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/Σ/ To discover what was dearly missed in schooling, this general will serve as a playground. >What do you mean? Why talk of mathematics? >I hate mathematics. https://www.maa.org/external_archive/devlin/LockhartsLament.pdf >Okay, but I don't know where to start. I still hate what I was put through. I'm just as lost, but I hope this is what this general will serve as. Don't be afraid to ask any 'stupid' questions here. Recount of your earlier math experiences if you wish. Maybe I should've tried better in improving the starting message, mhmhm. Do we want to make a books list? Anyhow, I'm starting this thread because I'm looking for supplemental material or a more advanced text that pairs up nicely with Axler, aka develops stuff without assuming that the field is R or C. Surely there are some math majors here that have some advice :)

Would you say Analysis is the core/essence of modern mathematics?

Hello Lainons, I am currently taking Calculus and am finding myself disappointed. Problem is, just like Lockhart said I have wasted that time memorizing random formulas and not really learning any math. I don't want the same thing to happen. Is there any resources that I could use supplementarily that teach Calculus conceptually and the underlying concepts in much greater detail. I'm also hoping to overcome some of the "math anxiety" that I faced during my social formative years. I'm not planning on taking anything after this but I want to learn math. tl;dr: I want to get good at Calculus but I don't know any of the underlying math required for it. >don't even understand quadratic formula What am I in for lains?

>>21256 Learn by going through Michael Spivak's calculus, or another beginner's analysis book such as Abbott, Tao or Pugh (or Rudin if you're brave enough). Consult Sheldon's precalculus or some other such resources for what you're lacking. Velleman or Hammack for introduction to proofs (although it's not really needed).

>>21248 How do I get into category theory as a computer scientist? I have little background in abstract math, but I've done some type theory before and have worked through Spivak's Calculus. I've been reading Milewski's 'Category Theory for Programmers', but is that the right path?

>>21259 Thanks Lain, Spivaks book is pretty easy to find so i'll start with that

>>21256 >don't even understand quadratic formula the quadratic formula is just a general way to complete the square. It helps to see how it is derived, I don't remember anyone teaching me this in school and I can't believe it's not standard to teach kids how to derive formulas they use. Suppose ax^2 + bx + c = 0 then x^2 + (b/a) x + c/a = 0 and it is not hard to show using the FOIL method that (x + b/2a)^2 = x^2 + (b/a) x + b^2/4a^2 so in the first equation, you can subtract c/a and add b^2/4a^2 from both sides then you end up with x^2 + b/a + b^2/4a^2 = (x + b/2a)^2 = b^2/4a^2 - c/a for convenience, you can multiply both sides by 4a^2, and there is a nice identity that for any numbers z and w z^2 times w^2 = (zw)^2, so you get (2ax + b)^2 = b^2 - 4ac then you take the square root of both sides, but importantly, if x^2 = z then (-x)^2 = z, so we use a plus and minus sign to indicate that there are actually *two* roots. 2ax + b = +- sqrt(b^2 - 4ac) subtracting b from both sides and dividing by 2a we get x = (-b +- sqrt(b^2 - 4ac)/2a which is the quadratic formula.

Excuse me for the dumb question but is a dimension just any amount of a set of values with rules attached? For example in the first dimension {x} and in the second {x,y}. you could say in 1d that x+2 where x is the previous number and that a line goes where x is, and as x gets closer to the center the line gets longer like left oekaki. Now supposing you made a rule that in the first line it follows a very specific color pattern and the same for all the other lines. Lets say this color pattern happened to be the same pattern as the one used in an anime girl. Then couldn't you also do something like this with 3d?

>>21261

>>21292 In any finitely generated vector space there is a minimum number of generators that span the space, and there is a maximum number of linearly independent vectors. These two numbers are equal. This common number is the dimension of the space.

>>21295 Thanks a lot for this post, I cannot express my gratitude enough in this format.

>>21292 I don't really understand what you're saying but I can try to explain what "dimension" means. There are actually lots of different meanings of "dimension" in math and they don't all mean the same thing. In a general sense though, the "dimension" of a space means the minimum number of coordinates you need to uniquely define a point in that space. So for instance, if you are in a two-dimensional space (such as a piece of paper), there is no way to uniquely define a point on that space using just one number. You can get close, but it will never be *exact*, no matter what you do there will either be two numbers mapping to the same point, or there will be points that are not mapped to by any number. However it is very easy to uniquely define any point on a piece of paper with just two numbers, with (x,y) coordinates.

>>21320 >there is no way to uniquely define a point on that space using just one number Take any bijection from R to R^2. https://math.stackexchange.com/questions/183361/examples-of-bijective-map-from-mathbbr3-rightarrow-mathbbr

>>21256 Use geogebra to visualize everything

>>21320

>>21330 >Take any bijection from R to R^2. such a bijection can never be continuous, which is a necessity for the coordinate system to have any geometrical meaning. I suppose I should have included that in the definition of what I was talking about. "continuity" roughly means that if you want to move the point you're referring to by a little bit, you can always choose a small enough change in your coordinates to get there. If the "coordinates" are discontinuous then you will have very nearby points with very different coordinates, which obviously makes things harder. >>21344 such an assignment is not unique. (0,0.999...) and (1,0) represent different points but map to the same number 0.999.... = 1. and before you tell me that 1 and 0.999... are different numbers, they are not by the typical definition of real numbers. If 1 is not equal to 0.999... then 1 - 0.999... does not equal zero, but if 1 - 0.999... = x > 0 then there is some k such that the k'th digit of x is not zero. However then 0.999...9 + x >= 1, where there are k repeated nines instead of infinity on the left. But since 0.999... > 0.999...9, we have that 0.999... + x > 1, so x > 1 - 0.999..., so x > x, which is not possible for real numbers. You can get around this by extending to "hyperreals" but in the end that will either not solve the non-uniqueness problem or will make your assignment equivalent to just listing two numbers in a different way.

>>21353 oh, I got that a little bit wrong, since your assignment is not very well explained. Instead it would be (0.111...,1.111...) and (1.111...,0.111...) having the same number 1.111... , you don't even need to correct a common fallacy to see that.

>>21355 or alternatively the point (0.999...,0) = (1,0) gets mapped to two different numbers 0.090909... and 1

>>21356 or (0.999...,9.999...) gets mapped to the same number as (1,0)

>>21353 >such a bijection can never be continuous Separately it is true that R and R^2 are not homeomorphic (in the standard way), but not only was there no hint of this in >>21320, which remains false, the notion of dimension does not require the notion of continuity. >I suppose I should have included that in the definition of what I was talking about. After stating "You can get close, but it will never be *exact*", which is false, you explicitly insist on the bijection condition with "there will either be two numbers mapping to the same point, or there will be points that are not mapped to by any number", showing that at the time you believed a bijection did not exist. More to the point, continuity requires a topology (open sets), but dimension does not. Dimension only requires generation/spanning and linear independence >>21307. A space with no specified topology, and therefore no notion of continuity, can still have dimension. So your attempted continuity save is also wrong. I will let >>21344 defend the interleaving scheme since he is the one who brought it up, there is a kink to defend it from, and with care it can be made to be formally correct. I will note however that all your attempted counterexamples so far have been false counterexamples. Either you simply misapplied the interleaving scheme >>21353 >>21355, or you are aware that 0.(9) is 1 but you insist on using them as distinct inputs >>21356 >>21357.

