Ask HN: How did you learn math notation?

https://news.ycombinator.com/item?id=29341152

by rullopat • 5 years ago

123 249 5 years ago

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Hi all! I'd really like to learn "higher level than highschool" math as a (long time ago) college drop out, but I find it really hard to read anything because of the math notations and zero explanation of it in the context. I didn't find on the web any good resource on the topic, do you any advice / link? Thanks!

ColinWright14• 5 years ago

I think a real problem in this area is the belief that there is "one true notation" and that everything is unambiguous and clearly defined.

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.

rullopat2• 5 years ago

I was trying to grasp some of the papers linked in the Valhalla DSP block, for example this one: http://www2.ece.rochester.edu/courses/ECE472/resources/Paper...

There is a formula with a triangle and I don't get what's that about, for example.

nickcw• 5 years ago

The triangle is a capital Greek Delta and is usually used to indicate a change in something. So ΔT means some change in T.

ColinWright14• 5 years ago

You say "There's a formula with a triangle ..." without telling me where. That's not real helpful, and you're making me do the work to find out what you're talking about. If you want assistance to get started, you need to be more explicit.

However, I have done that work, so I've looked, and in the second column of page 210 there's a "formula with a triangle":

t_c = 5 \middot 10^{-5} \sqrt( V / Dt )

... where the "D" I've used is where the triangle appears in the formula.

But that can't be it, because just two lines above it we have:

"For a pulse of width Dt, the critical time ..."

So that's stating that "Dt" is the width of the pulse, and should be thought of as a single term.

So maybe that's the wrong formula, or maybe it was just a bad example. So trying to be more helpful, the "triangle" is a Greek capital delta and means different things in different places. However, it is often used to mean "a small change in".

https://en.wikipedia.org/wiki/%CE%94T

FWIW ... at a glance I can't see where that result is derived, it appears simply to be stated without explanation. I might be wrong, I've not read the rest of the paper.

Jensson32• 5 years ago

That is a physics based paper, it used physics notation. Can find common ones here (including that triangle):

https://en.wikipedia.org/wiki/List_of_common_physics_notatio...

NamTaf3• 5 years ago

What you're looking at is calculus, specifically differentiation. This is pretty core to understanding physics, because so much of physics depends on the time-evolving state of things. That's fundamentally what's happening here.

The triangle, for example, is the upper-case greek letter delta, which in calculus represents 'change of'. You might have heard of 'delta-T' with respect to 'change of time'.

In calculus, upper-case delta means 'change over a finite time' vs lower-case delta meaning 'instantaneous change'. The practical upshot, for example, is that the lower-case is the instantaneous rate-of-change at an instant in time, whereas the upper-case is the change over a whole time (e.g. the average rate of change per second for time = 0 seconds to time = 3 seconds).

If you are trying to grok this, I would suggest an introductory calculus or pre-calculus resource. It doesn't have to be a uni textbook - higher-level high school maths usually teaches this. In this particular case, the Khan Academy would be my recommendation because it is about the right level (we're not talking esoteric higher-level university knowledge here) and it is eminently accessable. For example, this link may be a good starter in this instance:

https://www.youtube.com/watch?v=MeU-KzdCBps

forgetfulness2• 5 years ago

Looks like you need to grind through an elementary calculus book. With the exercises, you may think you build intuition by reading just the definitions, but half of the understanding is tacit and you get through the exercises.

If you're trying to get into signal processing, it'll involve calculus in complex numbers, and knowledge of that is often gained through plodding through proofs and exercises over and over.

pfortuny• 5 years ago

You probably want to try and read Courant-Johns Calculus (I forget the exact title).

kens• 5 years ago

Everyone is talking about the Δ symbol, but the real problem that you'll encounter will be later in the paper where they start talking about H(ω), which is the Fourier transform of the impulse function (equation 4 and following). You'll need to know a fair bit about Fourier transforms and impulse responses and filter design to get through this section. The notation is the least of the problems.

