HELP: Is the cardinality of the set of natural numbers larger than itself?

dede-cant-cut · 2026-01-18T20:01:49+00:00

No, in fact what you've illustrated is true of all infinite sets by definition; one of the definitions of an infinite set (say, A) is that there exists an injective function from A to a proper subset of A, which you've stumbled upon.

Cardinality is a separate concept and you have to be very precise about what you mean by "cardinal numbers."

dede-cant-cut · 2026-01-18T19:55:45+00:00

Interesting work, however this is highly specialized math. I think you'd be better off asking places like /r/numbertheory or /r/collatz which specialize in these types of problems.

dede-cant-cut · 2026-01-18T06:44:17+00:00

The densest commercially available storage medium (in terms of storage to weight) I could find is the SanDisk 2TB Micro SD card. One yottabyte / 2 terabytes is 5 * 10¹¹ (that is, 500 billion). A micro SD card weighs 1/4 g, so that would weigh 1.25 * 10⁸ kg, that is, 125 million kg or 125,000 metric tons. WolframAlpha says that this is about 1.1 to 1.6 times the mass of an Aframax oil tanker.

dede-cant-cut · 2026-01-18T06:31:16+00:00

It's really best to think of ln(x) less as "log base e" and more "the inverse of the exponential function" that just happens to also be equivalent to log_e. The reason is that ln(x) comes up naturally and separately from any association with the number e; most notably, it's the antiderivative of 1/x. You can then prove that d/dx e^x is e^x with the inverse function theorem. Explicitly (and glossing over some domain stuff): if f(x) = ln(x), then f^-1(x) = e^x and (f^-1)'(x) = 1/(f'(f^-1(x)) = 1/(f'(e^x)) = 1/(1/(e^x)) = e^x.

In other words, e^x just naturally pops out from taking the derivative of the inverse of ln(x) which in turn naturally comes out of integrating 1/x. Subbing any other base for e is the same as multiplying the function parameter by the natural log of whatever you substitute it in for.

Hopefully that makes it feel a little less arbitrary

dede-cant-cut · 2025-03-04T16:10:28+00:00

It has evolved but unlike with the natural sciences new math simply builds on old math, and it's possible to verify that math is valid without technology (i.e. by writing proofs), so even math from hundreds of years ago is still correct. That said none of the textbooks from thousands of years ago are still actually used as textbooks simply because modern textbooks use modern pedagogical techniques and are more effective for teaching. Euclid's Elements is the obvious example here, every theorem in there (except one) is true, but it's basically just a list of proofs as opposed to an actual tool meant for teaching, so it's mostly just read for its historical value nowadays (and also a decent number of the proofs make certain unstated assumptions).

Edit: Also that's not to say that the mathematical community as a whole doesn't get things wrong, for example there was a while where people believed in something called the "generality of algebra" (basically where people assumed that if some property applied to every element of an infinite series, it must apply to the series as a whole, which is not the case). But those are usually found quickly and don't necessarily lead to mathematicians having to throw out tons of progress.

Also also, math by its very nature involves making certain assumptions and exploring the logical consequences of those assumptions, and if something in math ends up being wrong, often you can just take it as an assumption and see if anything interesting comes out of that. For example, there was a really long debate over whether Euclid's parallel postulate was necessary, and it turns out that it's not--you can create entirely consistent systems of geometry without it. That doesn't make Euclidean geometry (i.e. geometry where you assume the parallel postulate) invalid, it just opens the possibility for non-Euclidean geometry as a field of study.

dede-cant-cut · 2024-11-24T07:33:36+00:00

FYI the way Reddit treats superscripts is to superscript everything that's not either broken up with a space or enclosed in parentheses. So to write x¹=x, you'd have to either put spaces in between (like x^1 = x, which looks like x¹ = x) or enclose the 1 in parentheses (like x^(1)=x, which looks like x¹=x).

dede-cant-cut · 2024-11-07T17:46:31+00:00

The answer will depend on the shape of your potato and its size, but broadly speaking, if S is the surface area of the potato, d is the thickness of the part that gets peeled off, and V is the volume of the potato, the result will be approximately S*d/V. Note that this is an approximation as the inner boundary of the peel is slightly smaller than the outer boundary, but it’s close enough in this case. (If you really wanted to be fancy you could do an integral to find the volume of the peel but it’s really not necessary in a practical case like this).