>>21353 (&al) >different points mapped to the same number (11,1) and (1,11) do (both being 1.111...) This is fixed by the new scheme which maps each point in the plane to a real in [0,1]. The issue here was that the location of the decimal place was lost. (X,Y) goes to the real whose base 10 representation consists of the following characters (line breaks added for clarity and not important): 0. <<number of digits before X's decimal place in base 10> in base 9> 9 <<number of digits before Y's decimal place in base 10> in base 9> 9 <1st digit of X in base 10> <1st digit of Y in base 10> <2nd digit of X in base 10> <2nd digit of Y in base 10> etc. The requirements stated in >>21320 are: >uniquely define a point in the plane using one number >no two numbers mapping to the same point >no points that are not mapped to by any number >inb4 there are multiple base-n representations of any number therefore this maps two numbers to the same point There exists an algorithm which, given the same real, will always produce the same base-n expansion. Assume I specified it's use. >inb4 0.1919009999999999... and 0.191911000000000... both represent (1,1). Only numbers produced in the specified way from a point are representations of points. For the above method to produce both implies the algorithm doesn't always produce the same base-n expansion. It is not stipulated in >>21320 that all numbers are mapped to a point. Example: >(0.314159..., 2.718281...) 0.191902371148125891... and breaks down as: >0. pointless prefix >1 0.314... has one digit before the deciaml place when written the normal way, one in base 9 is 1. >9 seperator >1 2.718... has one digit before the deciaml place when written the normal way, one in base 9 is 1. >9 seperator >0 first digit of 0.314... >2 first digit of 2.718... >3 second digit of 0.314... >7 second digit of 2.718... >... etc

>>21359 >but not only was there no hint of this in >>21320 yes, I literally pointed this out as an oversight of mine >the notion of dimension does not require the notion of continuity. dimension of a manifold, and the related notion of topological dimension, does. I was attempting to describe the dimension of a manifold because it is a case that I think is easiest to describe to a complete layman. The dimension of a vector space is different (but related). As I said, there are multiple versions of 'dimension'. Even for topological spaces there is a notion of fractal dimension that is different from the typical topological dimension, and there are all sorts of strange forms of dimension in group theory that have no physical analogue. It really makes no sense to ask "what is dimension", "what is the dimension of a _" is however a well-posed question. >After stating "You can get close, but it will never be *exact*", which is false, you explicitly insist on the bijection condition with "there will either be two numbers mapping to the same point, or there will be points that are not mapped to by any number", showing that at the time you believed a bijection did not exist. I knew that a homeomorphism did not exist, I forgot to mention that I was considering continuous coordinates. >More to the point, continuity requires a topology (open sets), but dimension does not. Dimension only requires generation/spanning and linear independence >>21307. A space with no specified topology, and therefore no notion of continuity, can still have dimension. yes. Every real vector space also has a natural manifold structure which has the same dimension as the vector space. >Either you simply misapplied the interleaving scheme >>21353 >>21355, or you are aware that 0.(9) is 1 but you insist on using them as distinct inputs >>21356 >>21357. I did misunderstand the interleaving scheme. However in >>21356 I show that the map RxR -> R is not functorial(one input maps to two different outputs if you change decimal expansion) and in >>21357 I show that (1,10) and (1,0) map to the same output. >>21360 >It is not stipulated in >>21320 that all numbers are mapped to a point. I suppose, I didn't want to say "bijection" because my audience supposedly doesn't know what that means. There are injective continuous functions [0,1] -> R^2 that have positive area, they are called Osgood curves and have many interesting properties. https://en.wikipedia.org/wiki/Osgood_curve I think this may indeed be an injective map RxR -> R, and an interesting one. However it is not an example of a continuous coordinate system for RxR using just one real number.

>>21353 >such a bijection can never be continuous true under the standard order topology. you can get continuous mappings with other topologies. not refuting you, just elaborating. >>21355 >(0.111...,1.111...) and (1.111...,0.111...) having the same number 1.111 presumably (0.11..,1.11..) -> 0.11.. and (1.11..,0.11..) -> 1.11.. the ones digit of the right coordinate becomes the tenths digit of the image. >>21356 >0.090909... and 1 it's not clear from the picture how terminating decimals work, but if you just stop interleaving once you hit a zero they map to the same thing. (0,0.99..) -> 0.99.. (0,1) -> 1 (0.99..,0) -> 0.099.. (1,0) -> 0.01 i'm assuming that's how it works because otherwise it wouldn't be a function much less a bijection.

>>21366 >(1,0) -> 0.01 sorry that should be -> 0.1 obviously >(1.11..,0.11..) -> 1.11 and this should be -> 1.011.. i think

>>21362 >I was attempting to describe the dimension of a manifold because it is a case that I think is easiest to describe to a complete layman. >manifold >to a complete layman And later: >I didn't want to say "bijection" because my audience supposedly doesn't know what that means So you go with manifolds but '''didn't want to say "bijection"'''. I believe this is what they call 'simply epic'. >However in >>21356 I show that the map RxR -> R is not functorial(one input maps to two different outputs if you change decimal expansion) "Change decimal expansion" is not an operation that is available to you, because the function's inputs are reals, not character strings. It is the function that decides what representation to use for the reals it is given as arguments. This is why you cannot use 0.(9) and 1 as distinct inputs. >and in >>21357 I show that (1,10) and (1,0) map to the same output. If you are going to go with implicit assumptions - continuity, manifolds - then interleaving schemes like >>21344 are meant to handle some convention of unit square and unit interval, so 10 is not available. Fractal constructions are cool.

>>21366 >it's not clear from the picture how terminating decimals work, but if you just stop interleaving once you hit a zero they map to the same thing. then you would miss some of the information. >>21376 >So you go with manifolds but '''didn't want to say "bijection"'''. I believe this is what they call 'simply epic'. I never actually tried to define a manifold, just describing the manifold dimension of Euclidean space. >"Change decimal expansion" is not an operation that is available to you the function is defined in terms of decimal expansion. I was proving that the function is not well-defined because using two decimal expansions of the same number, you can get two different outputs. You're not even being pedantic any more, you're just being wrong. >then interleaving schemes like >>21344 are meant to handle some convention of unit square and unit interval, so 10 is not available. the function as described didn't have an explicit domain or codomain.

>>21376 why are you so rabidly attacking this guy? Don't you have anything to contribute?

>>21381 They think they're a hotshot lol. Very arrogant.

>>21380 >the function is defined in terms of decimal expansion. This is false. The interleaving scheme is f:(0,1)x(0,1)->(0,1) and can be made to be a bijection with the adjustment descrbed in >>21330. The selection of the expansion is explicitly at the discretion of f, as >>21360 also tried to tell you starting at "There exists an algorithm which". Claiming that f is defined on character strings rather than reals is nonsensical because points in the plane are not pairs of character strings, they are pairs of reals, and these two differ essentially precisely because the real 0.4(9) is the real 0.5. You might be working under the misconception that the reals in [0,1] are the set given by the infinite strings of the regexp "0.[0-9]*", but they are not. See the link in >>21330 which offers two sources, "R.L. Wilder's Introduction to the foundations of mathematics" and "Theory of sets by Kamke (Dover reprint edition, p 31)".

