTC)

Those values are just an example: 23/6 rounded. --104.172.61.168 (talk) 02:54, 27 October 2014 (UTC)

You will need to find the equation for the curve in the form y = f(x). Once you have that equation, just plug in the values of x to find y. If you don't know the equation, some type of curve fitting program would be in order. StuRat (talk) 03:44, 27 October 2014 (UTC)

Yes, indeed, that would be perfect, but as far as I can tell, there is nothing like that easily available. Perhaps with more mathematical knowledge I would know how to find something like that. A simple formula is just y=xⁿ, correct? However, this doesn't include constants that would allow me to find a curve between (0, 0) and (23, y). --104.172.61.168 (talk) 05:03, 27 October 2014 (UTC)

There are infinitely many such curves, since I am assuming you are looking for an equation of the form of $y=a\ b^{x}+c$; you have specified only two conditions for three unknowns (the system of equations derived from two points is underdetermined), so you will need one more point or some other condition. But what I can say is that c ≠ 0, since b^x ≠ 0 for all x (including all complex x) but you have specified a point where y = 0.--Jasper Deng (talk) 06:20, 27 October 2014 (UTC)

Thanks, Jasper. That makes sense. Is there some equivalent of slope for exponential curves that would provide that third condition in addition to the two points? --108.38.249.92 (talk) 16:59, 27 October 2014 (UTC)

One would normally only consider curves with c=0 as exponential curves. But supposing it is of that form then practically any extra nugget of information like the slope at the start or a third point would be sufficient. Dmcq (talk) 17:39, 27 October 2014 (UTC)

For the slope or gradient you probably want the derivative of the function. For $y=a\ b^{x}+c$ the derivative would be ${\tfrac {dy}{dx}}=a\ b^{x}\ln(b)$ evaluating this at x will give you the gradient at that point.--Salix alba (talk): 18:07, 27 October 2014 (UTC)

Condition That Convergent Series of Functions Be Termwise Diff.

I can't seem to remember how this works: given a series that converges to a differentiable function in a neighborhood, what must it satisfy to be termwise diff in that neighbourhood (as in the derivative is the sum of the derivatives of the terms) ? If it helps you can assume the terms are analytic and real valued. Thank you for any assistance:-)Phoenixia1177 (talk) 07:14, 27 October 2014 (UTC)

Naturally, after taking the derivative termwise, the resulting series must still converge (this is evident with the Weierstrass function - try differentiating and integrating and you'll see how integrating each term still results in a convergent series while differentiating doesn't; that wasn't enough to show that the function isn't differentiable, however). This is the only sufficient condition I can think of, although it may be too strong.--Jasper Deng (talk) 07:37, 27 October 2014 (UTC)

Thank you for your reply - I realized in the last few minutes of mucking around on paper what the condition is: the derivatives of the partial sums must uniformly converge (assuming the original series converges somewhere). Which, as usual, brings me to a related question: given a series, are there any tools that make determining if it converges uniformly easier? For example, like if the terms are all of a certain kind, or satisfy some property, etc. Something besides working directly from the definition.Phoenixia1177 (talk) 07:45, 27 October 2014 (UTC)

There is the Weierstrass M-test that is useful in many cases of practical interest. Sławomir Biały (talk) 20:07, 27 October 2014 (UTC)

Thank you, that is very much the type of thing I was looking for:-)Phoenixia1177 (talk) 08:50, 28 October 2014 (UTC)

What is the highest known number?

Venustar84 (talk) 12:05, 27 October 2014 (UTC)

There isn't one but maybe Large numbers is of interest. PrimeHunter (talk) 12:09, 27 October 2014 (UTC)

The smallest positive integer not definable in fewer than twelve words. Well I though I knew what it was but somehow it is now definable in only eleven words. There must be an infinity of numbers definable in eleven or less words ;-) You might be interested in the article Interesting number paradox, I don't know what the current smallest uninteresting number on Wikipeda is. Dmcq (talk) 12:33, 27 October 2014 (UTC)

I think that the smallest positive integer that does not (yet) have its own individual Wikipedia article is 247. Gandalf61 (talk) 13:07, 27 October 2014 (UTC)

What a brilliant redirect, 247 ≈ 240 YohanN7 (talk) 13:18, 27 October 2014 (UTC)

(The OP already has the right answers above, so I thought I'd share some other info)

I once saw an '8' that was 20 feet tall, but surely some numbers are higher? There are also plenty of numbers on the ISS, which is pretty high. But I suspect the highest numbers are those on Voyager 1, which has just entered interstellar space. You might also be interested in "imaginary numbers" such as twentington [1], which by all accounts is very high. I also suggest reading The Phantom Tollbooth, which includes a whole chapter on an adventure in large/long/high numbers. Further reading on specifying very large numbers here [2] [3]. SemanticMantis (talk) 15:46, 27 October 2014 (UTC)

Graham's number is usually the number that's trotted out when one talks about things like "the largest number ever used in a serious mathematical proof". That claim is of course disputed, and the situation may have changed since that title was bestowed upon it in 1977, but if you were thinking of a particular number someone at some time had mention to you, whose name has now slipped your mind, that's probably the one you're thinking of. Otherwise the answer is, as mentioned, "there isn't one". -- 160.129.138.186 (talk) 23:31, 27 October 2014 (UTC)

Questions like these are a bit ill-defined. What is a "known" number?

