# Entering math through the side door [duplicate]

I am not really good at math, I'd say I'm a lot worse than good when it comes to math but I am a programmer so I have to learn to get over that fact.

A lot of times if I want to implement some code I have to read math formulas and it might as well be written in Russian... I speak no Russian.

Let's take "perturbation theory" for example. The formula looks like this:

now I know most of those symbols so that's not too bad.

but there's some formulas with like Sigma [summation] $\sum$ notation and a whole lot of other symbols that mean things in the math, physics world that I have no idea what they mean.

Like this squiggly $f$ or $s$ looking thing in this image:

$$\int_{-\infty}^\infty \sqrt{\frac{\sqrt{x^n}+1}{\alpha+\beta^\gamma}} \, dx$$

squiggly $f$ or $s$ square root of $x$ to the $n$th power plus one divided by alpha plus theta to the (?) power times a type of $d$ times $x.$

I know this might seem like a joke but i am very sincere with this question.

Is there a book or repository where these signs are laid out with their meanings? I've googled but can't find anything except maybe these images with names but that often leaves me more confused than I started.

So is there a singular book or at least a place that I can go to reference these symbols to get their meanings?

edit I don't think this is the same question as the one it's being marked as duplicate. I read that question and that's a very general question about how to get better at math.

That's not my question, I am asking more for a definition of terms. When you look at math papers there's often very cryptic symbols in the function. I am looking for a resource to say what those cryptic symbols mean and not generally how to do math.

I've tried to make this distinction clear. Let's use something simple for example. If you didn't know the rules of summation, how can you even start to decrypt what that formula means. What if you don't even know that that symbol is called sigma, where do you start?

I thought that there would be some resource for math that gave the names of those symbols and a way to use them. Back to the sigma example, if there was some resource that says that's called sigma and

Then I can say oh okay, I got it I can figure it out from here.

So while the other question was more general about learning math, this is more directed at how do I decypher those symbols. Hopefully that makes the differences between the questions more obvious.

• I don't mean to offend or to be derogative, I'm nearly curious: how is it possible that you are programmer without at least knowing the notion of integration? This is high school mathematics and you really ought to do yourself a favor and find your old books, or borrow them from someone before you move on college mathematics. – Ennar Aug 14 '15 at 13:14
• @Ennar I'm not trying to argumentative here, but I've written a lot of programs and never once used integration in any of them. – Colm Bhandal Aug 14 '15 at 13:23
• @ColmBhandal, I'm not saying there is any need to know integration techniques or that knowing any mathematics is prerequisite of being a good programmer, I'm just finding it very strange a programmer is not familiar with basic mathematics. I know quite a few and all have very solid mathematical backgrounds. EDIT: to draw an analogy, not knowing Newton's laws is hardly a sign of being a bad mathematician, but I've never met any that are not familiar with them. – Ennar Aug 14 '15 at 13:30
• guys I went to college for Art. I didn't even make it to 4 year university because of things. Nevertheless I wasn't a bad student but school was quite boring. Now I taught myself programming and since I use programs to solve problems that are concrete in my life, it's not so bad. I do programming because I like it, even though it's hard. – user1610950 Aug 14 '15 at 13:40
• @Ennar I see your point. I'm sure there is a strong correlation between the two. Though from my experience, there is also a divide between programmers and mathematicians, and it is particularly apparent when the subject of notation comes up. Programmers work mostly in a discrete world of bits, bounded memory, bounded time, function names that are full words with some hint of the function's job. Mathematicians work in a much "wilder" world e.g. where infinite quantities are allowed; they tend to favour shorthand, greek symbols, subscripts, superscripts etc. – Colm Bhandal Aug 14 '15 at 13:56

## 10 Answers

The bad news is that mathematical notation can be regarded as an incredibly dense kind of compression: you take a very subtle and carefully evolved idea and choose some symbol to represent it. The "quick meaning" of a symbol will typically lose all the subtlety.