>>21292 Mathematically there are as many dimensions as there are integers. You can think of each dimension as a span of all linear combinations of all vectors within a vector space. Spans can be though of as all the possible locations you could ever feesibly occupy given a linear combination, or it could be thought of as all the directions you could go in. I'm a bit of a newbie to this myself though so hopefully someone can provide a clearer definition. 3b1b has a nice video on this i think: https://www.youtube.com/watch?v=k7RM-ot2NWY&list=PL0-GT3co4r2y2YErbmuJw2L5tW4Ew2O5B&index=3

>>21380 >then you would miss some of the information. Sorry, I mean the terminating zero, like in the example I gave. >>21386 >Claiming that f is defined on character strings rather than reals is nonsensical The operation depicted in the picture is defined on sequences. What he (and possibly also you) are missing is how you'd actually formulate f:R^2->R. You would want to say something like >given (a,b) in R^2, f((a,b))=a1.b1a2b2... >where a1,a2,... and b1,b2,... are *any* decimal expansions of a and b respectively Of course you'd have to define what you're going to do with terminating decimals, and numbers bigger than 9, and negatives, but those aren't too hard to figure out if you think about it a bit. The key is the universal quantification. You write a function that works with any decimal expansion of a given real and then you let the choice of expansion be arbitrary in your function definition.

>>21392 >a bit of a newbie to this myself though so hopefully someone You have the right general idea, and you can take a look at >>21307.

>>21394 >The operation depicted in the picture is defined on sequences. The image the other anon posted is a hint for what interleaving means, not a definition. >how you'd actually formulate f:R^2->R >You would want to say something like >are *any* decimal expansions >The key is the universal quantification. >You write a function that works with any decimal expansion of a given real and then you let the choice of expansion be arbitrary in your function definition. You are repeating the same falsehood, and this time you also contradict yourself. You are correct here: "f:R^2->R". As you can see f takes a pair of reals. F does not take a pair of character strings. Decimal expansions are not the reals, and reals are not their decimal expansions. You cannot interchange these terms whenever you wish, because they are not the same thing. F takes a point in the plane - equivalently the open unit square - and produces a value on the real line - equivalently the open unit interval. These are all the restrictions on f. With the adjustment in >>21330 f becomes a bijection and falsifies your non-existence claim of >>21320. The contradiction is that you want to pass decimal expansions to a function that takes reals, which is a type error. I hope you are aware that decimal expansions are not the reals, and reals are not their decimal expansions, otherwise this is the source of your error.

Lainons I've started working on an isometric game and I quickly realized I need to understand the mathmatical underpinning to get things done in this life time. Does anyone have a resource or a set of resources on isometry so I know what I'm doing? Thanks in advance

>>21416 Just think about it. It's a grid, but each line of tiles is offset by some amount.

>>21433 I understand that much and I have naive solutions to certain things. But for common processes I'd like to have something more efficient. Take for instance telling which tile on the grid a mouse pointer is clicking on. I could cycle through each tile sprite and check for collision but I have the idea that if I could reverse the projection at the point the mouse cursor is on onto a grid of squares I could simply do a range check to get the coordinates.

I've had similar thoughts in the past and downloaded a bunch of PDFs, but truth be told there's so many things to learn in CS itself already that, as much as I want to dive deeper into mathematics, time constraints force me to ignore it unless necessary. Will this come back to haunt me in the future? I already feel as if I'm always running out of time, adding an entire area of study on top of it won't make things any easier.

>>21434 Hav fun

>>21436 Math time is never wasted time.

What are some good resources for learning about group theory? Any recommendations?

>>21436 >adding an entire area of study on top of it won't make things any easier I don't know how common this experience is, but I've found that learning math has made picking up the basics of pretty much any other technical field easier. On the other hand, learning math tends to be harder than learning any of those fields individually. Essentially you're trading difficulty for broad transferrable knowledge. >>21523 I like Fraleigh's abstract algebra book. It's very well structured and builds naturally through introductory group and ring theory.

>>21523 You didn't specify your level. You may either need an overview of algebra (introductory or slightly more advanced) If the guess of the above reader is right (you have no clue), then I would also like to add C. Pinter's book as a suggestion, and perhaps M. Artin's classic too

>>21535 either ... or something more specific to group theory such as Rotman. (sorry, somehow forgot to finish the sentence)

One of my recent projects was an RK4 integrator which takes a function f(x,y) for the slope at (x,y), and integrates a bunch of lines to make cool slope field plots. I had a lot of fun making it, and it's pretty neat to use. Maybe it'll be inspo for someone. https://pastebin.com/v4CS5Qwv You can either edit the FUNCTION macro manually, or use this little wrapper shell script ` # This script is a wrapper around plotter.c that changes the function for y', # recompiles plotter.c, makes a dir for the data if necessary, deletes any old data, and # cats the seperate forward and backward files together. The reason it doesn't # also run gnuplot is that loading scripts takes gnuplot out of interactive mode, # which means I can't set the scale of the plot for nice viewing. #! /bin/bash if [[ $# -ne 8 ]]; then echo "usage: ./plotter \"<function>\" <xmin> <xmax> <ymin> <ymax> <rows> <cols> <stepsize>" echo "<function> must be a valid C expression in x and y using math.h like cos(M_PI*x) + sin(M_PI*y)" exit 1 fi functiondef=$(echo $1 | sed 's/\//\\\//g' | sed 's/,/\\,/g') #this changes / from division to \/ for the next line sed -i 's,\(#define FUNCTION\)\s.*,\1 '"$functiondef"',g' plotter.c gcc plotter.c -o plotter -lm -pthread if [ -d "curves" ]; then rm curves/* else mkdir curves fi ./plotter $2 $3 $4 $5 $6 $7 $8 for row in `seq 0 $(($6 + 1))`; do mv "curves/curve_${row}_0f.dat" "curves/curve_${row}_0.dat" done for row in `seq 0 $(($6 + 1))`; do for col in `seq 1 $7`; do curve="curves/curve_${row}_${col}.dat" mv "curves/curve_${row}_${col}f.dat" $curve echo "" >> $curve cat "curves/curve_${row}_${col}b.dat" >> $curve rm "curves/curve_${row}_${col}b.dat" done done for row in `seq 0 $(($6 + 1))`; do mv "curves/curve_${row}_$(($7 + 1))b.dat" "curves/curve_${row}_$(($7 + 1)).dat" done `

>>21551 I forgot to include this header file which that code needs, the part that actually does the integration (lol) https://pastebin.com/NiFMXXts the main code should be called plotter.c and the header rk.h if someone messes with this I found it fun to have gnuplot open alongside a terminal, where I could run the script repeatedly to change the range or resolution then update the plot in close to real time

If you have 10 divided by 0, and division is just multiplying a number by x to get to another number, in this case 10, that means 0 amount or times of x is equal to 10, and since x does not exist because nothing would make it equal to 10, it doesn't exist. But what say you make a limit l = (l * x = 10) as l approaches 0. This means also that x is equal to 10/l, so l*(10/l) = 10. Multiply everything by l and you get l(l*(10/l)) = 10l. You are left with 10l = 10l. how do I force 0 to exist?

>>22361 >This means also that x is equal to 10/l, Until you actually go to the limit, while l is still nonzero, this is true. As soon as you actually go to the limit it becomes not-even-wrong because l no longer possesses a multiplicative inverse. >how do I force 0 to exist? You bootstrap the naturals as the free monoid of one generator, and assign the name 'zero' and the symbol '0' to its neutral element.

