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> Tadashi Tokeida

It is written up in "The Fishtank Proof of the Pythagorean Theorem" section in the book The Mathematical Mechanic by Mark Levi. In fact Chapter 2 of that book is called "The Pythagorean Theorem" and presents a collection of proofs that OP may be interested in (although certainly none of which would be called the "worst" proof!)

Most likely a vector graphics program like adobe illustrate or Inkscape, I reccomend Inkscape, because it's free, and with a little bit of setup you can even include mathematical formulae with LaTex.

It's quite time consuming to make graphics like this, but certainly worth it, in my opinion.

The book How to Read and Do Proofs by Daniel Solow is a gem. It may be just what you're looking for. Here's a link

A lot of times with a course I was studying there was just something I didn't get and it took a while for it to click, then once it did the rest of the course was cake. I had this problem with exterior derivatives. The text we were using didn't introduce or motivate them well so I ended up in the profs office for an hour once and he helped motivate it and after that the course was no problem.

With proofs there's a big jump to get over and often times the profs don't motivate what you're doing really well. I found a good book on proof technique was helpful because it helped me understand what I was reading. It turns out for most undergrad math you can break a proof down into one of a few patterns and once you know the patterns when you're reading a text you know what to look for and recognize the flow of the proof.

This is a good text on the subject

https://www.amazon.com/How-Prove-Structured-Approach-2nd/dp/0521675995

If you ever need a computer to solve it for you, there's an entire programming language/environment dedicated to this kind of math called Macaulay 2, although I think Sage math handles it too and it's based on Python rather than its own syntax

That's simply not true, as u/bluesam3 said before.

You can see there are two solutions here: https://www.desmos.com/calculator/z38fpr5b6y

Also, since you demanded that others show their working earlier, I would encourage you to do likewise in explaining why you think there is only one solution.

If you're ok with an extra axis then GeoGebra should do the trick.

edit: Ah, wait a minute, GeoGebra only goes to 10^9 . I don't think it's possible to have a zoom-in-able number line that goes arbitrarily large, considering that there are numbers that take an arbitrarily large amount of information to specify. (That is, provided you want a guaranteed level of precision, such as to the nearest integer.)

The app “Brilliant” is kind of like the “Duolingo” of math, I recommend checking it out.

And I think your idea is great.
I recently did something similar. I was coming form a stem background, but still had to learn and relearn a ton of math. It’s a really powerful thing. Math is like tools of thought — is one perspective. It has an alien and austere beauty unto itself, but because it is inherently abstract — the study of relationships unto themselves, so to speak — it creates new and powerful perspectives everywhere.

It’s also the best workout regimen to overcome a crutch of thinking — which is reasoning by familiarity(reminds me of) rather than what is actually laid in front of you. (Type II processing vs Type I processing to use Kahneman’s perspective (see: Thinking, Fast and Slow)).

Beside Brilliant a challenging, but excellent and friendly book I’d recommend is: Probability: For the Enthusiastic Beginner by David Morin. It’s not just a great book for learning probability and the ‘why’ of probability. It’s also an excellent example of how to think when learning math.

I've found Larson Calculus to be a bit less rigorous and more plain text.

At a small college I taught at, we used Calculus from Graphical, Numerical, and Symbolic Points of View. I think, for some, it's an excellent text. It mostly tucks away any rigorous math and tries to use graphical "proofs" when possible to explain. It's not perfect but for some students it was really helpful.

I would highly recommend this course on Coursera: https://www.coursera.org/course/maththink

It's called Introduction to Mathematical Thinking, and it's intended to fill in the gap that's left by our sorely lacking public school system. A good chunk of the course focuses on the writing and analysis of mathematical proofs.

I think it's this: https://www.desmos.com/calculator/h8e19rgp5o

(u/vierkantor is bang on - it's two circular arcs back to back. Mine really only differs in where it meets the y-axis; changing the width of the arch and the radius have pretty much the same effect)

Wilson's Theorem IIRC is 100% accurate and precise when it comes to prime detection. However, due to computational inefficiency, it's awfully worthless in that regard. It's more useful in things like Olympiad math problems or related problems involving modulus. It can be used alongside the Chinese Remainder Theorem to solve some pretty interesting problems.

I can go through a list that I used for my bachelors on the same topic. But I think this was one of the most useful https://arxiv.org/abs/1508.02312 and also a book https://www.amazon.com/Three-Body-Problem-Mauri-Valtonen/dp/0521852242. Try using your library tools for searching articles and also for books, google scholar is shite if you don’t know exactly what you want.

>I have attempted the problem and my instructor has informed both myself and another who worked on the problem separately that we're both wrong.

If you read my response, I specifically stated that it's not clear what help you need. If you detailed the context of the issue, including the work you've done and whatever reason your instructor gave for why your work is wrong, it would be easier for others to know specifically how to help.

>If you read my question, I specifically asked NOT to have this answered for me.