One place to start is https://en.wikipedia.org/wiki/Impulse_response

BeetleB• 5 years ago

A few points:

1. You're reading a journal article. They will assume you know the notation not just of the broader discipline (e.g. physics/electrical engineering), but of the subdiscipline and at times the subsubdiscipline. Journal papers are explicitly written not to be easy to comprehend by beginners.[1] Notation will be only one problem you'll face.

2. As has been pointed out, this is not a mathematics paper. Mathematicians have their own notation, as do physicists and engineers. As I mentioned in the above bullet, they can have their own notation even in subdisciplines (e.g. circuit folks use "j" for the imaginary number, and semiconductor folks use "i"). There is a lot of overlap in notation amongst these parties, but you should never assume because you know one notation that you'll easily understand the math written by other fields.

3. Most introductory textbooks will explain the basic notation. Unfortunately, I often do find gaps where you go to higher level textbooks and they use notation that they don't explain (i.e. they assume you've seen it before), but is not covered in the prior textbooks.

4. Finally, sorry to say this, but "delta" (the triangle) for representing change is used in almost all sciences and engineering. It was heavily used in my high school as well. If you're struggling with this you really need to read some introductory textbooks in, say, physics.

[1] I'm not kidding. I've spent time in academia and I've complained how obtuse some articles are, and almost universally the response is "We write for other experts, not for new graduate students". One professor took pride at the fact that in his field, one can comprehend only about one page of a paper per day - and this coming from someone who is an expert. These people have issues.

User23• 5 years ago

Edsger Dijkstra, who was a mathematician by training, wrote a wonderful little monograph on this subject called The notational conventions I adopted, and why[1]. I am particularly fond of his commented equational proof format.

[1] https://www.cs.utexas.edu/users/EWD/transcriptions/EWD13xx/E...

cabalamat2• 5 years ago

I think a real problem in this area is the belief that there is "one true notation" and that everything is unambiguous and clearly defined.

One main cause for this belief is that in a programming there is one true noation (or rather, a separate one for each language) that is unambiguous and clearly defined.

I dislike maths notation as I find it lacks rigour.

Jensson32• 5 years ago

there is one true noation (or rather, a separate one for each language) that is unambiguous and clearly defined.

This is such a disingenuous take. How many of the source code files you write are 100% self contained and well defined? I'd bet not a single one of them are. You reference libraries, you depend on specific compiler/runtime/OS versions, you reference other files etc. If you take a look at any of these scientific papers you call "badly defined", did you really go through all of the referenced papers and look if they defined the things you didn't get? If not then you can't be sure that the paper uses undefined notation. If you argue that it is too much work to go through that many references, well that is what you would have to do to understand one of your program files.

throwaway313384• 5 years ago

Came here to say the same thing harshly and laced with profanity. I guess I can back off a bit from that now.

I was filled with crushing disappointment when I learned mathematical notation is "shorthand" and there isn't a formal grammar. Same goes for learning writers take "shortcuts" with the expectation the reader will "fill in the gaps". Ostensibly this is so the writer can do "less writing" and the reader can do "less reading".

There's so much "pure" and "universal" about math, but the humans who write about it are too lazy to write about it in a rigorous manner.

I can't write software w/ the expectation the computer "just knows" or that it will "fill in the gaps". Sure-- I can call libraries, write in a higher-level language to let the compiler make machine language for me, etc. I can inspect and understand the underlying implementations if I want to, though. Nothing relies on the machine "just knowing".

It's feels like the same goddamn laziness that plagues every other human endeavor outside of programming. People can't be bothered to be exact about things because being exact is hard and people avoid hard work.

"We'll have a face-to-face to discuss this there's too much here to put in an email."

randomdata• 5 years ago

One main cause for this belief is that in a programming there is one true noation

And then there is SQL.

ColinWright14• 5 years ago

I dislike maths notation as I find it lacks rigour.

I see this a lot from programmers, but in essence, you seem to be complaining that maths notation isn't what you want it to be, but is instead something else that mathematicians (and physicists and engineers) find useful.

f38zf5vdt• 5 years ago

I'm glad I'm not the only person like this. I've never liked tradition math notation and found it about as useful as traditional musical notation, that is, hard to read for the layman and for no other reason than "this is how people have been doing it for a long time". Maybe I'm the minority, but when I read a CS paper I mostly ignore the maths and then go to the source code or pseudocode to see how the algorithm was implemented.

randomNumber72• 5 years ago

Formulas would also be easier to read if they would not name all their variables and functions with one character.

