by rullopat on 11/25/21, 1:24 PM with 249 comments
by ColinWright on 11/25/21, 1:31 PM
Yes, conventions have emerged, people tend to use the same sort of notation in a given context, but in the main, the notation should be regarded as an aide memoire, something to guide you.
You say that you're struggling because of "the math notations and zero explanation of it in the context." Can you give us some examples? Maybe getting a start on it with a careful discussion of a few examples will unblock the difficulty you're having.
by Syzygies on 11/25/21, 6:07 PM
Math notation is not math, any more than music notation is music. Notably, the Beatles couldn't read sheet music, and it didn't hold them back.
The best comparison would be is reading someone else's computer code. At its best computer code is poetry, and the most gifted programmers learn quickly by reading code. Still, let's be honest: Reading other people's code is generally a wretched "Please! Just kill me now!" experience.
Once you realize math is the same, it's not about you, you can pick your way forward with realistic expectations.
by tgflynn on 11/25/21, 4:22 PM
But it isn't just about the notation. You also need to understand the concepts the notation represents, and there aren't really any shortcuts to that.
These days there are online courses (many freely available) in just about every area of mathematics from pre-high school to intro graduate level.
It's possible for a sufficiently motivated person to learn all of that mathematics on their own from online resources and books, but it isn't going to be an easy task or one that you can complete in a few weeks/months.
by gspr on 11/25/21, 2:43 PM
I've been repeatedly called a gatekeeper for this stance here on HN, but really: notation is a red herring. To understand math written in "math notation", you first have to understand the math at hand. After that, notation is less of an issue (even though it may still be present). Of course the same applies to other fields, but I suspect that the question crops up more often regarding mathematics because it has a level of precision not seen in any other field. Therefore a lot more precision tends to hide behind each symbol than the casual observer may be aware of.
by ivan_ah on 11/25/21, 2:06 PM
That covers most of the basics, but I think your real question is how to learn all those concepts, not just the notation for them, which will require learning/reviewing relevant math topics. If you're interested in post-high-school topics, I would highly recommend linear algebra, since it is a very versatile subject with lots of applications (more so than calculus).
As ColinWright pointed out, there is no one true notation and sometimes authors of textbooks will use slightly different notation for the same concepts, especially for more advanced topics. For basic stuff though, there is kind of a "most common" notation, that most books use and in fact there is a related ISO standard you can check out: https://people.engr.ncsu.edu/jwilson/files/mathsigns.pdf#pag...
Good luck on your math studies. There's a lot of stuff to pick up, but most of it has "nice APIs" and will be fun to learn.
by todd8 on 11/25/21, 3:36 PM
Now, of course, you have the internet and it can tell you what the square root of 217 is. Consequently, the value of these used CRC handbooks is low and many are available on eBay for a few dollars. Pick up a cheap one and in it you will find many useless pages of tables covering square roots and trigonometry, but you will also find pages of formulas and explanations of mathematical terms and symbols.
Don't pay too much for these books because the internet and handheld calculators have pretty much removed the need from them, but that is how I first learned the meanings of many mathematical symbols and formulas.
You might also look for books of "mathematical formulas" in you local bookstores. Math is an old field and the notations you are stumbling over have likely been used for 100 years, like the triangle you were wondering about. (Actually the triangle is the upper case greek letter delta. Delta T refers to an amount of time, usually called an interval of time.)
Unfortunately, because math is an old subject it is a big subject. So big that no one person is expert in every part of math. The math covered in high school is kind of the starting point. All branches of mathematics basically start from there and spread out. If you feel you are rusty on your high school math, start there and look for a review book or study guide in those subjects, usually called Algebra 1 and Algebra 2. If you recall your Algebra 1 and 2, take a look at the books on pre-calculus. The normal progression is one year for each of the following courses in order, Algebra 1, Geometry, Algebra 2, Pre-Calculus, and Calculus. This is just the beginning of math proficiency, but by the time you get through Calculus you will be able to read the paper you referenced.