Any "hint" or "guide" contributes to the solution. Since you haven't detailed the work you've done to understand the problem, it's not possible for others to know what constitutes an answer w.r.t the context of your issue. Since you've explicitly asked that others "not answer for" you, there's a clear contradiction.

Bottom line: posting a homework problem without also providing contextual details about the work you've done to understand the problem is ostensibly motivated to have someone else do the work for you.

Answer for upright rather than arbitrary orientation, answer in two dimensions rather than three. I shudder to think what the actual answer to your question looks like. Maybe there's a mathematical framework which provides an elegant solution, but this is evidently not it.

[Cross-posted Comment with the discussion linked above.]

This question has been asked many times over since people first started playing games with dice.

Yes, the materials and density distribution of the composition of a die can affect how "fair" the die rolls, and yes, for most games & gamers small variations don't really matter.

But, if you want a mathematical answer, the best test or model to use is a Chi-Square Test. This is a statistics model that predicts the probability that what is chosen (rolled) is really random. There are a good explanations in several math web-sites online for this, just search. I first learned about this process when I was 12 from the October 1983 issue of Dragon Magazine, and the article "Be thy die ill-wrought?" by D.G. Weeks. Since it was presented there in a way that a 7th-grader could figure it out, I would recommend using that for the basis of an explanation. Actually, figuring the math out for rpgs and games like this when I was young was one of the things that inspired me to get degrees in science and math.

You can find a pdf copy of the magazine here, and the article is on page 62: https://the-eye.eu/public/Books/rpg.rem.uz/Dungeons%20%26%20Dragons/Magazines/Dragon/051-100/Dragon%20Magazine%20-%20083.pdf

So calculating the chi-square value for a die will mean a bunch of rolling in a row, and then a little math.

I hope this helped you!

>I see, would this really be preferrable to just writing it right away in html/js/mathjax?

I believe it would be very similar. Reddit comments, for instance, are written in a markdown format. Instead of having:

<ul> <li>item</li> </ul>

you can just have:

*item

pandoc is nice in that it lets you convert to and from many different formats (ie docx, pdf, html+mathjax, etc) and just convert to the one that you want as output. I like markdown because it's simple and you can just focus on content. If you wanted to write html+js+mathjax, I think it would still let you convert that to pdf (though I haven't tried).

>I don't have special knowledge of mathjax nor latex right now. Or are you saying that this is the only way to put latex directly on a website - to convert it to mathjax?

There is usually not just a single way to do something and this is not an exception. mathjax is just javascript that iterates the html DOM/through html tags and takes \[formula\] or $formula$ and converts them to images or mathml on the page. pandoc would output to that format if you wanted and wouldn't force you to write all the html, but if you prefer the html, you can certainly do that as well.

If you don't want to do latex at all, you even have alternatives of using MS Word and exporting your equations as MathML. I believe you can also directly export Word documents as html or as pdf if you don't want to do html either.

Alternatively, you can write pure latex, and convert from that with latex2html and not need any javascript, all of these things are preferences and have their own strengths and weaknesses.

Yes. On both a spherical surface and a flat 2D surface, you can move left, right, up, down, or any combination of those.

In a 3D space you can move left, right, up, down, forward, backwards, and any combination of those.

Like another commenter said, you should read about Cartesian coordinates and then vectors

They're used by Cloudflare for SSL encryption, supposedly for ~10% of the internet. Basically they take photos of a group of lava lamps and use the digital files to generate encryption keys. It's gimmicky in the sense that they could use photos of anything that's constantly changing, but they picked lava lamps because someone thought it would be cool (I assume).

I assume the periodic nature of one lava lamp becomes lost when you have ~100 of them, and use photos that are encoding all the light changes across the field of view.

(I would also dispute your claim that lava lamps are periodic... I feel like they're pretty chaotic – but it's a while since I've seen one, so maybe you're right)

https://www.cloudflare.com/learning/ssl/lava-lamp-encryption/

I wrote up a very simple version in Python:

http://codepad.org/nukVnpHM

The key "trick" is that if the library contains all possible books, the book id's must be as long as the books themselves and you can set up a simple mapping from id to book contents, and from book contents to id.

This means that you can simply generate the appropriate book id with the desired contents.

Nice work.

Incidentally, I tracked down what I guess is the source for OP's problem: https://www.scribd.com/doc/233399614/Geo-Ch-2-Lesson-3-Intro-to-proofs

and it appears the rules are stated correctly, it appears they really were assigned an impossible problem!

If you're interested in problem solving, and want to learn to code and do research on problems, project euler is a great place to start. The problems are difficult, and maybe obscure, but they'll show you how to work through finding the solution. I do them in my spare time; I'm a much better programmer and much more familiar with number theory, combinatorics, regular expressions, and discrete/finite math because of it. It's been a huge supplement to my education.