If programmers would write code like that (even fortran programmers use 3 characters), noone would be able to understand the code...

bigbillheck• 5 years ago

in a programming there is one true noation (or rather, a separate one for each language) that is unambiguous and clearly defined

Yes this is why we all use Hungarian notation and GNU indentation.

xtiansimon• 5 years ago

Where is the legend/key?

I recall my disappointment when as an artist I started studying different maths for use in animation. I would open a book from a university library and expected to find a page with summary of notation used in the book. Maps have this, I would grumble, why not math books?

bajsejohannes2• 5 years ago

I think a real problem in this area is the belief that there is "one true notation" and that everything is unambiguous and clearly defined.

Just to back up this point: In probably every university-level math book I've read, they introduce and explain all the notation used. In the preface and/or as concepts are introduced.

There are lists at wikipedia [1] and other places, but I'm not sure how valuable it is out of context.

[1] https://en.wikipedia.org/wiki/Glossary_of_mathematical_symbo...

nicoburns• 5 years ago

Ha, you've clearly used better books than me. I've read plenty where they glossed over the notation and expected you to guess.

readme2• 5 years ago

came here to say this

some people literally make the notation up as they go along

klibertp18• 5 years ago

Why are you telling OP what his problem is? Shouldn't you address his pain points, not your rationalization of them?

I wrote it many times already and am bit tired of it, so just a quick summary:

- programmers[1] also use cryptic notation and tend to think in concepts rather than syntax

- nevertheless, programmers spend a lot of time commenting the code, documenting it, specifying it, and so on.

- why can't mathematicians emulate it? What is so wrong about attaching additional few pages to every paper that nobody wants to do it? Pages with explanation of the syntax used, even the common bits. And you know what they could also do? Link to external resources with explanations! But no. This is not happening. Do their PDFs have a size limit or something? Is inserting a link into a paper considered some kind of blasphemy?

I don't know the reason, but in all the discussions on this topic mathematicians almost always underestimate the importance of knowing the syntax. It's much more important for comprehension than they tend to admit. And in the end they do exactly nothing to make the syntax more approachable for newcomers. And then newcomers are out-goers in a heartbeat. It's so obvious that I can't help thinking it's premeditated...

EDIT: [1] Among many others, of course.

Jtsummers5• 5 years ago

Mathematicians do document and comment, that's what papers and textbooks are: commentary on the math. They don't throw out formulae and equations and call it a day. Attaching a full tutorial for every level of reader is tantamount to attaching Stroustrup's C++ books to every C++ program, or K&R to every C program. You wouldn't do that, you'd expect the reader to ask you for references or to seek them out themselves.

TheTrotters• 5 years ago

Note that the OP is asking about college-level math, not cutting-edge papers.

Textbooks routinely have a list of symbols and their definitions.

But, from my experience, notation is rarely the problem. I’d bet that the root cause of OP’s frustration is lack of understanding of concepts, not notation. (But, of course, it’s hard to say more without specific examples).

Grustaf4• 5 years ago

There pretty much is one true notation. There could be some slight variations, like bolding vectors, putting an arrow over them or not distinguishing them at all from scalars. But 95% of the time everyone uses the same notation.

ColinWright14• 5 years ago

I don't know your background, but I wonder how broad it is in terms of mathematical topics. The notations used in Algebraic Topology vs Category Theory vs Algebraic Number Theory vs Analytical Combinatorics vs Complex Analysis.

This isn't a criticism, it's just that notations vary wildly in those areas, and there's lots of cross-over of notations, not all of which agree with each other.

I'm not an expert, but I've had some exposure to the problem(s).

kazinator6• 5 years ago

I think a real problem in this area is the belief that there is "one true notation" and that everything is unambiguous and clearly defined.

