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I liked the first math thread, but that hit the bump limit so I'm making another one.

Here is a neat tool posted in the previous thread that shows you how to do geometry the way the greeks did.


Here are a series of MIT OCW courses that will help you learn calculus:



Full MIT OCW Mathematics catalog:


Attached is the a Numberphile video about the seven bridges of Königsberg because I dunno what else to attach to this OP.
196 posts and 42 image replies omitted. Click reply to view.


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Let's discuss Max Tegmark. Did anyone know him before they read my post? The most important thing I would like to talk about is one postulate that goes like this: "all mathematically consistent structures exist in reality." This statement follows from the principle that the universe is entirely composed, "woven" from mathematics. Now, I have a question. Does this mean that somewhere there must be a universe in which an exact copy of me was lucky enough to live in a virtual reality with full immersion, in which the world perfectly adapts to the needs of each user? In the film Matrix, the virtual reality was the same for all people, but I'm talking about a private and individual, one that was created only for my copy. Because why not? Imagine what will be possible in a thousand years. I'm sure that computers and AI will be so advanced that everyone will be able to leave this world for the sake of diving into a simulation in which they will get everything they want. I even think that people from birth will immediately connect to the simulation, because this shitty real world will not be needed by anyone.
Generally speaking, does this mean that there are all sorts of universes that can only be imagined? I do not quite understand what do you mean "consistent". Contrary to what? After all, in another universe, there may be other mathematics that will not be comparable to our own, then what is it about? This is a really interesting point. I first started talking about virtual reality and my copy, because I am very dissatisfied with my life and would like to know if my second self (or third, or fourth, or fifth) exists in another world that gets everything it wants.
I came here because I don't understand mathematics at all, and I need the opinion of someone who has a deep understanding of the subject and can tell me how right Max Tegmark is.


How many here tried to study mathematics at uni and failed out? Count me in, I wonder who else went through the same experience.
It was just too intellectually hard for me personally.


STEM in university, at least at the undergrad level, is all about doing problem sets over and over again until the process of solving them becomes automatic and ingrained. The assignments and exams tend to be recycled from year to year with minor variations, so if you can confidently solve all previous problem sets, you will very likely do well on the course.



I think a major issue when it comes to "pop math" is the assertion that the universe is based from mathematics, rather, it's predominantly the other way around. We've taken reality as an inspiration for conjectures, theorems and sub fields. Euclidean geometry arose from observations of the real world, where you can consider geometric objects as being a set of points within a vector space having 3 orthonormal bases. From there came Euclid's postulates, and we discovered that the parallel postulate doesn't actually hold in spherical/hyperbolic geometries (like Earth) somewhat recently (it took Gauss, arguably the smartest mathematician in history, to figure it out). The need for calculus arose mainly because physicists wanted to explore the notion of instantaneous rates of change. Likewise Hilbert spaces (generalizations for euclidean spaces) were developed so that QM could work. Formulae like e^pi*i, for example, are more a consequence of notation rather than some fundamental property. Like 3blue1brown explains in the video, a substantial amount of mathematics is a consequence of reality, and there isn't much of a point to theorems that aren't useful (of course many "useless" ideas become incredibly useful in the future, like graph theory).


Perhaps but the amount and difficulty of these problem sets/past exams were too hard for me to overcome(topology and number theory kicked my ass).
Why argue about this though?


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Damn, am I dumb. Even the first chapter of "Basic Mathematics" is too complicated for me. I'll have to start relearning arithmetic because I don't remember how negative numbers work nor how to multiply effectively, seeing as I just use a calculator for everything these days. According to Khan academy that material is pre high school level. Looks like I have a long way to go.


I accidentally came across the following statement in one of the encyclopedias, but the proof was not there: adepts of abstract algebra can prove that 2+2=1. Can someone explain to me how this happens?


I don’t know about 2+2=1, but I’ve seen some for 1=2 and other such stuff. Usually things like these come down to disguising an error mathematical logic. For example disguising a division by zero by using variables, or applying both a 2nd power and square root to a negative value and assuming it comes out unchanged.



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I'm trying some competition papers online, the first question is pic related. The answer says that f(x)=1 for all x in Q>0 (aka f[x] = x^0), but surely that's not right? Can't f(x) also be √x? That leaves x√y = x√y, unless I'm missing something.


It can't be √x because the values of f(x) should be rational.


Oh, that's disappointing. God, I'm such a retard.