Primitive Roots on Brilliant.org is likely what you're looking for. In general that website is very nice at explaining things as it's meant for teaching, rather than Wolfram or Wikipedia which is meant more as a definitive article on the topic. That being said, I would suggest looking at the Wikipedia page since it provides a simple example of a primitive root (and writes out the math that shows it is one). You should also note that the page you found on Wolfram is correct for what you're looking for, it just might be complicated for a first introduction.

> I didn't know pdf had animated features.

I didn't either until I saw some Tikz examples.

> How do you suggest to go from knowing how to code in LaTeX to getting the content onto the web site?

There is the MathJax Javascript library for integrating some LaTeX features in the web, but for more complicated stuff you'll be best off generating an image or something and embedding that on your website normally. I've definitely used LaTeX to generate images then sliced them down to size for insertion into other documents. If you want to draw diagrams in the webpage that are interactive or dynamically generated then I can recommend the D3 JS library. It's fairly friendly and uses SVGs to produce really nice scalable graphics, but it's not as mature or feature-rich as Tikz and other LaTeX libraries. For the simple things that you need to do, I would say you can probably get away with MathJax and D3 without too much trouble if all you want is to embed it in a web page. It's probably easier than getting a LaTeX development environment set up.

Read this book if you haven't already. This is in my opinion the gold standard, and should close all the holes you feel you have in your game.

https://www.amazon.com/Understanding-Analysis-Undergraduate-Texts-Mathematics/dp/1493927116/ref=sr_1_3?dchild=1&amp;keywords=real+analysis&amp;qid=1612931232&amp;s=books&amp;sr=1-3#customerReviews

Not sure if these are the style of books you’re after, but maybe:

Mathematics for the Nonmathematician by Morris Kline

What Is Mathematics? An Elementary Approach to Ideas and Methods by Courant and Robbins

Hi there, first of all thanks for the input. I just want to tell anyone the follow up.
I got some feedback today from the email I sent to the program. The director is pretty honest with me that I lack quite some topics such as topology, functional analysis and measure theory. She gave me lecture notes from the very first compulsory course for the program. It's at the level of Probability with Martingales by D.Williams
https://www.amazon.com/Probability-Martingales-Cambridge-Mathematical-Textbooks-ebook/dp/B00AKE1U14
Well, it's totally above me right now. I feel it's at least a second course on measure. But I have a feeling that if I go study the first course somewhere I might be able to reach this level by the time. I probably will ask whether this is a good idea. Still thanks for the advice. At least I hear something back.

Hi there, first of all thanks for the input. I just want to tell anyone the follow up.

I got some feedback today from the email I sent to the program. The director is pretty honest with me that I lack quite some topics such as topology, functional analysis and measure theory. She gave me lecture notes from the very first compulsory course for the program. It's at the level of Probability with Martingales by D.Williams

https://www.amazon.com/Probability-Martingales-Cambridge-Mathematical-Textbooks-ebook/dp/B00AKE1U14

Well, it's totally above me right now. I feel it's at least a second course on measure. But I have a feeling that if I go study the first course somewhere I might be able to reach this level by the time. I probably will ask whether this is a good idea. Still thanks for the advice. At least I hear something back.

Get your combinatorics on with Combinatorial Problems and Exercises by Laszlo Lovasz.

Great text that ramps up pretty quick into what I consider post-grad material, but I love how it's arranged: Problems, Hints on the Problems, Answers to the Problems.

I didn't get very far (yet! Someday...) but it could be used as a good guide to looking into combinatorics problems, which I think are cool and underrated and sadly often relegated to some side-note in a discrete class.

If you really want to dig into this question (and it's a lot of fun to do so), there's a great history book that you can pick up on the cheap:

The History of Calculus and it's Conceptual Development

It's an extremely through telling of the build up, development, and immediate consequences of developing Calculus.

Also, if you type the name of the book into google, you may find a Pretty Dank File.

I think what you might be looking for is Schaum's Outline of Review of Elementary Mathematics or Mathematical Problems and Solutions for High School Students

Get him a new or good used copy of this specific book:

https://www.amazon.com/Calculus-Vol-One-Variable-Introduction-Algebra/dp/0471000051

He may not be quite ready for it, he may need to bone up on his algebra and trigonometry first, but if he ever wants to study black holes for real, he’ll be very glad that he studied every topic in that book. I studied black holes as part of my PhD and I wish I had gone through this book as a teenager.

Also tell him that studying math isn’t like studying history or biology. It’s not about lots of reading and memorization. It’s like studying how to play an instrument or paint pictures: it’s about practice.

If he can learn to do even half the problems in this book then he would be well-suited to study any STEM topic at any university.

Also tell him not to get discouraged when he hits a topic in the book that he thinks is too hard for him. With dedication and lots of effort he can push through. Sometimes it’s just a matter of getting used to the strange notation or just doing lots of problems. If he gets really stuck in spite of trying, he can move on and come back later.

Give him this he will be engaged forever.

I see, makes sense.

And apologies: I said Taylor, I meant Stewart.