How do ye take mathematical notes? Currently I only take notes on definitions and theorems, without any concern for heuristics, (e.g. when dealing with universally quantified conditionals prove the contrapositive, name an instance of the existential quantification, and then assume the antecedent) or proofs (my personal proofs I keep on file, and I simply follow along with others' proofs). My typical system for notes on other subjects involves taking bibliographic notes and then compiling them into smaller notes where I draw my own conclusions. This feels sort of odd to do in mathematics because there are no real conclusions. An alternative I've recently been considering is to create notes with intuitive interpretations of definitions, and the missing heuristic mentioned earlier, but at some level I worry about this systemic opposition to rigor. Any thoughts on this approach?

>>22865 >I've mostly stopped taking proper notes when I study math because I always tended to reference the textbook instead of my notes when reviewing. Now I keep a giant notebook on my desk and use it as an extension of my working memory. I intend to simply throw it out when it fills up. That's a very interesting approach, it does make sense, and this is how I treat many of my notes. >I'm pretty good with latex at this point and if I have a good template I can type it about as fast as I can write, but managing tex files and rendered pdfs gets messy and seems like overkill when the notes don't add much beyond what's already in the textbook. Drawing pictures is still an issue but I've been getting better at using gnuplot so maybe that can be solved some day too. What I personally use for this is org-mode in Emacs with org-latex-preview (C-c C-x C-l) to preview the latex in document, and LaTeX-math-mode so that I can type `e instead of \epsilon, etc. I use org-noter so that my notes are linked directly to the textbook page. I really haven't gotten into visualization as much as I should, even by hand but I have heard of people using PGF/TikZ, and gnuplot as you mention. Overall I'm not very satisfied with my methodology either, the only benefit over the textbook I can think of is searching, and more easily scanning for related theorems and definitions due to the lack of proofs and narrowing. >No need to copy down anything specific since the book is easy enough to reference, but writing forces you to organize the material in your mind and can expose gaps in your understanding. Thanks for your feedback on the ideas, I need to consider more the prospect of using the textbook directly as the reference or using something like the mentioned compendium further. >>22867 >I usually make notes of important-looking definitions and theorems & their proofs. Plus if an example is nontrivial enough I'll work through it in full detail in my notes. This is interesting as well. Perhaps I should give more thought to including proofs in my notes.

Does anyone here use structured proofs, as defined by Lamport? I stumbled upon this methodology while looking for a proof assistant which was viable for work in mathematics (which I could not find, if you find one viable do let me know). I doubt the methodology would solve many of my errors since these are typically relatively simple things like changing the order of some quantified variables, or not exhausting every possible case (I'm only a student). Regardless the argument against the use of English in proofs as we've eliminated it from the expression of equations does resonate with me. English seems to mostly operate as glue for what is meaningful in a proof or to cite some theorem. The use of prose does seem to discourage the proof of lemma where they would distract from the overall message of the proof, and can not be easily ignored as in a structured proof.

>>23304 Thanks for the links, those are interesting. I came to mathematics by way of formal logic, so I used to write proofs in a very terse way kind of like those examples (but with worse structure and more symbols). My professors told me to cut it out, so I started writing proofs in a more prose-like way, but I'd like to get back to something more elegant. I think a good proof is a bit like well annotated code. The symbolic parts express *what* you're doing while the prose expresses *why* you're doing it.

>>23309 That image is ugly to look at, though. PositiveReal instea of |e| > 0? Openball(a,e) instead of just B(a,e)? and so on. Structured proof is nice, but the fact the author only showed simple two or so lines proofs(in prose, that is), I can't say that works well when you are trying to prove something that in prose took over 129 pages(fermat's last theorem). Overalll, mathematicians are a lazy bunch when it comes to writing proof("left as exercise for the reader"), so you are not going to find that in this field.

>>23309 >I further liked the argument that it's not really feasible to at once make an intuitive English description of your proof, and a formal rigorous proof at the same time, but rather they have to be separated out. I'm not entirely convinced. I do think that this approach has the benefit of forcing the writer to clearly explain their intentions, which is something that is often neglected. However, I think this can (and perhaps should) be done inline. I've got a topology exam coming up, maybe I'll give some structured proof stratagies a shot and let you know how it goes. I won't write anything like that example pdf but I've always appreciated the elegance of a sequence of steps with clear, brief justification for each one, so I'll see if I can do that much. That's one thing I wish mathematicians had borrowed from logicians.

>>23335 oh, I'm sure I *implicitly* used LEM all over the place. I just avoided using proof by contradiction or contrapositive.

>>23340 >I just avoided using proof by contradiction or contrapositive. Oh, well by coincidence I guess I did too then.

>>23340 Do you need LEM to construct the reals? I thought I read somewhere you don't but all the constructions I know use it. >>23335 >especially, without the Trichotomy Law which I used in several places even if implicitly A lot of basic real analysis proofs have topological versions which might not need trichotomy. In particular a lot of properties of metric spaces are actually just properties of Hausdorff spaces. The real hardmode is doing analysis without the axiom of choice.

>>23348 >The real hardmode is doing analysis without the axiom of choice Can one even do functional analysis without it?

if modern crypto is based around the notion that prime factors is hard to calculate.... is there a way to find a polynomial time that runs faster than just checking every prime factor? or maybe having a rainbow table would help?

>>23348 >Do you need LEM to construct the reals? I don't think so. But I also don't do real analysis if I can help it.

If you create a function that divides ints by 3 and return the result, gcc will output code that multiplies the int by a certain magic number, to avoid doing division since division is slow. Since the whole int overflow thing is basically a field, with modulo(max size of the int + 1), how can you calculate the magic number? For 32 bit ints the number is 1431655766, 0x55555556 in hex, which seems suspiciously round, but I don't really understand how to calculate/generate these numbers. Some kind of bitshift stuff is also generated but I assume that's just to deal with the case when it isn't a multiple of 3.

>>23407 There is a neat result that if gcd(a,n) = 1, then a^phi(n) = 1 mod n, where phi(n) is the Euler totient function. It's called Euler's Theorem. So finding an inverse to a is equivalent to factoring n, you just let a^-1 = a^(phi(n) - 1). BTW this is actually the basis for the security of RSA. For n = 2^k, phi(n) is pretty easy to calculate, it's just the number of odd numbers less than 2^k, which is 2^(k-1). So for any odd number a, you can calculate a^-1 = a^(2^(k-1) - 1).

>>23407 https://stackoverflow.com/questions/30790184/perform-integer-division-using-multiplication

I feel like I'm not very good at calculating things. Particularly when it comes to differential equations and anything to do with Riemannian/pseudo-Riemannian geometry, I know a lot of the theory but have trouble doing even simple things like calculating curvature or finding geodesics. I'm also awful at calculating homology or cohomology and I usually resort to row-reducing a matrix. Does anyone know of a good sort of workbook or recipe book for getting better at these kinds of things, assuming a reasonably high level (1 year of grad school) of theoretical knowledge, mostly from Bourbaki-style teaching? Most differential equations books for example are not suited for this because they assume no knowledge past calculus and usually don't have good proofs or general formulas as a result.

>>23679 You must just feel need to exercise calculations. And you probably won't unless you will start doing stuff that requires actually doing them. Like writing simulations, engines, doing numerical methods etc. Theory on the other hand is just partially intersecting with doing calculations. You may be good at theory and slow at calculations. Math and doing calculations is very much like programming with few notable differences. It requires a lot of exercises like going to the gym and weight lifting. I have done a soykaftone of calculus exercises at my uni like solving really fuarrrked up Integrals, differentials, differential equations etc and didn't use that skills for years, yet I have some flashbacks when I look at example exercises. So doing exercises helps a lot. You need to do exercises in the way you exercise theory so you start to link steps of calculations with theoretical basis for every step. That way you are reinforcing ties in your brain of actual theory and actual application of that theory.