A Proof That The Square Root of Two Is Irrational


I took an undergrad graph theory course last semester and absolutely loved it! I think I want to do research projects in the topic to consider whether I'm fit for it at grad school. We followed Introduction to graph theory by Douglas West in class, it's an amazing book. You can also check out Reinhard Diestel's book, it's also nice.


Oh this was an IMO problem: https://www.imo-official.org/problems/IMO2018SL.pdf

Pretty neat solution and relatively easy to follow. I would have never been able to come up with that myself though, and considering it's an IMO problem I doubt you should get frustrated if you overlook something.


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i hope to read them one day.


Challenge accepted.
Seriously though, thumbed through Bourbaki and seems pretty decent. I might read it.
I can't imagine who in his right mind would read principia though. It looks like it was written for a computer (it kinda was). Perhaps one could reframe the whole work to be input to Coq or something like that. Of course imstead of sets it would require Type theory, which would be an improvement as a matter of course.
Makes me wonder about the nature of such work. Is it to be read by humans? Or perhaps only incidentally the way a computer program is?


Hey, Vsauce. Michael here.

Skeletons are scary and spooky, but you know what else is? Niggers.

According to the U.S Justice Department, in 2006, 32,443 succubi of Caucasian origin were raped by men of African origin.

That same year, the number of African American succubi raped by Caucasian men… was… zero.

In fact, 90% of all interracial crimes in the U.S. are committed by blacks… against whites…

So what if all blacks were to… suddenly… disappear from the U.S.?

Murder would go down 49.7 percent, welfare recipients would go down 40 percent, SAT scores would go up about 100 points, the average IQ would go up 7 points, and AIDS victims would go down a staggering… 67… percent.

Significant changes for race that only makes up 13% of the population.

In biology, races are distinct genetically divergent populations ‘’within the same species’’, with relatively small morphological and genetic… differences.

Populations can be described as ecological races if they arise from adaptation to different local habitats or geographic races when they are geographically isolated.

However, if sufficiently different, two or more races can be identified as… sub… species.

So how long do two races have to be isolated from one another before they're considered separate species?

Earlier this year, Archaeologists found artefacts in a cave on Western Australia’s Barrow Island dating back more than 50,000 years, making it Australia’s earliest known site of human occupation.

In contrast, the domestication of the dog began just 15,000… years… ago.

This distance, some several thousand miles between the mainlands of Australia and Asia also caused another kind of distance.

Genetic… distance.

Genetic distance is a measure of the genetic divergence between species or between populations within a species.

Dogs and wolves have been found to have a shorter genetic distance between each other than Negroids and non-Negroids. The distance between Eurasians and Africans being even larger than Homo Sapiens and Homo… Erectus.

Well, it kinda makes sense. I mean, can two races with diverging skin tones, eye colors, behavioral patterns, intellectual and athletic capabilities even be considered the same subspecies? Is it even a question? Is it even a question you're… allowed to… ask?

In October 2007, geneticist James Watson, best known as one of the co-discoverers of the structure of DNA, and the familiar double-helix model we all had to learn in High School, was lambasted by the scientific community for a response he gave in an interview regarding the divergence of intellect between geographically isolated populations.

"There is no firm reason to anticipate that the intellectual capacities of peoples geographically separated in their evolution should prove to have evolved identically," James wrote. "Our wanting to reserve equal powers of reason as some universal heritage of humanity will not be enough to make it so".

The response resulted with a suspension of Watson's administrative responsibilities and was forced to… cancel… his book tour. And tragically, in 2014, Mr Watson auctioned his Nobel Prize medal he won in 1962, stating that "no-one really wants to admit I exist".

Harsh. But people tend to react this way to ideas that go against their own personal world view and… cognitive… biases.

Confirmation bias. It's the tendency to search for, interpret, focus on and remember information in a way that confirms one's… own… preconceptions. This effect, stronger for emotionally charged issues, warps your interpretation of data in a way that keeps you from… being… wrong. There is something fundamental in our minds that makes us hesitant to question ideas that we've… already come to a conclusion to.

We all do it. But who could blame us? That's just part of being a member of the human… species.

And as always, around blacks, never relax.


I enjoy this pasta.


Why do whites never have ghettos?