No, that belief isn't the problem; that actual status quo itself is obviously the problem. There are numerous notations and authors don't explain what they are using, assuming everyone has recursively read all of their references depth-first before reading their paper.

hansvm• 5 years ago

Why does RTFM not apply to mathematics dependencies?

thorin• 5 years ago

You might be better picking an area, and trying to work out the notation relating to that area e.g. vectors / matrices / calculus etc. As Colin says below there are often multiple equivalent ways of representing things across different fields and timeframes. I seem to remember maths I studies in Elec Eng looking different but equivalent to the way it was represented in other disciplines

solmag3• 5 years ago

Practice, just like you learned programming. "The Context" gives you the meaning for the notation, sadly. You have to kind of know it to understand the notation properly.

solmag3• 5 years ago

You can also get sufficiently angry and just write out linear algebra books and what not in Agda / Coq / Lean if it pisses you off so much (I've done a bunch of exercises in Coq)

db48x3• 5 years ago

I like the approach they took in Structure and Interpretation of Classical Mechanics, where the whole book is done in Scheme:

    (define ((Lagrange-equations Lagrangian) q)
      (- (D (compose ((partial 2) Lagrangian) (Gamma q)))
         (compose ((partial 1) Lagrangian) (Gamma q))))

dqpb• 5 years ago

Ultimately I think this is the right answer.

anter• 5 years ago

Related question, does anyone know of any websites/books that have mathematical notation vs the computer code representing the same formula side by side? I find that seeing it in code helps me grasp it very quickly.

hdinh• 5 years ago

https://github.com/Jam3/math-as-code

hatmatrix• 5 years ago

I think the problem is that there is no authoritative text, that I know of, and as ColinWright says, the same ideas can be notated differently by different fields or sometimes by different authors in the same field (though often they converge if they are in the same community).

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.

smitty1e• 5 years ago

The Greek alphabet would like to thank all the scholars for the centuries of overloading and offer a "tee hee hee" to all of the students tormented by attendant ambiguities.

Tough love, kids.

cjfd• 5 years ago

Could it be that you are trying to read things that are a bit too advanced? Maybe look for some first year university lecture notes? In general, if you cannot follow something, try to find some other materials on the same subject, preferably more basic ones.

tclancy• 5 years ago

I’ve run into this problem as well and it’s put me off learning TLA+ and information theory, which bums me out. I assume there’s a Khan Academy class that would help but it’s hard to find.

ivan_ah• 5 years ago

As a starting point you can check out the notation appendices from my books: https://minireference.com/static/excerpts/noBSmathphys_v5_pr... https://minireference.com/static/excerpts/noBSLA_v2_preview.... You can also see this excerpt here on set notation https://minireference.com/static/excerpts/set_notation.pdf

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.

pmontra• 5 years ago

School, one year after another. Delta t stuff were probably in the last year before college.

dwheeler• 5 years ago

There is no single authoritative source for mathematical notation. That said, there are a lot of common conventions. You could do worse than this NIST document if it's just a notation question:

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!

analog31• 5 years ago

Maybe a problem is trying to learn it by reading it.

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.

CornCobs• 5 years ago

Mathematics is a lingo and notations are mostly convention. Luckily people generally follow the same conventions, so my best advice if you want to learn about a specific topic is to work through the introductory texts! If you want to learn calculus find an introductory college text. Statistics? There are traditional textbooks like Introduction to Statistical Learning. The introductory texts generally do explain notation which may become assumed knowledge for more advanced texts, or as you seem to be wanting to read, academic papers. If those texts are still too difficult, then maybe move down to highschool text first.

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

swframe2• 5 years ago

Naively, I would say the following:

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.

gspr2• 5 years ago

I hear this question asked quite often, particularly on HN. I think the question is quite backwards. There is little value alone in learning "math notation", even ignoring what many people point out (there is no one "math notation"). "Math notation", at best, translates into mathematical concepts. Words, if you will, but words with very specific meaning. Understanding those concepts is the crux of the matter! That is what takes effort – and the effort needed is that of learning mathematics. After that, one may still struggle with bad (or "original", or "different", or "overloaded", or "idiotic", or…) notation, of course, but there is little use in learning said notation(s) on their own.