>>23680 >You may be good at theory and slow at calculations. Yes, I am. But I also want to be better at doing calculations, because I think it's important for a mathematician to at least be competent at them. Do you know of good resources/books for more advanced calculation based exercises? Maybe once I start teaching more higher-level classes besides just calculus I'll get better at it? Usually explaining something helps to understand it.

>>23681 Read physics books, they assume the required knowledge and come with plenty of new stuff (physics, that is) and plenty of excercises. Math books usually ask you to prove stuff, they don't provide much in the way of drill, physics books on the other hand....

>>23681 >Maybe once I start teaching more higher-level classes besides just calculus I'll get better at it? Usually explaining something helps to understand it. Teaching or studying? If teaching it kinda is strange you had just calculus classes if you studied computer science. If studying then yes they may help. Integrals are typically part of physics related classes, statistics, probability, numeric methods, Monte Carlo methods, simulations and other. Algebra is part of computer graphics and physics simulations etc.

>>23681 >Yes, I am. But I also want to be better at doing calculations, because I think it's important for a mathematician to at least be competent at them. Do you know of good resources/books for more advanced calculation based exercises? I do but they are all in Polish. I don't know English literature on that topic. I myself just gathered all I could have find and were doing exercises from them.

>>23684 >If teaching it kinda is strange you had just calculus classes if you studied computer science. I study math. Once I pass the prelims I'll be allowed to teach upper-division classes like abstract algebra. >>23685 dang

>>23696 then train more ffs

>>23699 I was asking for materials to help with that.

I am worried finding counterexamples by bruteforcing each number could weaken my hability to find a property for each counterexample.

Last one i swear Can i trust this to factor integers floor(x/2)+y+ceil(x/2)+z for y+z=0? Those y and z concerns me a bit because for example 1+9999+1-9999 would give the same effect as 1-1+1+1 while it makes sense when the positive integer is smaller than x.

>>24151 >please don't write proofs in fully formal notation. use human language. a proof should read as dialogue. Please don't write equations in fully formal notation. Use natural language. An equation should read as a dialogue between the right and left sides... Jokes aside, it seems to me (a random bystander) dialogue is crucial for understanding in any discipline of mathematics with proof generated concepts. Dialogue can serve to justify definitions, and to give intuitive understanding where typically we would simply “Choose”, “Let”, or “Define”, leaving the reader to rediscover (or more than likely not) the process that generated these things. There is also value in giving an intuitive sketch of Theorems and Lemmas with their proofs so that the reader can more easily understand the truth rather than merely confirming the logic of the proof. On the other hand doing mathematics in natural language leads to error and discourages rigour by obfuscating the logic and structure of proof, along with making proofs of Lemma distract from the proof overall. I increasingly find myself agreeing with the idea that the form of good mathematics is more like that of a good program rather than a good piece of philosophy.

>>24175 A mathematical proof is an argument made to convince the audience that something is true. Of course it must joldmup tp scrutiny, but the use of natural language need not be a detriment. The proof must be correct at every step, but it would be overkill to outline each and every step of the way. For "formally correct" proofs you can use coq or something similar. For humans to read, you can make it a literary coq file. Propositional logic notation may be formally correct but still relies on the human head to parse it, and with such arcane, dry symbols it is more likely to gloss over or make mistakes. Lest you be willing to do the symbol manipulations on paper.

>>24175 >On the other hand doing mathematics in natural language leads to error and discourages rigour by obfuscating the logic and structure of proof, I disagree, it can help illuminate the logic and structure of a proof. A good picture can also be invaluable. >I increasingly find myself agreeing with the idea that the form of good mathematics is more like that of a good program rather than a good piece of philosophy. Perhaps, but it needs to be a very well documented program. I also think that formalizing *all* proofs is a big waste of time. Once you get past a certain point of so-called 'mathematical maturity', you can get a certain sense for what informal arguments are correct versus incorrect. Like how a programmer eventually gets an intuition for creating algorithms, such that they can know the algorithm is correct even if they make mistakes in the code. Most research papers in math don't fill out every step of their proofs, they just explain important steps and any novel reasoning/definitions they did. This is because the reader of such a paper is expected to be actively reading along to build their own knowledge, instead of just passively verifying. Textbooks are a little the same way but because their audience isn't an expert, they explain more.

Why is my automaton over >>19751 is outputting a cot^2x + cotx output despite having no trigonometric input in the source code?

>>24181 >A good picture can also be invaluable. I considered mentioning pictures, they are a significant aid in exposition. >Perhaps, but it needs to be a very well documented program. I also think that formalizing *all* proofs is a big waste of time. I never claimed otherwise. A poorly documented program is a poor program and even in algebra we are permitted to skip steps. What was intended by the claim that proofs should be similar to programs in form is simply that the proof should be symbolic with a separate exposition. >>24176 Algebraic notation may be formally correct but still relies on the human head to parse it, and with such arcane, dry symbols it is more likely to gloss over or make mistakes. Lest you be willing to do the symbol manipulations on paper. Therefore we should manipulate equations in natural language instead.

>>24371 >I sux at proofwriting. Do i have hope in other areas, like fol or set theory? Depends what you want to use it for. FOL and set theory IMHO are more needed as prelude to topology which is useful as heck for solving many problems or as prelude to automated proving systems.

>>27223 >You're taking a logarithm of both sides here i did not understand this. in school i learned that if we have two "power expressions" with a same base you can "cut" the base and solve an equation with an exponents, and that logarithm is an equation like [/code]a^b=c -> log(a,c)=b[/code], but i am kinda stiff on logarithms. >Likewise division by 0 is undefined this is why i think 0*0 is undefined. i called 0=a and divided both sides by a, like i would do with a normal number or an unknown value. by the way, i found another "proof:" aa=a a=a/a a=1 why i cannot get 0 by any manners? yeah, there is a way: aa=a aa-a=0 a=1 or a=0 but this is circular (0*0-0 .)

How does Stitz & Zeager Precalculus (https://stitz-zeager.com/) compare to Serge Lang's Basic Mathematics for self-learning?

>>26899 Dude most of your post is just complaining... While I understand, you need to temper yourself before jumping into asking questions. >Why do scientists pick letters from the Greek alphabet? It helps with understanding the context. >calculations on numbers below 100 takes me up to 10 seconds. Practice will help you "speed up". >I need to get better at that calculus soykaf, any tips for that? Do many calculus problems to help get better at it.

>>26899 >Why is the syntax in math so inconsistent and ugly? most of it was invented ad-hoc by some guy and everyone else went along with it because they couldn't think of anything better. Be glad, for instance, that the calculus notation is Liebniz's and not Newton's, because Newton notation is way worse. >Why do scientists pick letters from the Greek alphabet? It makes it harder for me to learn. Because there are thousands of important variables/constants and if they only picked from the Roman alphabet it would be way harder to avoid using the same variable name twice in some equations, which would naturally be very confusing. >yet I still feel mathematically disabled and making calculations on numbers below 100 takes me up to 10 seconds. I'm a graduate student in mathematics and I'm still pretty slow at calculations with big numbers. There is literally no shame in this. >Also why is the syntax of calculus so ugly? What's ugly about it?

>>27241 You might want to formalize your axioms to help you discover what 0 is. What exactly are you defining as "0" here? A common definition for 0 is the additive identity; i.e., we assume there exists a number "0" such that 0 + y = y for all y. With this definition and the basic properties of addition and multiplication, you can prove the even stronger statement: 0y = 0 for all y.

