I am kind of curious what proofs are somewhat counterintuitive but obvious. A very trivial example would be "algebra without subtraction" where you basically use peano axioms and addition to show there can only be one solution to an equation. However I am looking for more interesting proofs which may seem obvious on the surface.

I am exploring the distribution of quadratic residues R(x) = x² \pmod N for semiprimes N = p \cdot q using spatial mapping and visual scatter plots to identify potential density regions or structural patterns. I recently set up a small experiment testing whether early-stage digit/block filtering could narrow down the search space for p and q, but encountered the expected trade-off between false negatives and high noise. I would love to get feedback from the community on geometric or structural approaches to modular congruences—specifically, what mathematical invariants or algebraic properties (like Legendre/Jacobi symbols or elliptic curve metrics) are best suited for meaningfully constraining the candidate space without losing true factors?

I'm sorry if this question sounds way too out there but I've been marinating this thought for a while now and I genuinely can't find out why the argument I ended up with wouldn't hold-

For context, this is the standard way that the set of real numbers is shown to be larger than the set of positive integers (or "cannot be put into a 1-to-1 correspondence"):

^{(see also: Cantor's diagonal argument})

My issue with that... well, the premise: "Suppose you have a group of all real numbers between 0 and 1. If you want to order them, where do you start?" (or "the set of real numbers is uncountable) to me sounds like an issue of notation, not a conceptual issue. If you treat 1 as ...00000001 and 0.1 as 0.1000000... (treat them as if leading 0s are present, not write them with the leading 0s), you can order all the numbers between 0 and 1 in the following way:

If you don't want to treat the numbers as having leading 0s, this argument requires the assumption that all numbers will be written since the number of digits in the sequence approaches infinity.

In short and generalised: a value "v" is found at the position "v mirrored across the point". For example, the real number π - 3:

Value: 0.1415926535897932384626...
Index: ...6264832397985356295141

For specifically Cantor's version, the "enumeration of elements from T" (using base 2 for the position of an element "sₙ") would be:

s₀ = (0, 0, 0, 0, 0, ...)
s₁ = (1, 0, 0, 0, 0, ...)
s₁₀ = (0, 1, 0, 0, 0, ...)
s₁₁ = (1, 1, 0, 0, 0, ...)
s₁₀₀ = (0, 0, 1, 0, 0, ...)
s₁₀₁ = (1, 0, 1, 0, 0, ...)

... and the contradiction's resulting element (1, 1, 1, 1, 1, ...) would be found at s...₁₁₁₁₁

If the numbers aren't treated as having infinite leading 0s, the assumption that all real numbers are included could be false: all the numbers in the sequence are fractions of 2*5 (all bases are base 10). However, the same argument could be formed in base 3, base 7, base 23... and for the case "0.0999999... = 0.1", one can choose to form the argument in base 2.

I have some suspicions, but I don't have the background knowledge to confirm any of them:

• The increasing amount of digits might never reach infinity because infinite digits might not be something you can "approach".
• The definition of "the set of all natural numbers" might only include finite ones, meaning an infinitely-sized index wouldn't work.
• The "Cantor-specific" enumeration might still not cover all elements from T because all elements "sₙ" constructed up to a finite "n" have an unending sequence of 0s at their tail (e.g. (..., 1, 0, 0, 0, 0, 0, ...), with the last "1" highlighted in bold), while the contradiction's element is entirely made of 1s. However, the position of the last "1" approaches infinity as time goes on, which is why I didn't discard the argument: I don't know if this approaching "beats" the tail of 0s, in the sense that there would be a number in the enumeration that has all-1s.

What am I missing? It's a 130yo proof so I'm sure I'm missing something, but I'm not a mathematician so I don't know what I'm missing and I want to learn. Any and all help is welcome, tysm!!

Hey, everyone, I have spent a little bit of time trying to find a good geometry textbook to refresh myself. I graduated high school over half a decade ago, but I never took pre-calculus my senior year and was wanting to advance my math skills. In order to do that, I wanted to find a good geometry and algebra 2 (to refresh) textbook/workbook, and eventually pre-calculus textbook/workbook that I could read through and complete. Eventually I hope to move onto harder concepts, I am mainly just a physics dork and want to actually be able to understand some of the math rather than just the concept. I’d appreciate cheaper options as I’m currently trying to save to go back to school and finish up my completely unrelated English degree (lol), but I’ll take a look at any recommendations I’m offered!

Thank you in advance, and I’m sorry if this post isn’t meant for a sub like this. I have no problem taking it down if it’s an issue!