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Is this image trying to say that S is not a subspace of R³ because these reasons?:
>S is a set in which element 1 is any arbitrary real number, element 2 is any arbitrary real number squared, and element 3 is zero.
> multiplied by c is not necessarily the square of an arbitrary real number. In order to be in the set S, the second element must be a square of an arbitrary real number. Because the second element, when multiplied by an arbitrary real number may or may not be square after the operation, it may fall outside of S and so under rule one it is not a subspace of R³.
> plus another arbitrary real number squared is not necessarily another square. Because of that, it may fall outside of the set S and thus must not be a subspace of R³ under rule 2.
The image is a little ambiguous and my math skills aren't very good, so I want to be sure I've got it correctly.
When I say arbitrary real number or a I do not mean that there is a consistent value between them. a in the first element could or could not be different from the a in the second element, and so on. Just being clear.


>>S is a set in which element 1 is any arbitrary real number, element 2 is any arbitrary real number squared, and element 3 is zero.
I meant to say that S is the set of vectors in which element 1 is any arbitrary real number, element 2 is any arbitrary real number squared, and element 3 is zero.


Correct. And if the a in the first and second places are not to be the same, use different variables. Use a and b instead.


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Studying math by programming


Ehm… that's not stuyind math, that's studying programming, or, at most, numerical analysis, for wich you don't need to understand the maths behind the algorithms.


I have a very poor understanding of 3x(n>1) vectors and matrices in any other context than as an array of numbers that may or may not represent points and those points may or may not be positions. For 3x3 vectors I can intuitively imagine constructing something like a right prism or triangular pyramid, but more complex ideas are difficult to imagine in any other way than "yeah it makes sense."
So, what are some interesting concepts that can be represented by 2 or more 3-dimensional vectors?



Yeah it's the resource I used. It's good, I think I'm just too much of a brainlet for linear algebra. The calculations and formulae themselves are trivial but I can't visualize anything but NxM matrices where N<4 or transformations that aren't from ℝⁿ to ℝⁿ. It's making going any further a real slog when I have to imagine everything as abstract numbers, especially n to m-space transformations when they're used in nearly every problem.


It is kinda hard to help if you do not provide specific examples. Matrices are basically nothing more than arrays of vectors. What those vectors represent varies depending on the use.

I would venture to say that most people do not understand higher dimensional transformations, though. You know know what works and apply it. Only thing that is easy for me to grasp about dimensions > 3 is that they contain more space per unit cube.


As an example imagine a 3x3 matrix multiplied by a 3x2 matrix. The product is a 3x2 matrix. What is a way to physically interpret the loss of a vector and preservation of dimensionality? 3blue1brown explains that this can be thought of as a 2-dimensional plane mapped in 3 dimensions, but that doesn't explain the process of the transformation. What is a physical process by which a concept that can be thought of as a matrix with 3 3-dimensional vectors be transformed into a matrix with 2 3-dimensional vectors? It can't be projection, because projection would lose a dimension while preserving vectors.
Another example would be multiplying an MxN matrix by an Mx(N+z) matrix. How can a transformation like this be interpreted as a physical phenomenon? I understand they can be interpreted however is necessary for the application, but as I said I'm not okay with thinking of matrices and vectors as arrays of numbers and transformations as arcane functions to use on those arrays to get more useful numbers. I'm not really asking "what are matrices/vectors", but instead "what is at least one way that these transformations can be explained as a physical phenomenon." Knowing some real physical phenomena that can be represented as a linear transformation that increases or decreases the dimensions or vectors of something will make intuitively understanding linear transformations a lot easier.


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I think it is kinda hard to understand because matrix notation is backwards. You can think of matrix multiplication A*B = C as a transformation of the unit vectors in B's space. A tells you how to transform the unit vectors in B's space (standard notation is to every unit vector transformation be represented by a column. col1 = X_t, col2 = Y_t, col3 = Z_t), and the result C is where the vectors are transformed once you "stretch" the unit vectors by A. Example shown in my screenshot. That is me stretching the X unit vector by 2 so all of the X values are multiplied by 2.

If A is a 3x3 matrix and B is a 3x2 then B are the two points you want to transform using transformation matrix A. If A is 3x2 and B is 2x2 then A is describing which plane the 2d points in B should go. 2nd attached pic is me putting the points on the XZ plane. 3rd is putting the points on the XY plane.


Yeah, I can understand the mechanics of linear transformations when used in rotations, translations, projections, or other abstract numeric operations, but seeing them as physical processes is the part I am having difficulty with. I suppose by your answer it's really only going to get more abstract from here, isn't it?



In that case the matrices aren't really anything special, they are just a way of rewriting the equations in your system, if the system happens to be linear. You can probably think of them as a stretching and squishing of vectors in the state space, but what that actually means is entirely dependent on what you are trying to model. A lot of the time the columns do not actually mean anything other than that is how it must be for your equation to be represented in matrix form, though. I'm sure someone who is crazy about number theory could probably ramble on about it is related to pascal's matrix or some other obscure shit, though.