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.

conjectures• 5 years ago

Khan academy and Schaum's Outlines are your friends.

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.

phtrivier• 5 years ago

Do you mean all the introductory mathematics books you tried fail to properly explain the notation ?

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.

wenc• 5 years ago

I learned it by asking peers in grad school what stuff meant. And working through the math myself (it was a slog at first) and then writing stuff out it in LaTeX. When one is forced to learn something because one needs to take courses and to graduate, the human brain someone figures out a way.

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.)

fsloth• 5 years ago

I think good first resource would be the book and lecture notes in an introductory university course treating the specific domain you are interested in because often lots of things in notation are domain specific. Lots of good open university lectures out there, if not sure from where to start the MIT open courseware used to be a good first guess for accessing materials.

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...

excitednumber• 5 years ago

Read calculus made easy. It won’t solve all of your problems but damn is this book good.

https://www.gutenberg.org/ebooks/33283

amitkgupta84• 5 years ago

First, just to state the obvious, if you can accurately describe a notation in words, you can do an Internet search for it.

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).

ReleaseCandidat• 5 years ago

Well, the real fun is deciphering a lower case xi - ξ - when written on the blackboard (or whiteboard), specially compared to a lower case zeta - ζ (fortunately way less commonly used).

As all the others already told you. you don't learn by reading alone.

pbhjpbhj3• 5 years ago

xi vs epsilon vs zeta when chalked on a blackboard at pace and read from 30m away!

Learning the Greek alphabet pays off.

contravariant• 5 years ago

Ah, yes. I remember the time when I saw someone write something vaguely like the following

[0,ξ[={x|0<=x<ξ}

Which was fun trying to figure out when written in handwriting where ξ,{,} all look the same.

If you can't figure out what it's supposed to be, this equation starts with a half-open interval denoted: [ξ,0[. This notation has some advantages but can be make things hard to read.

734129837261• 5 years ago

If math was a programming language, all mathematicians would be fired for terrible naming conventions and horrible misuse of syntax freedom.

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.

todd8• 5 years ago

For about $5 you can find an old (around 1960-1969) edition of the "CRC Handbook of Standard Mathematical Tables. I've owned two of the 17th edition published in 1969, because back then hand calculators didn't exist and many of the functions used in mathematics had to be looked up in books, like what is the square root of 217. Engineers used these handbooks extensively back then.

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.

specproc• 5 years ago

As someone else who'd like to have a better understanding of formal notation, I think that's a great answer. Thanks for taking the time.

sumnole• 5 years ago

The rhino book is a good dead tree reference.

https://www.amazon.com/Mathematical-Notation-Guide-Engineers...

motohagiography• 5 years ago

I sometimes think math notation is a conspiracy against the clever but lazy. Being able to pronounce the greek alphabet is a start, as you can use your ear and literary mind once you have that, but when you encounter <...>, as in an unpronouncable symbol, the meaningless abstraction becomes a black box and destroys information for you.

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.

kaetemi• 5 years ago

Math notation feels like a write-only language somehow.

I can read and understand undocumented code with relative ease. Reading math notation without any documentation seems pretty much impossible, otoh.

CogitoCogito5• 5 years ago

You get better at it the more you do. A tip is also to actually change a mathematical exposition into a form you better understand (e.g. by writing it in a different notation and/or expanding it out in words to make the existing notation less dense). Basically convert the presentation into the way you would personally like to see it.

If you do this enough, the process becomes easier and the original notation becomes easier to understand. But it takes a lot of time and patience (as I'm sure it took for you understand undocumented code did as well).

the__alchemist• 5 years ago

It can be quite provincial. Could you please post a link to a paper or website that has notation you'd like to understand? Which domains are you interested in particularly?

sealeck• 5 years ago

Most textbooks come with a list of definitions.

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.

janeroe• 5 years ago

I find it really hard to read anything because of the math notations and zero explanation of it in the context.

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.

slipmasterflex• 5 years ago

Great recommendation!

tgflynn• 5 years ago

It sounds like you're trying to read papers that assume a certain level of mathematical sophistication without having reached that level. Typical engineering papers will assume at least what's taught in 2 years of college level mathematics, mainly calculus and linear algebra, and no they aren't going to be explaining notation used at that level.

