Wikipedia:Reference desk/Archives/Mathematics/2020 November 30

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November 30

Why is integration being used here rather than standard summation?

I am taking an online algorithms course that includes this video. At 13:55, the video shows the following equation:

$2(N+1)({\frac {1}{3}}+{\frac {1}{4}}+{\frac {1}{5}}...{\frac {1}{N+1}})$

The video then shows that the above equation can be expressed as:

$2(N+1)\int \limits _{3}^{N+1}{\frac {1}{x}}\,dx$

I don't know much about calculus (my only experience with it is a high school course), and I'm having difficulty seeing why an integral is appropriate here. Shouldn't this be simple summation like this:

$2(N+1)\sum _{3}^{N+1}{\frac {1}{x}}$

I think the fact that the equations were related with a $\sim$ sign and not an $=$ sign might be part of it, but I don't quite understand how the former sign works. --Puzzledvegetable^{Is it teatime already?} 00:16, 30 November 2020 (UTC)

It is not an equation but an asymptotic approximation. The article explains the f(x) ~ g(x) notation. Both the summation (see harmonic series) and the integral tend slowly to ∞ as N tends to ∞, but their difference is limited by a relatively small constant, related to the Euler–Mascheroni constant, and so their ratio goes to 1. An advantage of replacing the summation by an integral is that it is much easier to compute, since it has the log function as its primitive (antiderivative).  --Lambiam 01:45, 30 November 2020 (UTC)

Why is the end of a swinging chain diagonal when J_0(0) is flat?

I'm trying to understand the Bessel function J_0, and how it describes the motion of a vibrating hanging chain, and I'm confused by the behavior of the free end of the chain. The derivative of J_0(0) is 0, but the free end of a swinging chain points at a diagonal.

https://en.wikipedia.org/wiki/Bessel_function#/media/File:Bessel_Functions_(1st_Kind,_n=0,1,2).svg

https://www.acs.psu.edu/drussell/Demos/HangChain/HangChain.html

Black Carrot (talk) 08:15, 30 November 2020‎ (UTC)

The argument of the Bessel function is not the vertical length $x$ along the chain, but is proportional to its square root. Calling that argument $u$, we have that $x=au^{2}$ for some value $a.$ Then for any function $f,$ representing the displacement as a function of $u,$ its derivative (with respect to $u$) is

${\frac {df(u)}{du}}={\frac {df(u)}{dx}}{\frac {dx}{du}}={\frac {df(u)}{dx}}{\frac {dau^{2}}{du}}=0.$

To find the (non-zero) derivative with respect to $x$, use L'Hôpital's rule.  --Lambiam 11:54, 30 November 2020 (UTC)

Ok cool, I see how I missed that now. And I see that that change to the function makes it look a lot more realistic (https://www.desmos.com/calculator/rd0zyuggw3).

Black Carrot (talk) 00:29, 1 December 2020 (UTC)

A Real Number Pyramid

Here is a short list of my assumptions and assertions:

1) {∞,...,0,...,-∞} A unit infinity, ∞, extends the integers such that ∞>|x| for all x in $\mathbb {R}$. Like the number 1, ∞ is an invertible unit, thus 1/∞ is an infinitesimal greater than 0 and less than the absolute value of every real number such that |x| > 1/∞ > 0. A set has the cardinality of ∞ if and only if there is a bijective function h : A → $\mathbb {N} _{0}$ to the set of naturals.

2) There are two infinite binary trees B_Z and B_frac that represent the integers and fractional part of any real number in $\mathbb {R}$ respectfully both with the same tree depth ∞.

3) Every full tree path P_Z represents the completed infinity of an integer in B_Z. Every full tree path P_frac represents the completed infinity of a fraction in B_frac. Every node on every level has one descending path that duplicates its binary representation.

4) 4^∞-1 is the total number of real numbers. The product of the node counts per tree level can be enumerated, it is a sequence of powers of 4, {0,4,16,...,4^∞}. Half of the 4^∞ nodes on level ∞ are duplicates, and a sign bit only doubles the positive reals so the final count is 4^∞-1.

5) By construction, all possible reals are defined and accounted for. Choose any real number, including the deterministic and infinite path that defines pi. It is built into the model.

Modocc explains the Real Number Pyramid to an astonished mathematical audience

6) Cantor's algorithm constructs a list of binary strings from set T.

7) Cantor's diagonal argument claims a path P_s from R is not counted.

8) Assertions 4 and 7 cannot both be true. This is the actual contradiction.

9) The reals are countable.