There's a worse problem: typesetting is a pain, so it's pretty common for mathematicians to use the same notation to mean different (although often closely-related) things, depending on context. As an example,

$$\int_0^1 \frac{1}{x^2} dx$$ and $$\int_0^1 \frac{1}{1 + x^2} dx$$ denote rather different things. The first is called an "improper integral," and as a mathematician, I notice that because I see that the integrand (that is, $\frac{1}{x^2}$) turns out to head towards infinity as $x$ approaches the lower-limit of the integral (namely $0$), but I see that the second is not improper, because the integrand is nice over the whole interval $0 \le x \le 1$. If you happened to find a book that told you how to do "numerical integration" and tried to apply the ideas to the first integral, you might well get things wrong, and never know why.

In short: the way to learn enough math to program things responsibly is to make sure that the person who asks you to write the program either explains them to you, and how s/he wishes you to convert them to code, or to learn the math yourself.

I'm not trying to be snarky here, but if you said "I was never good at chemistry, but I want to do home electroplating, so can someone tell me what those chemical names all mean?", you'd instantly know that this was crazy (I hope). Perhaps a better instance would be "I want to build high-power electronics, but don't understand circuit diagrams." The main difference in this case is that in mathematics, the meanings are probably more subtle and the ambiguity generally far greater, and resolved only by context.

• I appreciate the explanation of the function but it sorta misses the point a bit. No one asks me to program, I was born an artist and taught myself programming. I didn't make it very far in college math, I only finished a 2 year AA degree. A formal education can helpful but not everyone takes that route. School especially now is very abstract and if I can't find some real world meaning I just can't focus on it. ex: I never cared about fractals until recently I wrote a program to draw calculate the mandelbrot set and draw it. – user1610950 Aug 14 '15 at 13:49
• The difficulty, I think, is one of generalization. To use another example: Sigma notation has a nice, straightforward definition that translates easily into code ("write a for loop and add this expression to the running total each iteration"). But then mathematicians generalize and give you things like infinite sums (your for loop won't end because there are infinitely many terms), not to mention insanity like Ramanujan summation. – Kevin Aug 14 '15 at 13:57
• Sorry...when you said "I am a programmer," I mistakenly interpreted that as "I am a programmer by profession, coding for a living and doing the kind of things programmers usually do", rather than "I sometimes write programs to help me do art". My comments still (mostly) hold, though: without some understanding of context, many symbols don't have a clear meaning. (Look at how many questions on math.stackexchange get answers that start with "can you please give more context?") Best of luck, though. One easy solution is a per-symbol question to MSE: just ask and we'll do our best to explain. – John Hughes Aug 14 '15 at 14:00
• @JohnHughes I program to make a living but from my background it would be hard to guess that. It's something that I always wanted to do but stayed away from until I just decided to start. I have no technical training in either math or programming it's just something that I found I had a passion for and keep doing it. Like I mentioned in another comment most programming can be done without ever seeing these complex notations but a lot of computer algorithms are based in physics, calc, trig, etc.. These come from academic white papers and that's where I personally run into the issues. – user1610950 Aug 14 '15 at 15:59
• I've come across this from the opposite direction, where mathematicians have asked me for quick understand of advanced programming concepts. Without the fundamentals, it gets really tricky to explain things well. – y3sh Aug 14 '15 at 19:34

These are all symbols that are usually learned in high school or introductory math in college. There is a wiki page that gives names and purpose for most symbols: https://en.wikipedia.org/wiki/List_of_mathematical_symbols

Wikipedia may not be a reliable source for some topics, but with math it is usually very good for a run down in an unknown or new topic.

• Also, to look up an individual symbol without knowing its name, Detexify (detexify.kirelabs.org/classify.html) may be helpful. You can draw any symbol used in LaTeX and it will at least give you the name of the symbol as used in the markup. – Milo P Aug 14 '15 at 19:53
• this is a cool web app, at least it gives me the names of the symbols. – user1610950 Aug 15 '15 at 19:22

This is a deep question raising a lot of issues about mathematical education. I could discuss it at length, but I'll try and keep this to some practical advice:

• Learn as you go, one symbol at a time. There are countless (but not uncountable) mathematical symbols, and each with different meanings in different contexts. For example, $\pi$ is usually taken to mean $3.14\dots$ but $\pi(n)$ means the number of prime numbers less than $n$. To try and learn the meaning of every symbol in every context is futile. So I advise you to just pick it up as you go, when you need it. Don't worry about other symbols. As soon as you get what you need, use it and move on. That said, learning some basic symbols e.g. $\Sigma$ for summation might be a useful exercise for you to do over a few days if you have the time.
• Learn from people when you can. If you don't know what a symbol means, ask someone. In person is best, because they can tailor their answer to your needs and make sure you understand it. If they're good that is. If there's nobody available in person, you can search this website. If you can't find anything relevant here, don't be afraid to post up a new question.