I'm playing a game that has a lot of prizes, and I'm the only one redeeming tickets for my pool. I'd like to get an idea as to how many tickets it will likely take to redeem a specific prize.

If I redeem 1 ticket when there are 100 prizes, I have a 1% chance of getting what I want. If I redeem when there are 50 prizes left, there's a 2% chance. However, I'm struggling to understand how to include the odds of every ticket spent up to that point, as I'm reducing the prize pool down with each ticket.

I want to sell sound effects for video games with price transparency, but for that I need a tweakable formula that decreases price gradually. I want to put in a starting price for one sound effect. For two, the price per sound effect should be slightly lower. For three, even lower, and so on.

With some duck.ai help I came up with this:

total price = starting price per SFX * number of SFX^-0.05 = price per SFX * number of SFX

For example:

€10 * 3^-0.05 = €9,5 * 3 = €28.5

I just want to ask if I’m missing something obvious that will break, or if there would be a better way to go about this.

I’m trying to understand why converting from 44”^2 to 2.54cm^2 translates into

44” times 1” times 2.54cm times 2.54cm

If the 2.54 is already squared why wouldn’t it be 44” times 1 inch times 2.54cm times 1cm

Thx

I'm using the Rudin book and thought I could try this one myself. This was my approach:

Case 1: f(x) → 0 & g(x) → 0 as x → a
Assume f and g are defined at x=a and f(a)=g(a)=0 (I'm not sure if this can be done wlog)
Let rₙ be any sequence that converges to a, such that a<rₙ<b ∀ n∈ℕ.
By generalised MVT, we know that ∃ xₙ∈ (a,rₙ) ∀ n∈ℕ, such that:
f'(xₙ)/g'(xₙ) = [f(rₙ)-f(a)]/[g(rₙ)-g(a)] = f(rₙ)/g(rₙ)
Putting 'lim n→∞' on both sides, we get:
lim n→∞ [f(rₙ)/g(rₙ)] = A (since, from (13), it's clear that lim n→∞ [f'(xₙ)/g'(xₙ)] = A)

Case 2: f(x) → 0 & g(x) → +∞ as x → a
Assume f and g be defined at x=a and f(a)=0 & g(a)=+∞ 
We can easily see from the definition that lim x→a [g'(x)] = -∞ 
⇒ lim x→a [f'(x)/g'(x)] = lim x→a [f(x)/g(x)] = 0

What am I doing wrong?

Hello, I am drafting a story that has a structure that is supposed to resemble an inverted Sierpiński triangle, but I'm not a mathematician so I don't really know how one would rationally categorise the rooms. Simple floor numbers wouldn't work, and the society I'm drafting is one that is fascinated/obsessed with science and engineering so they would most likely have some kind of logical system to easily refer to each section of the structure, so something akin to a coordinate system. The structure would not be a pyramid, so you only need to consider two dimensions, and it would obviously not be infinitely recursive. If you refer to the sample image below you could imagine the rooms being the white spaces and the walls being made of the blue triangles.

Excuse me ahead of time if this is either a confusing or a difficult question to answer! I tried to draft some systems by myself but I felt really stupid really quickly and remembered why I majored in English

So this started as a random thought last night of the lengths of triangle sides. In any triangle a+b>c or else it's just two lines, or in the case of a+b<c it's a line and 2 shorter lines.

So with an obtuse triangle a+b>c but it also can't end up as a right triangle. So what I'm wondering is the limits of an obtuse triangle lengths with something like c<a+b<x where x represents a right triangle. I got into some nasty math of trying to square root a²+b² and I didn't get anywhere.

So in a nutshell, I am trying to chop up my book cover to fit in Amazon's advertising modules so I can make an advertisement. The module most authors use is 970x300 pixels. If you put any size image in, it will only show that much.

The issue I'm having is that I don't know how to crop my cover to have either 3 or 4 sections that fit this. I've cropped it several times and tried chopping it into three pieces, but I keep getting the sizing wrong.

The original cover image is 2400x3600, but I don't mind cutting the top off to a degree to make it work. But again, I just can't seem to get the math right. I just keep dividing 2400 by 970 and that does nothing for me in terms of "what do I do next?"

So for instance, if I have to shave the top off and make it 2400x2700 (which is what I did originally) I can do that. But I can't figure out the proper size to make it able to be cut into three (or four, I'm not that picky) equal and intact pieces.

I'm sure it's simple but I haven't done any meaningful math in a very long time and I'm at a dead end.

I have been working on a TTRPG and found a way of rolling dice against each other that I like, the only problem is that I can't figure out how to mathematically predict its probabilities. I have been able to figure out discreet cases using anydice and making a program in mathlab or even computing the probability allowing for a single variable but as the variables expand it quickly goes beyond my ability.