So my recommendation would be to start with pretty much any edition of Stewart, like for instance https://www.amazon.com/Calculus-Early-Transcendentals-James-Stewart/dp/1285741552/ref=sr_1_4?crid=27TLBWSQDEBRF&amp;keywords=stewart+calculus&amp;qid=1666117266&amp;qu=eyJxc2MiOiIzLjAyIiwicXNhIjoiMi45NyIsInFzcCI6IjIuOTYifQ%3D%3D&amp;sprefix=stewar...

The later editions just exist to be more expensive but don't really offer any additional value.

Once you've mastered everything in that up to Taylor series, I would maybe switch to a book on Linear Algebra. There are many good ones, and you might just "try out" the books by Strang, Lay, Friedberg, ... if you're really ballsy you can attempt Axler's. Axler's is not recommended as a first book, but you might get something out of the attempt at least.

Anyway, it's hard to imagine you'll even get to the Linear Algebra book before class starts. But if somehow you manage to get through Calculus and Linear Algebra, a very profitable thing to do is look into logic and discrete math. (This would cover, for instance, combinatorics among other things.) A good book for that is by Rosen.

May I recommend this book - A Mind for Numbers

Or you could work through the course which is free for now - Learning How to Learn

For Maths specifically, how about working through Khan Academy? Their practice exercises are, to me, pretty good.

You might be interested in this book: https://www.amazon.com/Arithmetic-Paul-Lockhart/dp/0674972236

Goes back to basic, fundamental counting and arithmetic and explores it with an adult's interest. It's one of my favorite math books ever.

To fully understand this, it helps to study

• General linear algebra
• Change of basis
• Properties of matrices
• Orthogonal matrices
• Euler Angles
• Vector normalization
• Linear Interpolation
• Basic properties of quaternions

I taught a course for my co-workers on this topic. I found the following books helpful:

Rigid Body Dynamics For Beginners: Euler angles &amp; Quaternions

Quaternions and Rotation Sequences: A Primer with Applications to Orbits, Aerospace and Virtual Reality

You can check out our Math Riddles app on Google Play or App Store.
It is free of charge, and if you don't know the answer, you can tap on "Show answer" on top right.

> if it's taught in college, there must be something important about it, no?

Not necessarily, see Intellectual Impostures and the Sokal hoax.

😀

I think that this recent book by Steven Strogatz is exactly what you are looking for. It's excellent:

https://www.amazon.com/Infinite-Powers-Calculus-Reveals-Universe/dp/1328879984

Raj

Well... if you're not a trained mathematician at a PhD level with years of research, I don't think those problems are worth your time.

You can always try problems that are close to your level. You don't have to take on fights you'll probably lose. There are many problem books to sharpen your math skills and have some good time. For example:

https://www.amazon.com/USSR-Olympiad-Problem-Book-Mathematics/dp/0486277097

YouTube: 3Blue1Brown, Numberphile

Books:

Bell, "The men of mathematics"

Hardy, "A mathematician's apology"

https://www.amazon.es/Mathematics-Content-Methods-Meaning-Dover-ebook/dp/B00GUP46MC/

If you have some multi variable calculus then I think you should look at a book by Darling called differential geometry.

https://www.amazon.com.au/Differential-Forms-Connections-R-Darling-ebook/dp/B01LZ8QOAT

This book starts with considering manifolds as subset of Euclidean space and uses differential forms as it’s main tool. It aims to train people to understand enough diff geom to do physics. So the book ends with principal fibre bundles and gauge theory.

It is very explicit and has lots of examples and exercises. It is a gem.

Check out this text: Discrete Mathematics by Norman Biggs. I thought it did a decent job of introducing topics in a nice broad way.

for what it's worth, i found this series of books useful

https://www.amazon.com.au/Mathematics-Complete-Introduction-Learn-Maths/dp/1473678374/ref=sr_1_5?crid=KW4ZT79D5VCI&amp;keywords=mathematics&amp;qid=1652275571&amp;sprefix=mathmetic%2Caps%2C313&amp;sr=8-5

You can check out Gareth Loy's Musimathics, in two volumes.

You can also look into microtonality, which explores alternate tunings based on mathematics. Erv Wilson is probably the most influential mathematizing microtonalist.

You can also come at the subject via psychoacoustics, which is more of an ploration of the physics of sound. I don't what's best in this area, but one book to have come to my attention is Tuning, Timbre, Spectrum, Scale.

You should get this book of R. Smullyan:

https://www.amazon.es/Forever-Undecided-Raymond-M-Smullyan-ebook/dp/B008ED5G2M/

Here's a start: Mathematical notation.

But, as others have said, from a certain level on it becomes field dependent.

> evolution is more strange than we give it credit

I disagree with that. Once we have the elementary trifecta of reproduction, variation and selection (as rudimentary as they may be), evolution is unavoidable.

It's the consequences we see around us which are marvelous and, frankly, mind boggling. But that's because these consequences are historically contingent (they could have been completely different), the result of a branching process we don't know the details of and which played out over the unimaginable length of deep time.