Yes. It's called recreational mathematics.


I really am delighted by aspects of information theory and often basic number theory. It fills me with a sense of wonder learning about some of this stuff. Like I am learning actual fucking witch craft; Being able to pass along secret knowledge. Being able to touch the boundaries of order right up until the bounds of noise. There's so much useful stuff hiding under the surface and its all so elegant.

The sad part is practically none of it is taught in schools. Parts of it might be found in undergrad degrees. But mostly maths in undergrad degrees are about turning you into a good little robot wagie who can churn out mundane results like a calculator. In contrast: research math is all about abstract bullshit that will never see the light of day and often has contradictory 'results' anyway. So while you might cover some cool stuff there. It's definitely not the focus. The advantage you have doing this on the side is being able to learn the actual cool shit.

My favorite result is 1 + 1


File: 1617801015185.pdf (79.21 KB, lebesgueMeasure.pdf)

I wrote up some notes on Lebesgue nonmeasurability. I think I tried to make a separate thread about it some time ago. I'm still stuck thinking about why it's not a bigger deal than it is. In my mind it should be up there with the other 'popular' math paradoxes–or at least up there in popularity with stuff like Banach-Tarski or the Klein bottle.


Thanks, that was useful. I'd been wondering for some time what Lebesgue integration was, and never bothered looking into it.


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Practicing writing proofs. Problem is from Mathematical Proofs: A Transition to Advanced Mathematics, Fourth edition, by Chartrand, Polimeni and Zhang.


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Made it simpler.


This one is hardly readable, but it all boils down to a matter of style. I understand your phrasing in term of P(x) and Q(x) is a didactic device from te book, but you ought to state it more directly using prose.
The second step in your proof obscures the links between the data. Instead say x^2 lies in the interval (0,1) and -2x lies in (-2,0).
Lastly, on step 3, a double negation is always harder to read than a positive statement.
Actually, you can't use k for both numbers because you're constraining your proof for consecutive numbers (eg 4 and 5), so the right hand only
applies to such pairs of numbers.


Still, he's figured out how to use LaTeX and align things well, and he's actually got the idea of a proof down in that the proof is technically correct. For someone starting at this, he's starting out pretty good.

You're right about the second point on >>57945 , this proof is assuming consecutive integers.


(I just remember when I was first starting this and got shat on to high heaven and I pretty much just quit the field entirely, so I wanted to add some positive points)


Oh I do not mean to discourage anyone. But i myself found myself wanting someone to critique my proofs lest I miss anything important.


This isn't a proof as it does nothing to demonstrate that all even integers are equal to 2k where k is some integer, and the same for odd integers and 2k-1


This is 500 pages of chemical kinetics theory, examples, and data. I want to do my capstone on something similar, so I will dig through this to understand the basic theory, and retrace the author's steps in terms of math/computer stuff


It's a bit like me. I was also trying to study maths for a while. I didn't put enough effort and I had to quit.


The paradox at the heart of mathematics: Gödel's Incompleteness Theorem - Marcus du Sautoy


>it took Gauss
wasnt that Lobachevsky?


Well, it's assumed that Gauss already knew about it since when Bolyai developed it and his father, Gauss's friend, sent a letter to Gauss to explain about his son's discovery and Gauss reply was not kind:
"To praise it would amount to praising myself. For the entire content of the work…coincides almost exactly with my own meditations which have occupied my mind for the past thirty or thirty-five years."
And that he already knew about it. Gauss was an arrogant person, btw, he didn't even wanted any of his sons to do mathematics because he knew they would never be better than him. Anyway, Lobachevsky is another person who developed non-euclidian geometry(around the same time as Bolyai) and it's believed that Gauss after learning Russian(at an old age), decided to read Lobachevsky's works which he became interested(it's said that he learned Russian to be able to read this very work, but I have read a russian paper that disagree with that and has valid arguments using Gauss's letters and journal).
Now, the interesting part is that by the time they accepted non-Euclidian Geometry, they realized that they have been using non-Euclidian Geometry for a long time in astronomy: Spherical Geometry. Which is why for hyperbolic Geometry we know who developed it: Bolyai and Lobachevsky; for Spherical Geometry, you are not going to find "the man who started it" since it has been developed since Ancient Greek, they just never realized it was non-Euclidean Geometry.

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