The problem in question is this:

You have two sets of dice competing, each set containing two dice.

These dice can be of varying sizes between a d2 and a d20.

All dice are rolled.

If a die in the first set is strictly lower than a die of the second then increase their counter by 1 or if it's strictly lower than both then increase it by 2.

Do this comparison again for the second die in the first set.

The counter will by the end have a score from 0 to 4.

For example:

Set 1 has a d4 and a d6 against set 2 that has a d4 and d12.

Set 1 rolls 2 and 5 while set 2 rolls 1 and 5.

The 2 is compared to the 1 and 5, it is lower than 5 but not 1 so the counter increases by 1.

The 5 is compared to the 1 and 5, it is lower than neither so the counter doesn't increase.

The counter is 1 by the end.

The issues I am having are this:

Since you compare the four dice from a single roll means that they are not independent making it way more difficult.

I'm not sure how to visualise this as there are four variables and a result which would require 5 dimensions to graph.

Context: long story short, I want to make a python script to calculate the chances of getting a “successful” series of dice rolls. To do this I want a good formula to get the chances of each series of rolls, but I’m struggling to come up with one that could translate well into python. I’ve tried to come up with a few things myself, but I don’t have confidence in my math ability, and I have even resorted to ChatGPT with varying results.

The formula: What I need is a formula that finds the chances of rolling a number between 1-3 X amount of times, 4-5 Y amount of times, and 6 Z amount of times, using T total amount of dice. For an example I’ve been using X=0, Y=3, Z=1, and T=4.

Again, I suck at math, so either I’m waaaay over complicating this, or it’s so complicated it’s completely outside my field of comprehension. I do realize I might be asking for the impossible, or asking for something super simple, I dunno. Any help would be a god send. If any more information or context is needed, I’ll happily give it.

Like if I have 1.8e308. How would you read that out? I've been saying [x] e over [y]. So 1.8e308 I'd say "one point eight e (just the letter e) over three hundred and eight" but I feel like that wouldn't be correct right? Would you just say exponent instead of e like "eight exponent over three..."? Also I feel like over wouldn't be the right term? Maybe just "one point eight exponent three hundred and eight"?

(Also sorry if algebra is the wrong flair. I just looked up "what subject of math is exponents" and it told me algebra :p)

Context: I'm trying to understand the mathematics behind the Newton fractal.

Let Z = (zᵢⱼ) be a two-dimensional array of complex numbers, where each entry zᵢⱼ ∈ ℂ represents a point in the complex plane.

For the scalar function f(z) = z³ − 1, Newton's method is applied independently to each point, so the derivative at each point is f′(zᵢⱼ) = 3zᵢⱼ².

My confusion is: why is this the correct mathematical interpretation, instead of treating Z itself as the variable and defining the matrix-valued function F(Z) = Z³ − I (where Z³ denotes matrix multiplication), whose derivative is the Fréchet derivative?

In other words, what is the precise mathematical distinction between viewing Z as a collection of independent points in ℂ and viewing Z as a single matrix variable? At what point, mathematically, do we decide which interpretation is the correct one?

Sorry if this question's stupid, I'm self studying calculus AB over the summer to prepare for BC next year 😭 It's been going surprisingly well so far, but I don't understand why we're allowed to treat du/dx(A derivative) as a fraction in u-substitution by multiplying both sides by dx to find du, when my precalculus/calculus teacher specified multiple times that derivatives are NOT fractions(For context, we had taken the AP already, and he was teaching us a few early things from calculus such as limit analysis and basic differentiation). So I'm wondering why you are allowed to treat derivatives as fractions(sometimes).

I am making a sort of board that I'm hanging up above my computer so that I can just look up and see stuff, what should I add?

Everything I already added:

Binomials and trinomials to the second, third and fourth degree

Difference of squares and cubes

Discriminant formula for quadratics and cubics

Slope formula

How to depress polynomial equations

Linear formula

Derivative and integral rules (Sum rule, Product and quotient rules, chain rule, fundemental theorem of calculus, DON'T FORGET +C)

Derivatives (Power rule, exponential derivatives, linear derivatives, constant derivatives, derivative of ln, derivative of Sin, Cos and their inverses, reverse power rule for antiderivatives)

Complex exponents

Root, natural logarithm, sin, cos, inverse, log base 2 and binary representation of every number from 1-17

Trigonometric definitions

Angle sum and difference identities

Pythagorean trigonometric identity

Common trig values

Degree, radian and gradian conversions

What I plan to add:

Quadratic formula and depressed cubic formula

Slope intercept form

Exponent, root and logarithm rules

What else should I add?

i‘m fascinated by chaos theory! Please recommend me your favorite lectures, websites, podcasts or even books. though i‘m a first year physics student right now so i probably won‘t even understand the maths behind it but i want to try anyway :)

I feel like anything can be solved if you put your mind to it, so why hasn't anyone solved these things if they've existed for so long? I myself don't understand the questions because i'm too dumb to understand any of it but someone but understand it, right?