A relevant read I found pleasant: Dennett's Darwin's dangerous idea.

> maybe it seems more intuitively natural if we consider any system with some kind of energy flow, where internally concentrations of energy are permitted, even if the net result is a loss of energy.

Of course, life depends on thermodynamic disequilibrium: the sun shines, the planet is young and active (thinking of deep-sea volcanic vents and their awesome fauna) and animals feed on plants or each other.

> some kind of mathematical formalization

Perhaps Systemics would be the relevant field. But I'm completely ignorant about it, so no guarantee. 😔

Iif you define weird as eccentric then you have also Grigori Perelman

Grigori Perelman

There is a book on his life

Perfect Rigor

Perhaps Exploratory Galois Theory?

Haven't read it, so don't take that as a recommendation, but looks like a cool intro.

What skills do you want to learn? No matter what you're doing, I'd recommend for any software developer a sound understanding of linear algebra. I found the Hoffman/Kunze text very helpful (I also come from an engineering/software background): https://www.amazon.com/Linear-Algebra-Kunze-Hoffman/dp/9332550077/ref=sr_1_1?keywords=linear+algebra+kunze&amp;qid=1647906254&amp;sr=8-1

It’s a paid app/website, but I highly recommend Brilliant.org.

It’s sort of like the Duolingo of math and science stuff. Individual mini-courses vary from “solid” to “excellent”, imo.

Very focused on the why of what you’re learning as opposed to just the rules. (By contrast, Khan, from my occasional glimpses at it is much more focused on what to do, rather than why to do it.)

If you have trouble learning something on Brilliant you’ll likely want to use additional resources to focus on it (as it has a limited number of problems to solve).

But I really like it as a review or companion guide. And strongly recommend.

On the book side, I don’t know many books that aren’t at least partly proof based, which may not be what you’re ready for, but I can highly recommend : Probability: For the Enthusiastic Beginner by David Morin.
Very focused, again, on why and seeing how to arrive at results from multiple perspectives. And the underlying math it works with is very basic/accessible — with the challenge being understanding and perspective.

If you have sometime, take a look through this book (<— online pdf of A Book of Abstract Algebra by Charles Pinter, which is a great (and cheap) book to get a physical copy of if you want it).

I think basic abstract algebra, which that course seems to be — it doesn’t sound like it goes beyond intro. group theory (equivalent to Ch 16 in the book I linked above), is barely even a proof oriented course. Because the objects and rules are so clear cut. The proofs are, very literally, almost entirely algebraic.

That said, everyone has different challenges walking thr same course so I’m not one to say.

But off-hand I wouldn’t stress it too much. Especially if you’re a junior. I’d go with both. But, talk to the prof and some other students.

In the past I'd bought a small Dover book that was trying to teach anyone interested how to do fast mental math, with providing the needed methods/tricks and many many many drills to get sharp. I did try for sometime, but then I left it on the side.

Anyhow, this is the book. It might interest someone.

I liked this textbook: Games, Strategies and Decision Making

I feel like it struck a nice balance of being accessible without dumbing the material down too much. The trade-off is that it's quite verbose.

I have a suspicion it may not be for your class though. The focus is very much more on helping students gain fluency with the important concepts for the sake of becoming better social scientists than it is on helping students develop mathematical or algorithmic problem solving skills in the context of game theory.

I’m an economist. We used this textbook during the first year of my PhD program. I thought it was pretty good.

Game Theory https://www.amazon.com/dp/1108825141/ref=cm_sw_r_cp_api_glt_i_HY4AF4116V6XXD1QJQBT

I hope that helps!

This isn't what is meant by mathematical foundations in the West -- stuff like Frege, Russell, Hilbert, and Gödel are more what westerners think of when discussing mathematical logic.

This is more about abhidamma and the refutation of of "hinayana" philosophy by thinkers like Nagarjuna. The formulation of these insights in classical Indian philosophy is completely separate from Western philosophical tradition past the Ancient Greeks.

You'd do well to get a copy of the mulamadhyamakakarika as translated by Siderits and Katsura. Read the notes for relevant abhidhamma authors and follow up India philosophical works.

This area of study is fascinating, but flourished independently of European mathematics.

> I took a geometry class in high school and I thought it was very interesting.

You'd probably be interested in I. M. Yaglom's Geometric Transformations books (there are four of them): classical high school geometry from a completely different point of view. They are quite fascinating.

After that, as others have said, it's non-euclidean, riemannian, differential, manifolds, &c.

I can’t speak to what you need to get all the way to current literature, but given your background (and lack of any nonlinear dynamics course) I highly recommend:

Nonlinear Dynamics and Chaos - Steven Strogatz

In addition to being a well regarded text its also really well written and fun — very suitable for self-study. And I believe you can find solutions for most of the problems online (and theres a separate solution’s manual) so you can work problems.

You can join online websites like Fivver, Udemy other educational platform where you can give lessons regarding maths if you are good in math.