Is it really a year for each of those subjects? It can be done faster but math proficiency is a lot of work. Like learning to be a good golfer, it would be unusual to become a 10 handicap in less than 5 years of doing hours of golf each and every week.
Calculus is kind of the dividing line between high-school math and college level math. Calculus is the prerequisite for almost all other higher level math. With an understanding of Calculus one can go on to look into a wide range of mathematical subjects.
Some math is focused on its use to solve problems in specific areas; this is called applied math. In applied math there are subjects like Differential Equations, Linear Algebra, Probability and Statistics, Theory of Computation, Information & Coding Theory, and Operations Research.
Alternatively, there are areas of math that are studied because they have wider implications but not because they are trying to solve a specific kind of problem; this is called pure math. In pure math there are subjects like Number Theory, Abstract Algebra, Analysis, Topology & Geometry, Logic, and Combinatorics.
All of these areas start off easy and keep getting harder and harder. So you can take a peek at any of them, once you are through Calculus, and decide what to study next.
by rackjack on 11/25/21, 8:18 PM
- better see the structure of the problem; or
- reduce the amount of ink they need to write the problem
Very similar to how programmers use functions, in fact.
To this end, mathematicians in different fields have different notation, and often this notation overlaps with different meaning. Think how Chinese and Japanese have overlapping characters with different meanings.
As others have stated, there is no "one true notation" -- all notation is basically a DSL for that math field.
Instead, choose a topic you are interested in, find an introductory text, and start reading. They will almost certainly explain the notation. Unfortunately, even within a field, notation can vary, but once you have a grasp of one you will probably grasp the rest quick enough.
I will mention, though, that some notation is "mostly" universal. Integrals, partial derivatives, and more that I can't recall right now all use basically the same notation everywhere, since they underlie a lot of other math fields.
by srcreigh on 11/25/21, 8:18 PM
Learning everything about math is nearly impossible like knowing everything about all code that exists.
That course should teach some basics for proof strategies. Ex here on page 2, there are definitions with examples: https://cs.uwaterloo.ca/~cbruni/pdfs/Math135SeptDec2015/Lect...
Specialized math tends to have specialized notation. For ex Linear Algebra, Calculus, Combinatorics. Any decent textbook will have an appendix or table with what the notation means.
by cjfd on 11/25/21, 1:55 PM
by klodolph on 11/25/21, 7:32 PM
I suggest finding contexts first, and exploring math within those contexts. Different subfields have their own conventions and notation.
For example, you might be working in category theory, and see an arrow labeled “π”. When I see that, I think, “Ah, that’s probably a projection! That’s what π stands for!”
Or you might be in number theory, and see something like π(x). When I see that, I think, “Ah, that’s the prime number counting function! That’s what π stands for, ‘prime’!”
Or you might be in statistics, and see 1/2√π e^(-1/2 x^2). When I see that, I think, “Ah, that’s the number π! It’s about 3.14”
Or you might see a big ∏ which stands for “product”.
The fact that such a common symbol, π, stands for four different things in four different contexts can be a bit confusing. So if you want to learn mathematical notation, pick a context that you want to study (like linear algebra), and look for accessible books and videos in that subfield. The trick is finding stuff that is advanced enough that you’re getting challenged, but not so advanced that it’s incomprehensible. A bit of a razor’s edge sometimes, which is unfortunate.
by dwheeler on 11/25/21, 2:36 PM
https://dlmf.nist.gov/front/introduction
Of course, if the real problem is that you need to learn some mathematical constructs, that is a different problem. The good news is that there's a lot of material online, the bad news is that not all of it is good... I often like Khan Academy when it covers the topic.
I wish you luck!
by excitednumber on 11/25/21, 3:11 PM
by anthomtb on 11/25/21, 4:35 PM
I also used get hung up on “mathematical notation”. But it turns out the problem wasn’t the notation. I was just bad at math. Well, out-of-practice is more like it.