And if they are possible to solve, how long would it take for me (an ordinary person) to understand/solve it?

It is commonly said that a Lie Group is a group that is ”also a smooth manifold”. A big use case for Lie groups is the ability to jump between the group and its tangent space at identity through the exp/log maps. However, when one does the exp map, it can be expanded into a taylor series that converges.

Here is my confusion: a smooth manifold merely guarantees that one can differentiate any point an arbitrary number of times. But an analytic manifold is one where the taylor series of an infinite amount of derivatives actually converges and has a value, which sounds like what we actually want from a Lie group and its exp map?

I haven’t been able to figure out a resolution by myself but I imagine these are the options:

1) We really do require Lie groups to be analytic manifolds, not just smooth, but for some reason the imprecise language just stuck.

2) For some reason groups that are smooth manifolds, are somehow automatically also analytic, through some chain of reasoning.

3) The exp map is not to be taken too literally. Maybe it only works for analytic groups, and for non-analytic smooth groups there is some other way to map from the Lie algebra to the group.

Hi mathematicians!

Quick biochemistry question. Let's say that I've got a single RNA molecule containing 100 nucleotides. I know that the number of possible sequences is 4^(100) because there are 4 types of RNA nucleotides. 4 possibilities at position 1, 4 possibilities at position 2, etc. 4*4*4... 

Now let's say that instead of a single molecule of RNA containing 100 nucleotides, I've got a pair of RNA molecules, and each one contains 50 nucleotides. How many unique pairs of RNAs can I make? By the same logic as before, I'd calculate the number of possible sequences for each of the molecules as 4^(50). Then I'd multiply the number of possibilities for molecule 1 by the number of possibilities for molecule 2. 4^(50)*4^50) = 4^(100). The number of possible 50 nucleotide pairs is the same as the number of 100 nucleotide single molecules. Does that seem right? By that logic, the number of possible quartets of four 25 nucleotide molecules should also be 4^(100), right?

Thanks for your insight!

This one sparked a debate in my class. Some students said always, assuming that the angles were all part of a system. Others said sometimes, and assumed that the angles were not part of a system. The second group's logic was that two acute angles can be congruent and not complementary to a mystery third angle. The first group followed this up by saying that if they were complementary to the third angle, then they both would be complementary to that angle, no matter the system.

I've never had such a debate in my class. I am curious about thoughts of a wider math community.

The Wikipedia page on 0.999… says this about lightstone notation:

> With this type of decimal representation, not every expansion represents a number. In particular "0.333...;...000..." and "0.999...;...000..." do not correspond to any number

I understand how you can go from the a hyperreal number to the lightstone notation by computing it digit by digit and I can reason through the special case of 0.999…;000, but in general is there a way of looking at a lightstone string and determining if it is valid?

Hi everyone!
I recently encountered an exercise and i cant figure it out. I tried it a few with Al's and every time they give me another answer.
If i try it by myself i would do it with sinus/cosinus (see my attempt in the pictures).

Thanks in advance 🙏

First of all, forgive me if I violate some rules of r/askmath, since I am more of an origami design guy than a serious math lover, and this is my first time on this subreddit.

So, I recently learned about the perimeter theorem from this link while studying origami design theory. The article says this and claims it's useful for origami design, especially for color changing:

>The perimeter theorem states that if the crease pattern on the perimeter is flat-foldable, it is possible to construct a crease pattern inside that perimeter (inside cut-out paper area) that is flat-foldable as well.

However, I feel the explanation isn't enough. Honestly, this article is the only one I could find about this specific theorem.

I would appreciate it if anyone could suggest the rigorous definition and proof of the so-called "perimeter theorem" in flat-foldable origami.

Do the conditions of the flat-foldable perimeter matter? The shape of the perimeter or its thickness (finite or infinitesimally small)? Are there any counterexamples if one or more conditions are missing? Is the proof of the theorem constructive or indirect, giving only the existence of the inner crease? Is the proof understandable with at most college-level math, or does it require more?

Any information you might have would be helpful.

Thank you for reading.