Fractals are cool !!
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This is topology, no? I have 0 experience with it outside of quick fun facts. Couldn't help ya here 
That's ONE BADASS FUCKIN' FRACTAL!
(yes I know that's not a mandelbrot set, shush)
(nevermind, judging from the file name it might be. curse you, fractal geometry!)
It seems obvious, but to actually prove it I'd have to look up the formal definitions of all those terms and rub them together until a proof happens, and I was never good at that.
Intuitively: E^\mathrm{o} is E without its boundary points, so its complement (E^\mathrm{o})^c is everything outside of E plus those boundary points, which looks suspiciously like the closure of E^c.
But for a formal proof I'd have to talk about unions of open sets and intersections of closed sets and whatnot.
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Ah yes, I'm all too familiar with proof by "trust me, this works" 
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Yeah on my second attack on the problem I actually was able to prove it but not sure if its 100% correct
Prove that \overline{E^c} = (E^\circ)^c
\textbf{Proof:}
Suppose x \in \overline{E^c} = E^c \cup (E^c)'. If x \in E^c then x \notin E so x \notin E^\circ hence (E^\circ)^c. If x \in (E^c)', then every neighborhood N_r(x) contains q \neq x with q \in (E^c) (so q \neq E). Thus every neighborhood is not a subset of E, so x is not an interior point of E, so x \notin E^\circ hence x \in (E^\circ)^c. Thus \overline{E^c} \subseteq (E^\circ)^c.
Now suppose x \in (E^\circ)^c, so x is not an interior point of E. If x \notin E then x \in E^c so x \in \overline{E^c}. If x \in E we know that for all neighborhoods N_r(x) we have E \subset N_r(x). So, there exists a q \in N_r(x) such that q \notin E (so q \in E^c). Since q \notin E then q \neq x, so x is a limit point of E^c. Hence x \in (E^c)' \subseteq \overline{E^c}. Thus (E^\circ)^c \subseteq \overline{E^c}, so \overline{E^c} = (E^\circ)^c. \square
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technically both the image and the fractal equation given in the song are julia sets rather than mandelbrot sets so the song is doubly relevant
from a hacky julia set renderer i made a long time ago :3
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me when I still dont understand cardinality 
(send help pls)
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at most a countable many ,,,
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neeeerrrrdddsss
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Numbers got me like
real analysis is killing me
I used to like math to the point i made my own procedures, i discovered by myself repeated addition at the age of 7 and i smoked kids using calculators just by mind calculation, but teachers couldn’t handle me and would make it slow, boring, and any type of creativity i tried to have them graded less or even 0s on my exam exercises because of “not following the procedure” even if the result was correct, so i stopped to care and eventually went from a 10 math student to a 1 math student.
But one of my favorite “math is mathing moment” is when i made my own procedure for the arithmetic rule on class while everyone else was still doing some baby exercises that i finished early.
it is basically a justification of why the arithmetic rule works but without fractions, which makes it easier to do by memory and on paper.
I am not sure i remember it fully, but if i am correct, for obtaining the middle of 2 numbers, the bigger B, and smaller A, bigger minus smaller gives the total distance from B to A, divided by 2 for obtaining the half of it, and then it is “added” to the smaller digit, making it the middle of the 2 previous numbers, here are 3 examples all with different symbol combinations, ++, +-, - -.
I always think is funny that to annoy my math teachers for not giving a fuck the last years that i approved math exams i went as a “roge mathematician” because i would deny completely to use a calculator, making my teachers follow complex calculations made by pencil and paper that were more than 1 or 3 pages both faces in total, completely ruining attleast 1 hour of their day every month, and if they made a mistake on their corrections it would be another extra 30 minets.
Conclusion, math is fun as long as you control your pace for learn it and not some dinosaur who doesn't care at all about your personal enjoyment.