Once you have the fundamentals clearly explained and you’re doing some math on a regular basis the notation, even obscure non-standard notation becomes relatively intuitive.
by hatmatrix on 11/25/21, 1:52 PM
Wikipedia has been helpful sometimes but otherwise I have found reading a lot of papers on the same topic has been useful. However, this is kind of an "organic" and slow way of learning notation common to a specific field.
by swframe2 on 11/25/21, 2:43 PM
1) Search youtube for multiple videos by different people on the topic you want to learn. Watch them without expecting to understand them at first. There is a delayed effect. Each content creator will explain it slightly differently and you will find that it will make sense once you've heard it explained several different times and ways.
I will read the chapter summary for a 1k page math book repeatedly until I understand the big picture. Then I will repeated skim the chapters I least understand until I understand its big picture. I need to know the terms and concepts before I try to understand the formulas. I will do this until I get too confused to read more then I will take a break for a few hours/days and start again.
2) You have to rewrite the formulas in your own language. At first you will use a lot of long descriptions but quickly you will get tired and you will start to abbreviate. Eventually, you get the point where you will prefer the terse math notation because it is just too tedious to write it out in longer words.
3) You might have to pause the current topic you are struggling with and learn the math that underlies it. This means a topic that should take 1 month to learn might actually take 1 year because you need to understand all that it is based on.
4) Try to find an applied implementation. For example photogrammetry applies a lot of linear algebra. It is easer to learn linear algebra if you find an implementation of photogrammetry and try to rewrite it. This forces you to completely understand how the math works. You should read the parts of the math books that you need.
by analog31 on 11/25/21, 2:38 PM
I was a college math major, and I admit that I might have flunked out had I been told to learn my math subjects by reading them from the textbooks without the support of the classroom environment. It may be that the books are "easy to read if a teacher is teaching them to you."
Talking and writing math also helped me. Maybe it's easier to learn a "language" if it's a two way street and involves more of the senses.
Perhaps a substitute to reading the stuff straight from a book might be to find some good video lectures. Also, work the chapter problems, which will get your brain and hands involved in a more active way.
As others might have mentioned, there's no strict formal math notation. It's the opposite of a compiled programming language. In fact, math people who learn programming are first told: "The computer is stupid, it only understands exactly what you write." In math, you're expected to read past and gloss over the slight irregularities of the language and fill in gaps or react to sudden introduction of a new symbol or notational form by just rolling with it.
by yongjik on 11/25/21, 6:51 PM
Also remember that math notations are meant for people. If you learn the sigma summation notation, and if you wonder "So I understand what is \Sigma_{i=0}^{10}, but what is \Sigma_{i=0}^{-1}?" then you're wondering irrelevant stuff. If a math notation is confusing to use, good mathematicians will simply not use it and devise an alternative way to express it (or re-define it more clearly for their purpose).
Also, don't skip exercises. Try to solve at least 1/3 of them after each chapter. Exercises are the "actually riding a bike" part of learning how to ride a bike.
by solmag on 11/25/21, 1:38 PM
by CornCobs on 11/25/21, 2:41 PM
Think about it this way. A scientist, wanting to communicate his ideas with fellow academics, is not going to spend more than half the paper on pedantics and explaining notations which everyone in their field would understand. Else what is the purpose of creating the notations? They might as well write their formulas and algorithms COBOL style!
Ultimately mathematics, like most human-invented languages, is highly tribal and has no fixed rules. And I believe we are much richer for it! Mathematicians constantly invent new syntax to express new ideas. If there was some formal reference they had to keep on hand every time they need to write an equation that would hamper their speed of thought and creativity. How would one even invent something new if you need to get the syntax approved first!
TL;DR: Treat math notation as any other human language. Find some introductory texts on the subject matter you are interested in to be "inducted" into the tribe
by wizardforhire on 11/25/21, 5:35 PM
2] dive deep into the history of math.
3] youtube… 3 blue 1 brown, stand up maths, numberphile, kahn academy. These channels are your friends.