Ok so I've been struggling with this for long enough. I went to a soykaf tier high school (worst in state for math and science) and when i got to uni wasn't prepared and did poorly in math. i want to learn precalc namely trig and also calc really well but i don't know how to start. i have my textbooks and some other supplementary material but it's way too fuarrrking boring and slow. does anyone know of simple projects that involve trig and calculus so i can learn while actually making something? I find that to be the most efficient way for me to learn. i want to dedicate about 3-6 months to it so i can finally learn Electronics.

>>27564 you can cover a standard precalc curriculum in a couple weeks. id just do that. calc is harder but also has more direct applications in physics and such so you could feasibly pick it up as you go. precalc is just a grab bag of concepts from algebra, trig, and calculus thrown together, and exists as a class purely to bridge the gap between high school algebra and differential calculus without spending a bunch of time on more foundational calculus concepts, like limits and continuity. you could feasibly just skip it if your algebra/trig is solid and you want to start calc from first principles instead of the standard "applied" curriculum of differential calc followed by integral calc.

>>27544 i am going to say something dumb, but i thought we assumed there was a number zero that means nothing and start to think of the properties, like "you cannot divide by zero because you cannot divide something by nothing" or "a+0=0 because zero means nothing and you are adding nothing to something," and after that we proof that this number exists. after all, i understand why the first proof is stupid: i am literally dividing both sides by zero. but the another proofs still bugs me.

>>27219 Define 0 by x+0=x for all x. Then x-x=0 for all x, and also 0*0=(0+0)*0 = 0*0 + 0*0 implies 0*0 = 0*0 - 0*0 = 0

>>27785 No. Generally, sqrt(a+b) != sqrt(a) + sqrt(b) Try with a = b = 2 to convince yourself.

i am learning maths in my native language. how to learn all the terms in english? i am thinking on reading maths texts in english, but where do i find it?

>>27812 try this, goes from arithmetic all the way to differential equations >Engineering Mathematics by Dexter Booth, Ken Stroud (2020).pdf https://libgen.is/book/index.php?md5=E7C02E31F357F6DAC3DDD29C8CC7C00F

damn, i was tricked by my textbook. nigga said cos1=1 and i went nuts because i was trying to discover a when b=c and Â=1 and i got a=sqrt(2b^2-2b^2). the calculator on my cellphone said that cos1=.9998476952.

>>27841 i did this. notice i traced a rectangle triangle with base d and height h, and another "inside" the parallelogram with base b+d.

>>27840 >nigga said cos1=1 Avoid bad books. cos(1º(this is 1 degree)) = .9998 something. cos(1(this is not 1 degree)) = 0.5403 something.

>>27843 it is a good book, at least compared to the education of my country.

how to use Angle() in geogebra? ABC is equilateral, so i do not think Â=60.

>>27849 What angle do you think it should be? The answer is 60°. An equilateral triangle is also an equiangular triangle. If you are unsure, there is an theorem that says opposite to the bigger side is the bigger angle(and the opposite) in a triangle. If all sides are equal, what that means about all angles?

>>27877 fuarrrk myself. i swear i have read in some hole that the angle of an equilateral triangle is 30. i have only noticed this when i was going to sleep. i want to die.

>>27903 You should have realized that 30° wouldn't sum up to 180°.

>>24151 I agree with you. The problem with the way we teach math is the formalism. Math should be a process of continuous abstraction, and by extension, of imagination (try reading mathematics and the imagination by Kasner and Newman). A proof, at least the fundamental ones as in calculus, should be explained with the right balance between words, notation and geometry.

Problem in pic related This is more of a self imposed challenge that I've made up because i felt like my Pong game was going a bit too slow sometimes. I want to move the circle on the origin to wherever direction i want, while I want that circle to touch a vertical wall in the same amount of time it takes every time. I can't wrap my head around it right now. It may touch the wall once the middle of the circle makes contact, or, preferably, when the perimeter of the circle hits the wall. That I can figure it out, but not the time issue... Hints would be much appreciated. If you have an answer, spoiler it. I want to discover it by myself first. (by units i mean pixels, but I don't think that's relevant(?))

>>28105 Maybe the last sentence of the question isn't well formulated. >What is the speed_x and speed_y for t1, for t2 and for t3.

>>28084 s/evolutionary adopted/evolutionarily adapted/

Hello, I am mostly failing math in pretty much all subjects no matter how hard I try. Where do I start so I can (mostly) re-learn math from scratch? Sorry for the vague question but I'm extremely lost too, everything is incomprehensible alphabet soup to my mind. Just want some books I hope.

>>28166 you'd have to give more information on what you're trying to accomplish. have you at least attempted khanacademy.org/math ? >Just want some books I hope. that's why you'd need to give more information. the material that mathematicians, statisticians, engineers, computer scientists and so on go through diverge considerably, and within those exist many subfields.

>>28166 A popular book for absolute beginners is Serge Lang's Basic Algebra, maybe it will be useful for you. Don't rush through it, read carefully, make sure you understand things well, do all the exercises. Take breaks and let things sink in. Drink plenty of water. If you find a section is still too heavy after trying to read it a few times, skip ahead and keep reading, and go back to it later. You might also want to read George Polya's How to Solve it. And come here and ask questions when you need to.

k^2=(l*sqrt(3)+2hk)/(l*sqrt(3)-2hk) l*sqrt(3)/h=(-2k^3-2k)/(1-k^2) is there a way to isolate k here? first one is discovered by myself, second one is discovered by wolframalpha by writing the first one. (that i am not sure if it is right. https://www.wolframalpha.com/input/?i=k%5E2%3D%28lt%2B2hk%29%2F%28lt-2hk%29)

>>28228 >is there a way to isolate k? probably not.

>>28228 write it as a third degree polynomial. then use the general cubic formula to find the roots. https://en.wikipedia.org/wiki/Cubic_equation it will be ugly.

>>28238 there might be a "nicer" factorization than just applying the cubic formula. I'm not deft enough at those to be sure though

>>28170 >have you at least attempted khanacademy.org/math I did not, will check it out thank you. >That's why you'd need to give more information I apologize, I'd say high school level math and stuff like that, I'm not very sure. >>28175 Thank you very much, will definitively read those. Very kind advice as well. I hope everyone here has a good day.

>>28238 >write it as a third degree polynomial how? there are k, k^2 and k^3 , but k^2 is dividing k and k^3 .

>>28249 multiply both sides by (l*sqrt(3) - 2hk)

from math import sqrt,atan class Point: def __init__(self,x,y): self.x=x self.y=y def __str__(self): return f'({self.x},{self.y})' def __lt__(self,A): return self.x<A.x or self.y<A.y def __ht__(self,A): return self.x>A.x and self.y>A.y def __eq__(self,A): return self.x==A.x and self.y==A.y #no idea on how to implement >= and <=. def distance(self,A): if A>self: return sqrt((A.x-self.x)**2+(A.y-self.y)**2) elif A<self: return sqrt((self.x-A.x)**2+(self.y-A.y)**2) else: return 0 def angle(self,A): if A>self: return atan((A.y-self.y)/(A.x-self.x)) elif A<self: return atan((self.y-A.y)/(self.x-A.x)) else: return 0 >>> from plane import * >>> from math import pi >>> A=Point(1,3**(1/2)) #side of an equilateral triangle of side two. >>> O.distance(A) 1.9999999999999998 >>> O.angle(A) 1.0471975511965976 >>> _*(180/pi) 59.99999999999999 is there a way to make the numbers more precise or i am doomed by the floating point size of my machine?