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.

arcbyte• 5 years ago

The author explained his problem and asked for resource recommendations.

Your response is to scold him for having the problem he already said he had and instead of recommending resources you told him to go look on the internet.

And you implied he doesn't have motivation.

readme2• 5 years ago

the notation you need to know should be defined somewhere in the book or paper you're reading

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.

kergonath5• 5 years ago

if it's not, try intuition

If it’s not, the book is badly written. Most of the time, you can’t rely on a specific bit of notation to be consistent across books or articles. Smart arses who try to impress the readers with their fancy unique notations are the bane of scientists doing literature reviews.

90% of the time, there needs to be a keyword when a symbol is introduced, e.g. “where Λ is the time-dependent foo operator” so you can get a textbook to find what the fuck a “foo operator” is. Then, the first time you spend a day learning what it is, and the next million times you mumble “what a stupid notation for such a straightforward concept”.

OneTimePetes• 5 years ago

Through a really nice and helpful math prof who took time out of her day to explain it to those in the "im in trouble" additional course. Forever grateful for that, would have failed otherwise.

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.

anthomtb• 5 years ago

This block post, which has been referenced several times on HN, was a god send for me: https://www.neilwithdata.com/mathematics-self-learner

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.

zwerdlds• 5 years ago

My suggestion to you is going to sound pithy, but its what worked for me: do problems. Lots and lots of problems.

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.

zoomablemind• 5 years ago

... I'd really like to learn "higher level than highschool" math...

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!

amelius• 5 years ago

I have a notation problem. I want to write "approximately 24 volt" on my printed circuit board, but I have little space. I could write "≈24V", but the wavy symbol makes it look like it is AC instead of DC. How to solve this without adding more characters or changing my circuit?

pbhjpbhj3• 5 years ago

Use =c.24V (read as 'equals circa 24 volts', circa is Latin for 'about').

Use the 3 line version of approximately equal (looks like tilde above an equal sign, ≅).

Zolomon• 5 years ago

I asked this on Mathematics StackExchange some time ago and got good responses:

https://math.stackexchange.com/a/13281

b20000• 5 years ago

get some english math textbooks used in high schools across europe.

gammalost• 5 years ago

I learned most of my university math through "Calculus a Complete Course". But it's a bit expensive so I would recommend you buy an older version of the book where you can find a free solution pdf.

But you'll have to be a bit realistic when going through the book, it's going to take a good while.

wizardforhire• 5 years ago

1] learn the greek alphabet if you haven’t already.

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.

Tycho• 5 years ago

I think the Princeton Companion to Mathematics covers a lot of it at the start of the book.

Syzygies2• 5 years ago

I'm a math professor, and my students find it revelatory to understand math as I talk and draw.

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.

eointierney2• 5 years ago

Super answer! I wish you were one of my professors, and I had excellent professors.

If I may humbly add, try making your own notation and playing around with it. Very rapidly one realizes just how hard a problem good notation is.

BSOhealth• 5 years ago

Richard Feynman did just this and described it at least in his great autobiography, “Surely you must be joking, Mr. Feynman!”

Syzygies2• 5 years ago

Actually, making my own notation is how I start puzzling out any problem. When one reformulates a problem, the appearance keeps changing but the problem is still there. It often takes multiple reformulations to "see" a problem, unencumbered by artifacts of birth.

The other day an EE grad student came to my office to show me his lab's research. After changing notation a couple of times, I was able to recognize that their novel combinatorial problem was an instance of graph coloring.

j7ake• 5 years ago

Great insight! I’ve definitely encountered mathematically inclined people but who cannot read or write math. Now it makes sense to me.