4] don’t give up and make it fun. Once you’re bit by the bug of curiosity and are rewarded with understanding you’ll most probably be unstoppable but still, its a long road. Better to focus on the journey.
Lastly, the notation is what it is because of the nature of math itself coupled with the history of who was doing the solving exacerbated by the cultural uptake. There have been and will continue to be new notation. Its unfortunate that often to learn a new concept the barrier is with parsing the syntax. Stick with it and stay curious and those squiggles will take on new magical and profound meanings.
by Koshkin on 11/26/21, 1:35 AM
by sumnole on 11/25/21, 3:38 PM
https://www.amazon.com/Mathematical-Notation-Guide-Engineers...
by ReleaseCandidat on 11/25/21, 3:13 PM
As all the others already told you. you don't learn by reading alone.
by sealeck on 11/25/21, 3:53 PM
Try to read it aloud.
"The Probability Lifesaver" has a lot of good mathematics tips (which are not even mathematics related) most of which are not probability-specific. It's a goldmine.
by fsloth on 11/25/21, 3:07 PM
As a sidenote I have MSc in Physics with a good dollop of maths involved and I am quite clueless when looking at a new domain so it's not as if university degree in non-related subject would be of any help...
by OneTimePetes on 11/25/21, 4:27 PM
Math notation becomes very readable, as soon as the teacher writes a example out on the black board, and that is why i will never forgive wikipedia / wolfram / latex for not having a interactive "notation to example expansion". They had such a chance to reform the medium - to make it more accessible to beginners and basically forgot about them.
by merlinran on 11/25/21, 8:25 PM
Let me explain a little bit. Just like a foreign language you stopped learning and using after high school, what prevents you from using it fluently is not just the vocabulary and grammar, but also the intuition and the understanding of the language as a whole. Luckily, math is a human designed language, with linear algebra and calculus being the fundamentals. And again, learning them is about building intuition on why and how they are used, so whenever you encounter transformation, you think in terms of vectors and matrices, and derivative for anything relevant to rate of change. By using carefully designed examples and visual representation, Grant Sanderson greatly smoothed the learning curve in the video courses. Try it out and you'll see.
Beyond that, different fields do have slightly different notation. When you first encounter them, just grab some introduction books or online courses and skim over the very first chapters.
by wenc on 11/25/21, 3:06 PM
A lot of it is convention, so you do need a social approach - ie asking others in your field. For me it was my peers, but these days there’s Math stack exchange, google, and math forums. Also, first few chapters of an intro Real Analysis text is usually a good primer to most common math notation.
When I started grad school I didn’t know many math social norms, like the unstated one that vectors (say x) were usually in column form by convention unless otherwise stated (in undergrad calc and physics, vectors we’re usually in row form). I spent a lot of time being stymied by why matrix and vector sizes were wrong and why x’ A x worked. Or that the dot product was x’x (in undergrad it was x.x). It sounds like I lacked preparation but the reality was no one told me these things in undergrad. (I should also note that I was not a math major; the engineering curriculum didn’t expose me much to advanced math notation. Math majors will probably have a different experience.)
by erichocean on 11/25/21, 7:55 PM
Almost all hand "proofs" in math papers have minor bugs, even if they're mostly correct in the big picture sense.
Even math designed to support programming (e.g. in computer graphics) is almost always incomplete/outright wrong in some meaningful way.*
But with a struggle, it's still largely usable/useful.
I've used advanced mathematics most of my career to do work (i.e. read a paper, implement it), but the ability to actually use math to do new things in computer science that mattered only to me only happened after I learned TLA+, which took a few weeks of solid study to click. Since then, it's been a pleasure. My specs have never been this good!
Lamport's video course on TLA+ is pretty good, but honestly I've read everything I can find on the topic so it's difficult to know what helped me the most.