>>28294 In your distance method, checking if self is lower or higher than A is just a waste of time: (x-y)^2 gives the same result, doesn't matter if x or y is higher: If x > y : x-y = z If x < y: x-y = -z And z^2 = (-z)^2.

Hey /λ/, is it possible to get a computer able to work with infinitely large topologies. >>28294 Use a big number library that represents digits individually in memory.

>>28324 I don't see why not.

>>28323 Higher in what sense, anon? Your definition of higher is important here.

>>28329 Because some infinite topological spaces are quite large. The discrete topology of the naturals, for instance, is of size aleph_1.

>>28330 never mind. i wrote an angle function[1] and tought only in a rectangle triangle, and then my dumb mind said "let's return the angle of the higher point considering the lower point." [1] def angle(self,A): if A>self: return atan((A.y-self.y)/(A.x-self.x)) elif A<self: return atan((self.y-A.y)/(self.x-A.x)) elif A==self: return 0 else: raise Exception('unexpected error.') this uses only one triangle rectangle. (0,0).angle(-1,-1) raises the exception. i plan on tracing four rectangle triangles, but maybe the trigonometric circle will come handy here.

>>28166 Read Precalculus Mathematics in a Nutshell: Geometry, Algebra, Trigonometry by George F. Simmons

>>28343 i give up. how to find the angle between two points? every tutorial that i saw traced a rectangle triangle linking both points and assumed the reference was always at the left of the 90 degrees angle. i have to "move" the point to above the reference, adding pi/4 for each time it needs to be moved, and using arctan to complete the result. a=0 if self.y>A.y: A.y=A.y+(self.y*3) a+=pi/4 if self.x>A.x: A.x=A.x+(self.x*3) a+=pi/4 a+=atan2(A.y-self.y,A.x-self.x) return a O is the center of the cartesian plane. >>> O.angle(Point(1,1))*(180/pi) 45.0 >>> O.angle(Point(-1,1))*(180/pi) 180.0 >>> O.angle(Point(-1,-1))*(180/pi) -45.0 >>> O.angle(Point(1,-1))*(180/pi) 0.0 do not ask me how it moves the point when x and y are zero nor why i did this thinking on O.

>>28425 There is no angle between two points. Angles are between two lines, so unless you define the lines passing those two points and then finding their intersection, your question is meaningless. What one usually does is find the angle between two position vectors(that is to say if A and O is the origin, then OA is the (position) vector at the point A). So in this case, if A and B are two points, the angle between vector OA and vector OB will be the same as the angle between two lines L and M, where L pass through the point O and A and M pass through the point O and B, point O being the intersection of the line L and M. Using the vector case, the angle between OA and OB(that is the angle between the lines that pass through A and B) is theta = arc cos(<OA,OB>/(||OA||*||OB||)), where <OA,OB> is the inner(or dot) product of the two vectors and ||OA|| and ||OB|| their magnitude(or length and in this case, the distance of the point to the origin of the system). <OA,OB> can be simple: Ax*Bx + Ay*By, where Ax = the x-coordinate of the point A, Ay = the y-coordinate of the point A, the same for the point B, Bx and By.

>>28426 >There is no angle between two points then how do textbooks use things like BÂC=90, for example? i want to find the angle between two points because i think it will come handy when i need one specific angle in a polygon. geogebra has this, but only with the center as reference. >What one usually does is find the angle between two position vectors i do not think this will work in my case. if there is one point above another, and i use this to calculate the angle between both using the above point as a reference, this would not return 90. but i might implement this.

>>28431 Also >geogebra has this, but only with the center as reference. Because geogebra is using a line from the center to the points, after all, there is no angle between points, only between lines. Since there are infinite lines that pass through a point, you could have infinite angles between two points.

>>28432 >just translate the two points so that the first point is now on the top of the coordinate system origin i did this in >>28425. >>28433 >One thing you could do is suppose you have two point A and B and make the vector AB, now the angle of the vector AB in relation to the x-axis is theta = arctan(By-Ay/(Bx-Ax)). this is what i called "tracing a rectangle triangle and measuring the angle of the left." >there is no angle between points, only between lines you are talking about perpendicular and oblique lines, right?

>>28457 >you are talking about perpendicular and oblique lines, right? No, I mean any kind of lines, including parallel lines, but in this case the angle between two parallel lines can be any multiple of 180[including 0] or undefined. >this is what i called "tracing a rectangle triangle and measuring the angle of the left." The problem is that you can trace any rectangle triangle with two points(since those two points could be in any of the sides). Using the vector example, you can see that one line is the one touching both points(the vector, so to say) and the other line is the x-axis, so the angle is pretty much the intersection of those two lines. So the case for 90º is easily seem that the line touching A and B is perpendicular to the x-axis. Keep in mind that you could easily make the y-axis be the other line, instead of the x-axis, and your two points (one above the other) would be parallel to the y-axis and not give you the 90º value. A reminder that angles between points makes no sense by themselves. Two lines will always be required for angle to make sense.

>>29113 If you wanna go for Math side be (very) familiar with First Order Logic, Set Theory and Topology. In exactly that order as it's an order of building one on top of another. If you need book on that subject - look for books by Kuratowski - as he was one of pioneers in the area and his works were translated to several languages (English included) from Polish. After that you could go for Graphs - especially proof-wise side.

>>28345 I also recommend Calculus: An Intuitive and Physical Approach by Morris Kline. Be warned though: the book uses an unorthodox style which may not suit everybody's preferences.

>>31372 Do not make the mistake of taking up a "proofs book" such as "Book of proof" or "How to prove it", these are a waste of time. I suggest you begin reading naive set theory, Halmos' book is good, but any other source will do. Once you get the hang of it (you can do the exercises, and/or you can guess beforehand how a proof is going to turn out) you'll have familiarized yourself with the most basic patterns behind proofs, and you should move on to something else to apply your new knowledge. I suggest you read a book on basic number theory like the one I've attached, but a basic college algebra course could also be useful. In no time reading proofs and proving theorems will become second nature to you, just stick to it and ask for help whenever you need it. Have fun!