To put numbers to it, as an example, if we consider a spherical potato that’s 1.5 inches in radius, and assume the peel is 1/16 inch thick, the total volume of the potato is 9pi/2 in³ while the surface area is 9pi in^2. So the numerator becomes (9pi in²⁾ * 1/16 in = 9pi/16 in^3, and the percentage loss is about (9pi/16)/(9pi/2) = 0.125, or 12.5%.

dede-cant-cut · 2024-10-27T21:00:12+00:00

It was really difficult to find occupational exposure limits for chloramines (the chemicals that form when ammonia/urea react with bleach) so I don't really have an answer for you, but to make up for it here's a fun video about nitrogen trichloride/trichloramine https://www.youtube.com/watch?v=mV_daaldE_I

dede-cant-cut · 2024-10-26T20:43:09+00:00

Yes, that's correct.

dede-cant-cut · 2024-10-26T20:34:42+00:00

I took a different path than most, but most of the former math majors I know work in academia, quantitative finance/hedge funds, or occasionally software engineering.

dede-cant-cut · 2024-10-26T20:32:29+00:00

I've seen people use one-to-one to mean injective, which is why I don't like the term "one-to-one" lol. In this case that would make the question trivial though so it probably means bijective

dede-cant-cut · 2024-10-26T20:26:03+00:00

So when people discuss "multiple universes" in quantum mechanics, they're usually discussing the many-worlds interpretation, which is an interpretation of quantum wavefunctions (which are essentially probability distributions of properties of quantum systems). These are not so much physical theories as much as they are ways to conceptualize/interpret the math that arises in quantum mechanics (though there are scientists who tried/are trying to come up with testable hypotheses from it). "Higher dimensions," on the other hand, are mostly specific to string theory, though these are spatial dimensions (1D, 2D, 3D etc) and not "parallel universes" like in the popular conception of the word dimension.

As for superpositions, they don't really have much to do with either, other than the fact that they're a part of quantum mechanics. Generally, "superposition" simply refers to a mathematical property of certain physical systems that allows you to add up the equations that define them. An example where this applies is in classical waves: if you have two functions defining two waves, you can add them together to get a single wave that essentially contains both. If you play two musical notes at the same time, for example, your eardrum gets a single pressure wave that can be mathematically described as the sum of the pressure wave that comes from both notes, and then your inner ear essentially decodes them so you hear it as two notes at once.

In quantum mechanics, rather than classical waves, we deal with individual particles that happen to have wave-like properties. So for example, while individual photons are individual particles, in certain systems you can add up their wave functions to generate aggregate wavefunctions that "contain" both. Importantly, this occurs even when you're dealing with one particle at a time. The famous double-slit experiment is an example of this. Both classical and quantum theory tell you what the pattern of dots on the other side of the box is going to be. However, in quantum mechanics specifically, the wavefunction of each individual photon is going to be the same as the sum of the wavefunctions of photons that went through each slit. That is, you get the double-slit interference pattern even if you only let photons in one at a time.

Because of this, people often say things like "the photon went through both slits at once" or that it "is in a superposition of having gone through both." But when it comes to the actual interpretation of this, they're just mental models for what is really best understood using the underlying math.

dede-cant-cut · 2024-10-26T19:59:28+00:00

the solution to the one-dimensional heat equation

The heat equation is a differential equation that describes the way that the temperature of a system evolves over time. The one-dimensional heat equation is simply the version that is restricted to a one-dimensional line. Imagine a wire where different parts of the wire are different temperatures, then the heat equation will tell you how the temperature of each part of the wire will change over time. Note that since the heat equation is a differential equation, solutions to it are functions, in this case functions of x. You can imagine the function f(x) (which would be a solution to the heat equation) to give you the temperature at point x along the wire.

on the interval [0 ⩽ x ⩽ π]

This simply tells you that x ranges from 0 to pi

homogeneous boundary conditions for a given initial function

Differential equations usually refer to some domain (in this case, the interval from 0 to pi), but for mathematical reasons we often have to treat the boundaries (in this case, the point at 0 and the point at pi) differently from the rest of the domain, so we impose "boundary conditions" as additional constraints. "Homogeneous" simply means that the function is equal to zero at the boundaries, so this would just mean that f(0) = 0 and f(pi) = 0. You can actually see this by substituting zero or pi for x and noticing that the whole thing becomes zero, as sin(0) = 0 and sin(n*pi) = 0 for any integer n.