*I think this is because, short of doing formal mathematics, there's no way to "test" your math. It's the equivalent of expecting programmers to write correct code the first time with no tests, and without even running the code.
by thorin on 11/25/21, 1:37 PM
by zoomablemind on 11/25/21, 4:38 PM
This sounds somewhat abstract, as the math field is vast. If you consider the next level from where you believe your present standing is, I would try to revisit the college-level math which you probaby experienced back in time.
Generally, the textbooks rely on previous knowledge and gradually feed the new concepts, including the math notation as needed in the new scope.
I find it easier to get the feel for the notation by actually writing it by hand. Indeed it's just an expression tool. Also, you may develop your own way of making notes, as you go on dealing with math-related problems.
But in the core of this you are learning the concepts and an approach to reasoning. Of course, for this path to have any practical effect, you would need to memorize quite a bit, some theorems, some methods, some formulas, some applications. Internalizing the notation will help you condense all of that new knowledge.
Picking a textbook for your level is all that is needed to continue the journey!
by amitkgupta84 on 11/25/21, 3:12 PM
When that fails, math.stackexchange.com is a very active and helpful resource. You can ask what certain notation means, and upload a screenshot since it’s not always easy to describe math notation in words.
If you don’t want to wait for a human response, Detexify (https://detexify.kirelabs.org/classify.html) is an awesome site where you can hand draw math notation and it’ll tell you the LaTeX code for it. That often gives a better clue for what to search for.
For example you could draw an upside down triangle, and see that one of the ways to express this in LaTeX is \nabla. Then you can look up the Wikipedia article on the Nabla symbol. (Of course in this case you could easily have just searched “math upside down triangle symbol” and the first result is a Math Stackechange thread answering this).
by whatsakandr on 11/25/21, 7:12 PM
by CogitoCogito on 11/25/21, 7:54 PM
It's kind of a cop-out, but to be fair it's basically what I would say for programming as well. Try to simultaneously write code that clear to yourself and clear to others. There's no perfect method. Just constantly self-critique and try to improve.
by anter on 11/25/21, 1:48 PM
by Zolomon on 11/25/21, 4:52 PM
by xemdetia on 11/25/21, 10:01 PM
by andreskytt on 11/26/21, 6:42 AM
by phtrivier on 11/25/21, 3:05 PM
Or that the notation differs from books to books ?
(In my case, I learned the notation via French math textbooks, and in the first day of college/uni we litteraly went back to "There is a set of things called natural numbers, and we call this set N, and there is this one thing called 0, and there is a notion of successor, and if you keep taking the successor it's called '+', and..." etc..
But then, the French, Bourbaki-style of teaching math is veeeeeeeery strict on notations.
by tclancy on 11/25/21, 2:00 PM
by kaetemi on 11/25/21, 3:49 PM
I can read and understand undocumented code with relative ease. Reading math notation without any documentation seems pretty much impossible, otoh.
by the__alchemist on 11/25/21, 3:51 PM
by zwerdlds on 11/25/21, 4:36 PM
Pick a direction (maybe discrete math, if you're trying to do CS) and get a book (I like EPP, as it is super accessible) and go, in order, through each chapter. Read, do the example problems, and do EVERY SINGLE PROBLEM in the (sub)chapter section enders.
Its a time commitment, but if you really want to learn it, this is one way to do so. IMO finding the right textbook is key.
by bradlys on 11/25/21, 6:51 PM
I’d highly recommend this book. It’s what I had for my intro to proofs class in college and it was the best book I found for understanding. I found many other books on this topic to be kinda garbage but this one was amazing.
by janeroe on 11/25/21, 4:00 PM
So many answers and no correct one yet. Read and solve "How to Prove It: A Structured Approach", Velleman. This is the best introduction I've seen so far. After finishing you'll have enough maturity to read pretty much any math book.
by gammalost on 11/25/21, 5:07 PM
But you'll have to be a bit realistic when going through the book, it's going to take a good while.
by amelius on 11/25/21, 4:52 PM
by pgcj_poster on 11/25/21, 9:18 PM
by pmontra on 11/25/21, 2:14 PM
by readme on 11/25/21, 4:25 PM
if it's not, try intuition
if that fails, email your mathematician friend and ask
don't have a mathematician friend? there's your next goal, go make one.
by conjectures on 11/25/21, 2:54 PM
Then some textbooks with exercises (e.g. Axler on lin alg).