>>31395 >Do not make the mistake of taking up a "proofs book" such as "Book of proof" or "How to prove it", these are a waste of time. Second this. Set theory is of course a prerequisite to all modern math so of course one would have to read it. But any math book whatsoever (that uses proofs, ie. all of them) should give readers exposition into reading proofs and the exercises require one to prove things, so really, any book, as long as it's within your level (you wouldn't want to read algebraic geometry if you haven't even taken calculus right?) Though there may be an adventurous soul, actually some books such as Baby Rudin are pretty much self-contained so one could conceivably use it without any prior knowledge though it does require mathematical maturity without which reading any such book wouldn't work, so what you really want to develop is mathematical maturity and some problem solving since writing proofs is also just that, try to find a path from A to B and fill in the details so maybe read How to solve it? Now that other book called How to prove it is named after Polya's book but that doesn't mean the book is at the same level or that they are peers. I tried reading it (or was it the other, book of proof?) and it was pretty much the same basic stuff as in set theory, actually, that's exactly what it teaches. And anyway there are many books that don't even use set theory, notably geometry books, which is weird because the notion of set applies most naturally there unlike in some other branches, but I guess geometry uses the notion of loci which is more or less the same, on the other hand analysis is heavily set-theoretical and it kind of makes sense but is also a strange way to frame some oftheir proofs so often, it depends of course on what book you're reading. A lot of people seriously abhor set theory, I don't really care but I don't like it that much either. Nor do I love analysis, I guess it does go hand in hand. But then again to understand most modern mathematics you need just that. It does play heavily on the whole requirement for 'rigour' and the formalist methods but I'm becoming increasingly suspicious of those methods, of course they have inherent value but when they become the primary concern for mathematicians then stuff start to turn pretty weird and you get geometry books with no figures and everything is explained in terms of sets even when they do not fit and everything looks awkward and completely opaque, like that one paragraph in Knuth's TAoCP where he defines a computation as a bunch of sets and none of it makes much sense nor is it used anywhere else in the book nor it iluminates or provide a really insightful definition of a computation, instead it feels like fitting a square peg in a round hole, that's how modern mathematics often feels like but of course everybody has to live up to the logicians' standard of R I G O U R as if the paper (or the people reading it) were computers. Of course there is a need to provide a complete proof that covers every possible edge case but more often than not proofs are never perfect and they are subject to revision and to critique and refutation. Speaking of refutation, here is a very very good book on the subject by Imres Lakatos called proofs and refutations, hopefully you didn't read my rant because there's nothing of value here but do look at the pic attached, look at it look at it and go read Lakatos' book and then go read Homotopy type theory and tell me how it is if it's really that good now they develop the reals using type theory does that make more sense than using set theory? Tell me please because I haven't read it, I haven't read any of the books mentioned, I've never even read a math book I take all my opinions from the internet, from videos of people ranting aimlessly and making baseless claims.

>>32610 If you find self study daunting maybe it's because you need a more strict routine, the more "squared" my life and I became the easier it became to do things, but that's just my personal perspective and it could easily be full of soykaf lel

What do you think about this video? https://youtu.be/pTnEG_WGd2Q

Consider a circle 10 centimetres in diameter and let us draw 2 000 000 different points in its interior. Show that there is a (straight) line such that there are a million points on either side of it.

>>32641 no. start with calculus. >>32840 Let L(t) be a continuous family of parallel lines that cover the circle, and let N1(t), N2(t) count the number of points on either side of L(t). Clearly there exists some t0 such that N1(t0) = 0, and another t1 such that N1(t1) = 2000000. The value N1 must increase monotonically from 0 to 2000000 as one travels from t0 to t1, so there is some T such that t<T <=> N(t) < 1000000, and t>T <=> N(t) >= 1000000. If N(T) = 1000000 then we are done. If not, take L' = L(T). Since N(t) changes by more than 1 at T=t, L' has some number of collinear points. Pick some pivot in L' so that half of the collinear points lie on one side of the pivot and half lie on the other side, then rotate L' through the pivot so that L' does not cross any other points while rotating (this is possible since there are only finitely many points, so there is a minimum angle between L' and any point not on L'). Let the new rotated line be L''. L'' now encloses exactly half of the points in the circle.

>>32845 >Advanced Calculus Show there are no functions f:R->R continuous only on the rational numbers.

>>32842 >no. start with calculus. Why?

>>32875 If you want a really deep understanding of any math that is even tangentially related to physics(which is almost all of math) you will need a rock solid understanding of calculus. You will also need to be good at algebraic manipulations, and calculus is great for building that skill.

>>32882 Do you want to learn math and physics or do you want to get to learning electronics as fast as possible? If it is the latter, it may be best to simply find an electronics book that covers the bare minimum of what you need to know. I believe that Practical Electronics by Paul Scherz is such a book.

This is extremely random, but is there any real benefit to getting a PhD in physics?

>>41279 its a great way to meet physics PhDs if you want to move to antartica, the best part of new mexico, or switzerland there are worse ways to go about it

Much of the modern world was made possible by physics. Then mathematicians with no clear understanding of how science works invaded it. The new mathematicians posing as physicists began studying worthless things like string "theory". That brought most physics to a screeching halt. The only way to fix it is to get rid of all the mathematicians who don't really know what science is or how it's done.

>>28105 >while I want that circle to touch a vertical wall in the same amount of time it takes every time. So if I understand correctly: - speed_x is constant - speed_y is variable in which case: Given: - constant speed_x - direction vector D = (Dx, Dy) Determine: - speed_y 1. The speed vector V is (speed_x, speed_y) and normalized(V) == normalized(D), hence 2. speed_x / speed_y = Dx / Dy <=> speed_x * Dy / Dx = speed_y >>28294 - Those __lt__ and __ht__ implementations will be confusing if they're ever used in an ordered dict, so make them explicit methods (or remove them). - x**2 is always >=0 for any x since x * x = -x * -x = x**2, so the branch is not necessary (and not correct for what you're trying anyway) - Ditto for angle(), but you should use atan2 which will deal with 0 denominators. >>28431 >then how do textbooks use things like BÂC=90 What you want is not the angle between two points but two vectors In general: a point is some absolute position in a given space while a vector represents a direction. What BÂC really means is >angle between vectors <AB> and <AC> Where <AB> = B - A and <AC> = C - A As a rule-of-thumb: - point + point is nonsensical - point + vector and vector + point gives a point - vector + vector gives a vector Many math libraries reuse their vector type as a point type, but there are also some which provide separate definitions for vector and point and will enforce the rules above. === Something important to realize about math is that abstractions apply as much as they do in programming. Taking spatial transformations with matrices* as example: - You can create matrices to perform specific transformations such as translation, rotation, scaling... - You can compose transformations like T = A * B * C - You can apply transformations like Q = P * T (depending on implementation you may need to reverse the operations) Instead of reusing the multiplication symbol you could replace it with some other name, say: - You can compose transformations like T = A `then` B `then` C - You can apply transformations like Q = P `transform-by` T (though I suggest just sticking with the multiplication symbol) Some (many?) math courses make you solve matrices manually, which is retarded and error-prone. You should do it a few times just to understand how it works, but computers are much better at dealing with such low-level details, let it help you! * I'm a proponent of quaternions since they are more compact and it's much easier to invert transformations with them. I also believe 3-axis scaling should be "considered harmful" but YMMV.

>>41282 here's your award you silly goose

>>41285 Here's a book for you that might (or might not) help you overcome your willful ignorance.

>>41286 the only worse thing than someone who hasn't read any books is someone who has read ONE book.

>>41282 I could've made your post say "I'm a dumb nigger" but I can't do that because of cryptographers, a.k.a mathematicians. The only thing that I can do now is say it You're a dumb nigger

>>41282 It's the opposite, nigga. The physics thinking they are mathematicians created string theory.

>>41282 Attempts to mathematically describe some phenomena in physics historically almost always preceded actual experiments, and either turned out to be true, or was invalidated. Displacement current was literally a guess that made the equations fit. It's natural that as physics get more complex more theory is needed first.

>>41282 i get that you're probably not a physicist or a mathematician, but someone interested in both that is looking in from the outside. as someone who is a physicist, let me tell you clearly that mathematicians did not bring physics to a screeching halt. physics has not ground down to a 'screeching halt.' much of it has been completely captured by market forces both in terms of how research is funded and how research is evaluated, which (for better or for worse) has changed both what it produces and what it prioritizes. being so closely tied to the market in the post ww2 order, it is not nearly as immune as one would hope to hype cycles and dumb capital, and several of these cycles have already played out. they are growing bigger as the industrial footprint of science grows bigger and as the broader reach of the hype economy expands.

>>41306 say, have you been developing some sort of... object oriented quantum mechanics, lately?