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Hi :D

I ended up completing my graph by replacing my failed formula with a very different one I found online. It is animated and the points can be dragged anywhere! Here it is: https://www.desmos.com/calculator/ayef9u04dg

I’m done!

I ended up completing my graph by replacing my failed formula with a very different one I found online. It is animated and the points can be dragged anywhere! Here it is: https://www.desmos.com/calculator/ayef9u04dg

It’s reducible over the reals, as a special case of Sophie Germain’s identity. It is not reducible over the rationals, by applying the transform x = 1+t then appealing to Eisenstein’s criterion.

I have a physics bg, but become increasingly interested in mathematics as years passed by. So I got into the habit of self-studying, mostly from translated Soviet Dover books.

I found that the following provides a nice overview of the basic math sub-fields, and it should be helpful for you too:

https://www.amazon.es/Mathematics-Content-Methods-Meaning-Dover/dp/0486409163/

You might like my goodreads list too.

Keep up the good work, believe in yourself, look out for friendly math communities either online (eg. this one) and offline, and never again let others define what you like and what you don't like :).

This is the book we’re using. I don’t have much knowledge on what makes a good abstract algebra book but I think this one is pretty shit. https://www.amazon.com/Abstract-Algebra-John-Beachy/dp/1577664434

I got my administration to get every student the xp-pen 4x6 graphics table (https://www.amazon.com/StarG640-Ultrathin-Graphics-Battery-Free-Pressure/dp/B078YR2MTF) and then they had a MUCH better time.

I began using peardecks with google slides because if you set up the 'draw on' slide, they can draw with their digital pencils and you can see every mark they put on their work. Whereas with nearpod you have to wait for them to finish and press submit. You see them drawing in real time.

Try using Inkscape, there are some templates online that have circuit basics. It can be added into word as a picture. You can download Inkscape here: https://inkscape.org/ just look up a file online with some circuit presets drawn.

I figure you'll get more out of this by playing around with a visual than just asking an internet stranger, so here's an animated desmos image where you can configure p,q, and see what happens when time is positive/negative.

https://www.desmos.com/calculator/ghfkyjx2lc

No problem. It's a good idea to keep desmos handy while you work out problems like this. If you have an answer key, then you can just do a quick graph to see if your answer is equivalent without working out all the algebraic manipulations. And if you don't have an answer key, there's a slope field generator that can be useful to check your work. This one's a bad example though, because the interesting part of the graph doesn't fit on the slope field that's shown when c=25

The two solutions are equivalent. This is actually an important lesson about solving differential equations. Depending on the path you choose to integrate, you will often get different looking solutions (with different constants) that turn out to be equivalent. This happens a lot particularly when you have logarithms in your solutions.

https://notability.com/n/1r9BP36Zt507ijpb2LnVdc

Thanks! Please use the above link to see my working for Q2a. As for 2b I think I really need to redo this topic from scratch coz I'm a bit lost here.

In the above link you can also have a look at my Fourier series of Q1. I'm doing mechatronics engineering Year 1, this is a coursework of engineering mathematics module. There are 2 more questions after this related to the electrical part

Since your objective seens to be general relativity, I suggest https://www.amazon.com.br/Gravitation-Charles-W-Misner/dp/0691177791, but if you're interested in a more mathematicaly rigorous approach, or need to study it deeper, like tensor fields, first study advanced linear algebra, real analysis and differential geometry, than I suggest reading using the book you're already familiar on advanced linear algebra or differential geometry, most probably one of then will cover tensor calculus.

Maybe check this one:

https://www.amazon.com/Philippe-Dennery/dp/0486691934/

It has a chapter on linear operators, covering a bit of tensor analysis too. It uses the Dirac notation.

Generally a pretty nice book, if you're into the intersection of maths and physics.

Also you might as well check some other Dover book that is specifically for tensor analysis.

Great question!

I used to tutor GED/high school/secondary school-equivalent students using a book called Pre-GED Connection. I found the explanations in that book quite clear and the examples pretty thorough. It starts with some number sense problems so that you gain an intuition on why negatives act the way they do, where reciprocals come from, and so on. It moves into some nice algebra I and geometry-level material. The last chapter, I think, is on very basic statistics. It's colorful, graphical, and manageable if you're looking to get the fundamentals down.

It's a little bit dated now, and PBS, who originally sponsored the book alongside the publisher KET, discontinued it. As a result, it's a little pricey for a small book and can only really be found used. It does looks like they're publishing a new one for the same adult education initiative. Full disclosure: I know nothing about the new book, but I trust PBS and KET to produce good content.