The notation is usually an expression of a mental model, so just approaching via notation may cause some degree of confusion.
by lukeplato on 11/25/21, 6:12 PM
by aabaker99 on 11/25/21, 6:18 PM
by thuccess129 on 11/25/21, 6:09 PM
by dekhn on 11/25/21, 11:20 PM
E Psi = H Psi
and we all joked you could just cancel the Psi and so E=H.
several very kind people explained vector calculus to me ("bold means a matrix, and this dot means matrix multiplication") but to be honest, I still can't read math notation but if you show me anything in numpy I'll understand it immediately.
by Grustaf on 11/25/21, 7:34 PM
by motohagiography on 11/25/21, 3:42 PM
Smart people often don't know the difference between an elegant abstraction that conveys a concept and a black box shorthand for signalling pre-shared knowledge to others. It's the difference between compressing ideas into essential relationships, and using an exclusive code word.
This fellow does a brilliant job at explaining the origin of a constant by taking you along the path of discovery with him, whereas many "teachers" would start with a definition like "Feigenbaum means 4.669," which is the least meaningful aspect to someone who doesn't know why. https://www.veritasium.com/videos/2020/1/29/this-equation-wi...
It wasn't until decades after school that it clicked for me that a lot of concepts in math aren't numbers at all, but refer to relationships and relative proporitons and the interactions of different types of things, which are in effect just shapes, but ones we can't draw simply, and so we can only specify them using notations with numbers. I think most brains have some low level of natural synesthesia, and the way we approach math in high school has been by imposing a three legged race on anyone who tries it instead.
Pi is a great example, as it's a proportion in a relationship between a regular line you can imagine, and the circle made from it. There isn't much else important about it othat than it applies to everything, and it's the first irrational number we found. You can speculate that a line is just a stick some ancients found on the ground and so its unit is "1 stick" long, which makes it an integer, but when you rotate the stick around one end, the circular path it traces has a constant proportion to its length, because it's the stick and there is nothing else acting on it, but amazingly that proportion that describes that relationship pops out of the single integer dimension and yields a whole new type of unique number that is no longer an integer. The least interesting or meaningful thing about pi is that it is 3.141 etc. High school math teaching conflates computation and reasoning, and invents gumption traps by going depth first into ideas that make much more sense in their breadth-first contexts and relationships to other things, which also seems like a conspiracy to keep people ignorant.
Just yesterday I floated the idea of a book club salon idea for "Content, Methods, and Meaning," where starting from any level, each session 2-3 participants pick and learn the same chapter separately and do their best to give a 15 minute explanation of it to the rest of the group. It's on the first year syllabus of a few universities, and it's a breadth-first approach to a lot of the important foundational ideas.
The intent is I think we only know anything as well as we can teach it, so the challenge is to learn by teaching, and you have to teach it to someone smart but without the background. Long comment, but keep at it, dumber people than you have got further with mere persistance.
by Jugurtha on 11/26/21, 11:00 AM
It takes you from the concept of number.
by 734129837261 on 11/25/21, 3:15 PM
Honestly, most math formulas can be turned into something that looks like C/C++/C#/Java/JavaScript/TypeScript code and become infinitely more readable and understandable.
Sadly, TypeScript is one of the languages that is attempting to move back to idiocy by having generics named a single letter. Bastards.
by caffeine on 11/25/21, 7:43 PM
by Tycho on 11/25/21, 6:02 PM
by teawrecks on 11/25/21, 6:38 PM
by mixmastamyk on 11/25/21, 10:17 PM
by housu on 11/25/21, 7:43 PM
by b20000 on 11/25/21, 4:57 PM