Also I’ve found the converse true. There are people who can manipulate mathematical symbols very well but actually don’t understand the big picture or general direction. The analogy would be that there are people who can write and read music notes (even transpose to different keys) without hearing it in their head (I was one of them).

thuccess129• 5 years ago

Look for an etymology dictionary on math notation? The biographical sketch of the person who introduced the equal sign is an interesting read.

lukeplato• 5 years ago

I found that there is a physicality/motion to the progression of notation that you learn by solving a lot of problems, especially solving them quickly during tests

aabaker99• 5 years ago

Is there any particular topic? I agree with other posters though that the notation is a short hand for the concepts and you need the concepts, not the notation.

teawrecks• 5 years ago

By doing math. Khanacademy has a lot of higher than highschool math courses you could check out.

yongjik• 5 years ago

Try reading a good undergraduate calculus textbook. It would be hefty and a bit wordy, and it may take a few months to go through, but calculus requires surprisingly little amount of prior knowledge - even the concept of limit should be defined in the textbook (the famous epsilon-delta).

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.

bradlys• 5 years ago

https://www.amazon.com/Introduction-Mathematical-Reasoning-N...

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.

whatsakandr• 5 years ago

I have a masters in engineering, but there was a lot of pure math things that I never understood until recently. I found the same approach to learning software concepts and APIs. Just start at the one you don't know and recursively explore the concepts until you find stuff you do know.

klodolph• 5 years ago

[…] I find it really hard to read anything because of the math notations and zero explanation of it in the context.

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.

Grustaf4• 5 years ago

If you don’t remember notation, surely you don’t remember the material either, so why not just skim through the basic textbooks?

caffeine• 5 years ago

Khan academy would be a great place - generally in high school you learn enough to get through basic notation.

housu• 5 years ago

I got a bachelor's degree in math

CogitoCogito5• 5 years ago

My advisor's advise was basically "find a notation that you yourself like and understand well" and stick consistently to it. He said this in a context of having seen many standard notations before (so he's not saying to re-invent the wheel), but his point was just that notations and ways of thinking are personal. Try to be clear and precise (for yourself and others), but realize that you are crafting something that reflects you and your way of thinking.

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.

erichocean2• 5 years ago

Math papers can be pretty sloppy, and you don't realize this until you start working with formal mathematics—then it's obvious.

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.

rackjack• 5 years ago

All math notation was created by mathematicians who wanted to quickly represent something, either to:

- 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.

srcreigh• 5 years ago

First math course at university of Waterloo: https://cs.uwaterloo.ca/~cbruni/Math135Resources/courseNotes...

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.

merlinran• 5 years ago

Had been in the same situation for years. Read a paper, encounter the first equation, scratch my head and search around trying to understand it, give up. That changed half a month ago, after watching the Linear Algebra and Calculus course at https://www.youtube.com/c/3blue1brown/playlists?view=50&sort....

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.

pgcj_poster• 5 years ago

If you haven't already, I would start by learning the Greek alphabet and the sounds that the letters make. Conventions like Σ for sum and Δ for difference seem much less strange when you realize that they're basically just S and D.

xemdetia• 5 years ago

One of the best things I figured out that at least in the last 70 years or so ago it's pretty easy to find the "first" or foundational paper for a particular construct where they have to explain their notation for the first reader or they have the vibe of working with the new idea in the raw rather than 40 years later where it is matured. One example I use for this is hamming codes where some of the recent examples or explanations don't build it from first principles, but the original articles do explain it very clearly.

mixmastamyk• 5 years ago

Community college is a good way, low commitment as well.

dekhn• 5 years ago

after starting to take quantum chemistry the professor wrote up on the board:

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.

Koshkin• 5 years ago

To be honest, I have never understood this question. You learn notation as you go, along with the subject itself (whether it’s math, chemistry, or electronics). You definitely do not learn “notation” for its own sake - it just doesn’t make sense.

andreskytt• 5 years ago

I learned it the hard way at college. The prof just blasting off in a flurry of greek and us trying to keep up. Eventually he resorted to using the gothic alphabet and at that point I gave up. Every single field of math appears to have their own conventions and even these are but guidelines smart people often bend for perceived benefit. So I can’t really recommend anything other than learning the notation with the field. If you find yourself at loss, chances are you have missed some of the context and should backtrack.

Jugurtha• 5 years ago

Vladimir Smirnov's "A Course of Higher Mathematics" is excellent.

It takes you from the concept of number.

Ask HN: How did you learn math notation?

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