If we say that all positive integers are "known", the answer IP 160.129 gave, "There isn't one", is correct: there is no greatest integer. (Proof: There is an axiom that says, "If n is an integer, n+1 is an integer", and n+1 is greater.)

It gets tricky because in a sense, numbers do not "exist" at all: a long time ago, somebody looked at several similar things (for example, berries) and thought, "They are so similar that I can generalize. For all intents and purposes, they are the same, and I only need to know how many there are. Rather than this berry and that berry and another over there, I'm calling them three berries." That, or something similar, is now known as the concept of numbers. (Nobody knows when that happened, but it must have been in prehistoric times.)

In that sense, numbers don't "exist"; they are only abstract concepts. - ¡Ouch! (^{hurt me} / _{more pain}) 07:20, 28 October 2014 (UTC)

It is not universally agreed upon that numbers do not exist; and there was, at least, one group that held that numbers have some form of physical existence (I don't remember what group, nor am I asserting agreement, just saying that your description is not the only legitimate one). I would say that your narrative about berries is describing the discovery of numbers and that the quantity of berries is an instance of the "more real" number that quantifies that quantity - I'm not asserting that I'm right, either, but that it is not a straight forward matter.Phoenixia1177 (talk) 09:00, 28 October 2014 (UTC)

Yes. We're into philosophy of math here (and ontology too), but I'll add this for the OP: Belief in the existence of _all_ the integers (or rationals, reals, complex, etc.) relies only on belief in the existence of the empty set, and the concept of set inclusion. Once you have those, you can construct all integers. So rejection of the reality of e.g. '42' is logically equivalent to rejection of either the empty set, or set inclusion... Of course those are abstract concepts that Ouch mentions above, but existence need not be limited to physical instantiation. SemanticMantis (talk) 15:18, 28 October 2014 (UTC)

WP:WHAAOE strikes again. MIND = BLOWN. - ¡Ouch! (^{hurt me} / _{more pain}) 12:25, 30 October 2014 (UTC)

There is no single largest number that has been known, but there are several other ways one could take this. For example, we can talk in terms of operations: n! grows large enough that n!²⁰ isn't a meaningful improvement. So, we could, in a sense, rephrase the question as asking about what produces large numbers; Fast-growing hierarchy and Grzegorczyk hierarchy are of interest, in that vein. Looking at "known" in a computational sense, Busy beaver may be of interest - and, from another direction, Church–Kleene ordinal. Ackermann function is also interesting, for size reasons; as is Paris Harrington Theorem. You may be interested in contests to specify large numbers: [4], [5], [6], and, related, loader.c [7], and Rayo's number. This may hold some interest: [8]. Finally, this [9] and the Googology Wiki in general, [10].Phoenixia1177 (talk) 08:45, 28 October 2014 (UTC)

At Mathematical universe hypothesis you can see an example where mathematics is considered if anything more real than the physical world, as opposed to digital physics where computer science is the basis of the universe Lots of people must think their way of thinking is the foundation of all things. ;-) I'm sure I saw a thing once where a person was saying we should consider Mathematics as having maximum finite number. Our brains are finite and the accessible universe is finite, therefore there is a maximum finite number which can be represented in any form within our brains or in the universe. Add one to the number and it can't be represented - just saying 'add one' uses too many more bits of information. Dmcq (talk) 17:15, 28 October 2014 (UTC)

While obviously not the largest number that can be expressed, the number of photons in the visible universe is one of the largest numbers corresponding to a count of physical things at approximately 10⁸⁹. Dragons flight (talk) 17:57, 28 October 2014 (UTC)

A number that not nearly as large as Graham's number but is one of the largest numbers that is the solution to a problem and that was actually computed is the solution to Archimedes' cattle problem. It is larger than the known universe. (There aren't that many cattle, because there aren't that many atoms.) Robert McClenon (talk) 14:52, 30 October 2014 (UTC)

The history of this problem is uncertain, but it is said that Archimedes was annoyed by the pride of the mathematicians of Alexandria and posed the problem as a "malicious riddle". Archimedes, who had invented a form of scientific notation, at least had an idea of the order of magnitude of the solution, although actual solution of the problem requires electronic computers. Robert McClenon (talk) 15:00, 30 October 2014 (UTC)

From your link -- "The general solution was found in 1880 by A. Amthor. He gave the exact solution using exponentials" -- now, if I can prove the solution to a given problem is exactly e.g. a^b^c^d^e^f^g, then I'd say that I actually solved it, even if I was unable to compute every digit in the base 10 representation. SemanticMantis (talk) 15:35, 30 October 2014 (UTC)

• G1 = Graham's number. G2 = G1**G1, repeated G1**G1 times. G3 = G2**G2, repeated G2**G2 times. Continue this process indefinitely. Take a snapshot at a random point. By now you've probably got a reasonably large number, certainly large enough for most practical purposes. -- Jack of Oz ^{[pleasantries]} 20:34, 2 November 2014 (UTC)