Persistence here is key, next time you take it don’t be the old you, study don’t cram if that applies. Also, go to your school’s tutoring center and get a group tutor for calc 3, I used to run a few of these groups back in my college days, it really helps! If you’re in HS and your school doesn’t have a tutoring center buy the Schaum’s Outline of Calculus or Advanced Calculus and read parallel topics in there that should help (runs about $12 - nix that they raised to $20 for Xmas — those greedy bastards on Amazon

Math Girls is a three book series by Hiroshi Yuki that explores interesting topics. The first deals with integer partitions, the second with Fermat’s last theorem, and the third with Gödel’s incompleteness theorem. Here is link to the first one:

https://www.amazon.com/Math-Girls-Hiroshi-Yuki/dp/0983951306/ref=mp_s_a_1_3?crid=3FGN0D5699N95&amp;keywords=math+girls&amp;qid=1639844932&amp;sprefix=math+girls%2Caps%2C56&amp;sr=8-3

>OneMeterWonder · just now
>
>Would you please read back to me the first seven words of the article you’ve linked?

I'm not going to do your homework for you young person. I'm classically trained - in the pre-computer era. Boolean logic in the context of debate and classical structure is not as wiki-era accessible as you might prefer.

I asked you to do "the math" - that is why I am here. Are you here to merely dispute a common word?

A synonym for what I am referring to is "Logical Pyramid" - but it all stems from language as the basis for all thought, including mathmatics.

The conceptual foundation of Boolean logic as applied in academics of thought have been grossly polluted by their natural application to computer science, particularly data retrieval vis a vis search engines.

Naturally, this growth has cast a shadow over the functional use of Boole's process by semanticists, logicians and philosophers.

Let us agree that I am using the word Boolean (ie: from George Boole) as an outlier, and set that aside. Yes?

This might be something you would wish to review:

https://www.amazon.com/Pyramid-Principle-Logic-Writing-Thinking/dp/0273710516

Haven't looked into trig specifically for quite some time… Anyway:

• free book: https://ia601601.us.archive.org/14/items/in.ernet.dli.2015.141418/2015.141418.Elementary-And-Advanced-Trigonometry.pdf

• non-free: https://www.amazon.com/dp/0486441776

Pick your poison.

Topology Illustrated is a very interesting option. Especially if you like lots of images. I find imagery doubly helpful when self-studying as they help make sure you’re understanding correctly.

The book is mostly about algebraic topology, but honestly, still super accessible assuming you’ve taken a modern/abstract algebra course.

Also Introduction to Topology is good and very short. A quick book makes for a nice first pass at a subject imo.

Rudin has published many books on analysis each quite different. It is important to know which you are talking about.

Check out the Amazon review “like drinking math from a fire hose” https://www.amazon.com/Principles-Mathematical-Analysis-International-Mathematics/dp/007054235X#aw-udpv3-customer-reviews_feature_div

It is a book for maths people. Maybe you have done enough math to get something from it, but I suspect that it is a poor choice.

Indeed Mathematica can do that.

Otherwise you can freely use Nyquist (a Lisp) or Processing's sound library (a Java dialect).

But I never used either for sound production, so take it with a grain of salt. ;)

Trefethen and Bau: Numerical Linear Algebra https://www.amazon.com/dp/0898713617/ref=cm_sw_r_awdo_navT_a_GJF92SDSY2WWJD0AZFKN

Honestly numerical linear algebra, done right, is just as rigorous. Numerical and applied mathematics doesn’t lack the rigour. What it does have is substantially more detail which makes the math “messier”.

As an example application of linear algebra to numerical analysis looks up papers on “the distribution of floating point numbers”.

There are a couple books on this subject

Why "extensively"?

Start with the basics. It will get you very far and will take you a weekend. Check out the tutorials at https://www.overleaf.com/learn/latex/Tutorials

It's not so easy to generate a Voronoi diagram (although if you're up for a challenge here is the most direct algorithm.

But if you can handle basic JavaScript, there are various graphics libraries that can do it. I recommend Paper.js.

If you're on a *nix box then it's pretty easy, just use your package manager to get LaTeX installed. If you're on windows the way to go is MikTex, but you'll probably have to manually make sure that it gets added to your path correctly. When you get beyond just generating PDFs you need some extra tools to automatically generate the images, ImageMagick is a popular choice. Again, on *nix boxes you can just install from your package manager and on Windows you have to do a more manual install. If you're used to developing software of any kind on windows then this won't come as a surprise to you!

You can use a proof assistant such as Coq for that purpose. Example:

assert Syllogism {

all Socrates: univ, Man, Mortal: set univ |

-- every man is mortal

Man in Mortal

-- Socrates is a man

and (Socrates in Man)

-- implies Socrates is mortal

implies Socrates in Mortal

}

check Syllogism

Most real-world examples, however, will send you crying for your mother because of their complexity and incomprehensible formalisms ...

Other's have suggested Khan. I agree that it is good. But you may find it better to try Brilliant.org. It's paid for, but instead of teaching a concept and then just doing questions on it (which is great for a solid foundation) Brilliant teaches you how to think about things in a mathematical way.

Yeah... especially if you are doing online classes. Fact all my expensive tuition classes used web assign and the teacher didn't teach me a thing. They only grade... discussion posts and that's it. They don't even make the posts they assign. Math classes today are a joke. If you are lucky to get a face to face class, ask as many questions as you can. Take full advantage.