--Modocc (talk) 08:05, 30 November 2020 (UTC)

• If I understand correctly, assertion (2) implicitly assumes that real numbers are countable. It is therefore unimpressive to deduce (9) from it.

More generally, if you think a well-known mathematical fact is false (and the uncountability of reals dates back to 1891), nobody will take you seriously if you cannot show where the previous demonstration is false. Tigraan^{Click here to contact me} 14:08, 30 November 2020 (UTC)

At least these trees in assertion (2) represent the number parts respectfully. You have no idea how disrespectful some of the other trees can be.  --Lambiam 15:25, 30 November 2020 (UTC)

Yes, I concur with Tigraan on this; assertion (2) is where everything falls down, by implicitly assuming the conclusion in advance. -- The Anome (talk) 15:35, 30 November 2020 (UTC)

2 also contains an inaccurate statement. It states that all real numbers contain a fractional part. This is not true. Irrational numbers (and their subset transcendental numbers) cannot be reduced to a fraction. There is no "fractional" part of these numbers. Your statement in #2 is only true for rational numbers and not real numbers. --Jayron32 15:37, 30 November 2020 (UTC)

I think Modocc is thinking of the "fractional" part as being x − floor(x) for all x in the reals -- The Anome (talk) 15:41, 30 November 2020 (UTC)

Yes, but 10*(x−floor(x)) is just another irrational number (assuming base 10) of the format he's trying to separate. There's no meaningful difference between x and x−floor(x) in the way they are trying to imply. --Jayron32 15:49, 30 November 2020 (UTC)

Agreed. My point is that that was probably what they had in mind, not that it was correct or that "fractional" was actually meaningful in that context. -- The Anome (talk) 16:33, 30 November 2020 (UTC)

I believe the term "fractional part" is standard for this, whether or not it's a rational number. "Fraction" in this context just means a number between 0 and 1; it doesn't have to be the ratio of two integers. --Trovatore (talk) 22:57, 30 November 2020 (UTC)

To see why (2) doesn't work, you can think about how just one of those infinite binary trees represents the real numbers in the unit interval, and apply Cantor's diagonal argument to it to demonstrate that that tree, just by itself, must have an uncountable number of paths within it. If a subset of the reals is uncountable, so are the reals. -- The Anome (talk) 15:39, 30 November 2020 (UTC)

See fractional part. I'm missing a qualifier. I messed it up further down too. I meant the binary expansion part of unit interval. What would that be called?-Modocc (talk) 22:01, 30 November 2020 (UTC)

@Modocc: I somewhat doubt (1) is well founded. When you say A set has the cardinality of ∞ you need to define what the cardinality of ∞ is. As the Cantor's theorem on power sets shows, for each set Q there exists a set Q' whose cardinality is strictly greater than that of Q – hence if there exists 'an infinite cardinality', then necessarily exists an infinite hierarchy of ever greater infinite cardinalities; and your definition does not specify, which one of them your cardinality of ∞ is. --CiaPan (talk) 22:42, 30 November 2020 (UTC)

@CiaPan:"When you say A set has the cardinality of ∞" you need to define what the cardinality of ∞ is." To be clear, I assumed only a single infinity defined by any bijective function with the naturals because Cantor's other cardinalities assumes his diagonal argument shows R is uncountable, but it need not, as I intend to demonstrate on my next attempt. --Modocc (talk) 00:21, 3 December 2020 (UTC)

@Modocc: You're wrong. Again. The Cantor's theorem about the power set's cardinality does not assume anything about a set. It does not distinguish between an empty set, a singleton, a set of integers, set of reals or set of all real-valued functions with real variable. That's the beauty and the power of the theorem: it applies to literary every set. Consequently, it does not 'assume' anything about the diagonal argument used for a set of binary sequences. What's more, Cantor did not use that argument to prove uncountability of reals – that was done about 15 years before the diagonal argument in Cantor's first set theory article#Second theorem. You seem to lack the most basic knowledge from the area you argue about. PLEASE, learn the definitions of notions you use. Otherwise your efforts are doomed to fail due to inconsistencies in your examples and holes in your reasoning. --CiaPan (talk) 18:37, 5 December 2020 (UTC)

You appear to be talking at cross-purposes here. You are referring to Cantor's theorem while Modocc is referring to Cantor's diagonal argument – which is another, simpler proof of his earlier theorem about the uncountability of the real numbers. Cantor's theorem generalizes this.  --Lambiam 23:38, 5 December 2020 (UTC)

You appear to miss my first entry in this talk. Please read a thread from the first level of indentation, not from the last one. --CiaPan (talk) 11:51, 8 December 2020 (UTC)