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I've been thinking a lot lately about whether or not there's anything that can be done to help my severe dyscalculia - even basic operations are becoming harder to do in my head lately, and I feel like I'm running entirely on memory for certain basic calculations I have to do on the daily. Kind of scary.
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Don't let numbers make you scared of maths
hmm. I don't have personal experience with dyscalculia [1], though i know of a person that got through a physics related PDH with it. Though what you struggle with can differ between people, it's worth considering.
If you struggle specifically with arithmetic [2] i would't let that bother you. I personally still don't quite grasp the 3,6,7,9 multiplication series without pen and paper or a calculator to offload working memory; but i got through a computer science degree without problem.
I largely disliked maths before choosing engineering, because i struggle with numbers. But if i could go back in time i would have chosen a degree with heavier maths. Maths ended up one of my favorite subjects and it became more and more fun the higher i went.
Abuse tools
Contrary to what i was told in school, we will always have calculator around. I was not allowed to use a calculator when in school, but i were given access to multiplication lookup tables and slide rulers. We learn to do things manually mostly to properly grasp the concepts, not because that's how it's actually supposed to be done. So there is no shame in offloading mental memory with paper, tools, or math machines.
Tiny steps
Struggling with algebra can be harder if you're aiming for higher maths, but often the expectation to write down the work helps. You can do smaller steps at a time to not loose track. Much or algebra can be split down into almost arbitrarily small steps at the cost of paper and time.
Tools all the way down
For work related math, consider learning a analytical mathematics language. Turns out clever people figured out how to make a computer do symbolic mathematics a few decades ago, and that has been the norm in actual engineering and physics for years. Personally i'm familiar with Matlab and wolfram mathematica [3]; but there are many other free tools (many swear by desmos).
Approximate is the real maths
finally, consider learning approximating. This is a big thing in engineering fields, where quick sanity checks can catch a vast range of common errors [4] in the much longer proper calculation. Just knowing if the answer is positive or negative, or within a rough 10x range is vastly helpful.
167 * 182 = ... ish 16 * 20 * 100 = 32000
if my calculation now ends up with a extra zero, or in the 50000 range, i know something may be up. A proper calculation may be a page of letters, but require just 1 minute to approximate.
How the heck is this relevant?
To be a bit more out there; I took a year before starting uni to study art and illustration. Questions can be raised about how much it actually helped my art, but i noticed a spooky amount of improvement in my algebraic skill. Drawing trained my visualization, which helped me keep track of larger expressions in my mind at once. This is not scientific conclusions with a subjective sample of one; but it was certainly interesting.
Summary of alot of yapping
So to summarize my yap; Doing math on mentally is unrealistic anyway; so abuse tools to your hearts content. Be it just extra paper, lookup tables and formula sheets, slide rulers, calculators, or formal analytical math software. We only to do math manually to learn the concepts in the firstplace.
Though i am dyslectics and regularly loose track of a minus sign or variable at times ↩︎
numbers, operators, division and multiplication ↩︎
mathematica is the formal language and tool behind wolfram alpha which is free website. If you ask wolfram alpha to solve a problem, you can actually view the mathematica equivalent syntax it translated to before it solved the problem ↩︎
a minus sign again! Gaaah! ↩︎
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i like to get creative with my notation let me know what you think of what i came up with to represent rotation
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I quite like the polar vector notation
r \angle \theta
Failing that, vector meshes better with my brain than complex numbers. (so many years yet e^{2\pi i} still gives me occational spooks when it does something weird)
r \mathcal{R}_\theta\hat v = \mathcal{R}_\theta\vec v = \mathcal{R}_\theta\begin{bmatrix}a\\b\end{bmatrix}
rotation matrix for 2 dimensions
\mathcal{R}_\theta = \begin{bmatrix}
\cos(\theta) & -\sin(\theta) \\
\sin(\theta) & \cos(\theta)
\end{bmatrix}
Eeh... has latex rendering pooped itself? Mabie just me.
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