Otherwise, 3blue1brown, khan academy, and brilliant.org got me through my bachelors with A's in all my math classes from algebra 1 to differential equations & linear algebra.

In the end, you will learn math no matter what if you really care about it. Read about mathematicians like J. Lambert who learned maths on their own at young ages during poverty and no internet. If you really love math, you will learn it without school or tuition.

This is a stars and bars problem https://brilliant.org/wiki/integer-equations-star-and-bars/

Take a string with 5 positions for “slots” and 3 positions for “dividers” between your elements (after a divider you switch to the next element). The number of ways to create a string with 5 “slots” and 3 “dividers” is 8!/(5!*3!)=56

This is just the same as the sum of squares: 5*5,6*6... 5^2,6^2.

So using the formula to sum series of squares from 1 to n

∑ k^2= n(n+1)(2n+1)/6

Derivation here https://brilliant.org/wiki/sum-of-n-n2-or-n3/

Summing up to 1000 gives 333833500 then take off 1+4+9+16 gives 333833470.

In adition to a tutor, you could try the brilliant.org app or website, which teaches math and is also supposed to feature a social learning aspect;

"Brilliant is a network.

Problem solving together is the foundation of all great relationships in the sciences. You can write your own solution and get feedback from people around the world. Contribute a question, edit the wiki, or mentor someone who is learning. Other people will do the same for you."

You could look into Sage. Sage (or sage-math) is a computer algebra system using Python as its language. Computer algebra systems do symbolic manipulation of formulas. They can do things like factoring (multivariate) polynomials, Taylor series expansions, calculating with finite fields and so on. They are very useful for "experimental mathematics", which imho is an unfairly neglected approach of doing mathematics. You can for example create algorithms to calculate topological invariants, and conjecture things based on the results.

All this said, I personally dislike Python, and especially given your interest in type theory, there is a chance that you will dislike it too. It's a language designed to be easy to get into, but it's not a "good" language.

Unfortunately, computer algebra systems tend to be "dynamically typed", probably because it's just too hard to make them both work and be convenient to use within a proper type theoretical framework.

Feel free to tag me if you make another post about the topic, sounds interesting. I modeled the thing up in geogebra https://www.geogebra.org/3d/fmvpdv7c essentially you "only" have to find the function f (or g, currently they're the same of course) such that the area requirement is statisfied. Formalizing this area thing probably results in a kind of ugly expression (involving some integral where the bounds depend the inverse of f to "hop down to the next turn" of the spiral or something like that), but it should be doable. By using the inverse function theorem and calculating the so called "variation" of the resulting functional (or using the euler lagrange equations) you may then be able to at least numerically find that function f.

So it sounds like you want to model the delays between consecutive "taps" as a smooth convex function that starts out high, then falls, before slowly turning around and rising ever faster to fade out.

The simplest function that comes to mind would be a parabola

f(t) = t^2

Then you just shift and scale it so that the minimum occurs exactly where you want it, for example:

f(t) = k * (t - t_mid)^2 + m,

where k controls how steep/shallow you want the curve to be, m is the minimum value you want it to hit, and t_mid is the midpoint in time of your sequence, so effectively t_end - t_start

You can play around with a graphing calculator to get a feel for the values, like here: https://www.geogebra.org/calculator/tfk8xnmc

If you for some reason need the time of each tap as a duration elapsed from the beginning of the sequence, just add all the delays together. Alternatively, you could calculate the integral of f(t) (which will be a cubic function) and calculate the times directly from there. I'll let you figure that out yourself.

If you consider the cross section, the top bit of the cylinder will appear to be a a triangle. The angle is 36°, and the bottom side of the triangle is 14 inches. With simple trigonometry, you get that the length of the ellipse is 17.3 inches - and so the length of your string should be 17.3 inches.

For simple things like this, I point you to the program geogebra - it's very powerful, and gives you want you need to make precise drawings and precise measurements from those drawings. Plus it's free and works in your browser.

geogebra is a CAS system and kinda programming languages you can build quite advanced applications with - I mainly use it for geometry, visualizing surfaces etc. for example around christmas I did this https://www.geogebra.org/m/teezhtce do visualize and play with curves and surfaces and some calculations I made.

Does anyone here know enough about how GeoGebra works to actually implement this as a nonparametric function? I've messed around with it using conditionals, but I can't seem to get it to implement correctly.

The fox/rabbit problem in differential equations has a system of linear equations and differential equations. You can solve for the critical starting points, which probably counts as finding zeros. Differential equations inherently include integrals and derivatives (e.g., count the total number of rabbits born and dead in time span t). I'm not sure what is meant by approximation. What class is this for?

I think this can be done with any elementary coupled differential equations example.

https://www.geogebra.org/m/FwQfAxqE

https://en.wikipedia.org/wiki/Lotka%E2%80%93Volterra_equations