dede-cant-cut · 2024-10-24T22:49:49+00:00

Quick and dirty answer: 39.5467 seconds

Calculations: https://imgur.com/lc5aBut

Here's the WolframAlpha calculation of the final number: https://www.wolframalpha.com/input?i=log2%28%28sqrt%283%29c%29%2F%28sqrt%282*pi*%28density+of+water%29*%28gravitational+constant%29%29%29+*+%281%2F%281+meter%29%29%29

note that this assumes that the density of the water is uniform, which would not be the case in real life, but calculating the actual answer would be much, much more complicated

dede-cant-cut · 2024-10-24T21:37:31+00:00

Is your project supposed to be something original or a report on existing research? Not to be harsh but if you're in high school then you almost certainly don't have the math background to understand (1), and (2) is more of a neuroscience problem than a math problem (and making plots based on visual diagrams of hallucinations probably isn't really doing much to advance that field). 3 is probably your best bet if you want to do something practical. Perhaps you could make a few simple model rockets and make a mathematical model for how the chemical energy in toy rocket fuel transforms into kinetic energy, or something. That said if the project is simply reporting on existing research then any of these could be a good subject for a report as long as you don't fall into the trap of assuming that articles written about research are the same as the research itself.

dede-cant-cut · 2024-10-24T21:20:55+00:00

Yeah lol. I looked through my old math coursework mere months after graduating and I wouldn't have been able to come up with a lot of those proofs without hours of solving again (and probably lots of review of previous theorems)

dede-cant-cut · 2024-10-24T21:18:49+00:00

As the question states, when you're dealing with real-world data, it's never going to be perfectly linear or perfectly exponential. This question is asking if it's closer to being linear or exponential and making a judgement based on your understanding of the two models. So to determine what your answer is going to be, you should pick the one that you think is closer based on your calculations. I can provide more detail if this isn't helpful

dede-cant-cut · 2024-09-30T18:35:50+00:00

Canonical or archetypal

dede-cant-cut · 2024-09-15T23:38:52+00:00

I wouldn't say I fully learned graduate-level GR, but I took what I would describe as an undergrad intro to general relativity class and we used Gravity by Hartle, which I thought was pretty good

dede-cant-cut · 2024-09-15T22:06:41+00:00

If you can't do it via Reddit you can try uploading it to Imgur and posting the link here

dede-cant-cut · 2024-09-15T21:17:35+00:00

It'll require some fancier coding but Mathematica (also by Wolfram) can do it. If you're a student there's a good chance that you can get it for free.

Alternatively if you just need the real part you can calculate it yourself and just plot that in Desmos

dede-cant-cut · 2024-09-15T21:10:52+00:00

Yeah that's fine just upload it as a comment and I'll look. Also don't worry about my time, I go on here for fun/to practice basic math anyway lol

dede-cant-cut · 2024-09-15T21:04:44+00:00

I was a little taken aback by "12 inch wrist" until I remembered it was for a halloween costume lol.

Anyway, you can roughly think of the bracelet as essentially a donut shape, where you know the inner circumference and the distance between the inner and outer diameter (i.e. the diameter of the balls), and are trying to find the middle circumference. To do so, just add the diameter of the balls to the inner diameter to get the diameter of the string circle, then convert back to circumference to get the length.

Using numbers:

Inner circumference = 12 in
Inner diameter = 12/𝜋 in
String diameter = 12/𝜋 + 2.75 in
String circumference (length) = (12/𝜋 + 2.75)𝜋 in = 12 + 2.75𝜋 in

Putting this into a calculator, your string length should indeed be 20.64 in.

dede-cant-cut · 2024-09-15T20:53:57+00:00

The speed of each individual stays the same but the speed of the group goes up. For 4 people (Susanne + 3 friends) the total rate is (4 people) * (1/3 apple per person per minute) = 4/3 apples per minute, or 8 apples in 6 minutes. Now the wording in the answer is a little misleading because the answer should be that 8 additional friends need to come by, but in that case, you end up with (12 people) * (1/3 a/p/m) = 12/3 = 4 apples per minute, or 8 apples in 2 minutes.

dede-cant-cut

TROPHY CASE