• Break it down. When you come across a new symbol, e.g. $\Sigma$ for summation, zero in on what it means. Like a dictionary definition, define it in terms of things you already know. You've now expanded your knowledge. Try some examples. E.g. $\sum_{i = 1}^4i$ just means $1 + 2 + 3 + 4$.

• Put the time in. You can only learn if you apply yourself. You get out what you put in. Still, if you're doing it the smart way and taking it in manageable chunks, you'll learn a lot more efficiently.

If you feel bogged down trying to cram all this knowledge into your head, here's a thought: nobody in the world knows all the mathematical symbols and all their meanings in all contexts. So just learn what you need to do what you're doing. You'll find yourself naturally picking things up as you go.

• You are right about education, I could and would not have attempted any of these topics in school because they are too divorced from anything that seems tangible so it's just pie in the sky. This isn't a slight against math, I love the stuff and I want to delve deeper into it. This has been my path already, learn what I can from where I can, I actually though about asking questions on here before but looking at the answers, complex equations are explained with more complex equations and it should be as this is the math SE. I could follow, if I had some guide and that's why this question. – user1610950 Aug 14 '15 at 16:03
• @user1610950- I am confident that if you ask your questions on this site the right way, people will help you out. A big problem we have here is figuring out people's experience level. So the more explicit you are in the question, the better equipped we are to answer. If you find an answer to your question already posted, but do not understand the answer, there is absolutely no problem in you re-asking the question, quoting the original, stating clearly your background and what you do know, and asking for a more elementary explanation. – Colm Bhandal Aug 14 '15 at 16:11

Please forgive what should be just a comment, but is too long to post as such.

The OP's plaint brings to mind a passage from Thomas Mann's novel Royal Highness:

What he saw made his head swim. A fantastic hocus-pocus, a witches' sabbath of abbreviated symbols, written in a childish round hand which was the obvious result of Miss Spoelmann's peculiar way of holding her pen, covered the pages. There were Greek and Latin letters of various heights, crossed and cancelled, arranged above and below cross lines, covered by other lines, enclosed in round brackets, formulated in square brackets. Single letters, pushed forward like sentries, kept guard above the main bodies. Cabalistic signs, quite unintelligible to the lay mind, cast their arms round letters and ciphers, while fractions stood in front of them and ciphers and letters hovered round their tops and bottoms. Strange syllables, abbreviations of mysterious words, were scattered everywhere, and between the columns were written sentences and remarks in ordinary language, whose sense was equally beyond the normal intelligence, and conveyed no more to the reader than an incantation.

• wow, this is pretty awesome! – user1610950 Aug 14 '15 at 16:07

So is there a singular book or at least a place that I can go to reference these symbols to get their meanings?

Without being abstract, there is, and it won't/shouldn't take more than a few months of applied effort.

Calculus: "Calculus" by James Stewart. Read and quickly (don't waste a lot of time getting bogged down by textbook wordiness) work through Chapters 1-4; complete other sections later if/when you need. Learn what it means to differentiate, integrate, and solve a number practice problems.

Linear algebra: "Introduction to Linear Algebra" by Gilbert Strang. Work up to the sections involving matrix diagonalization. Practice solving a system of matrix equations by matrix methods, and calculate some matrix eigenvalues and eigenvectors.

Complex analysis: Find a .pdf introduction to complex numbers. Know what it means to square, and take real and imaginary parts of complex numbers. Thoroughly understand Euler's formula.

Move quickly through these outlined concepts, and you should have enough math to e.g. understand a large part of most undergraduate physics books, as well as be able to tackle basic problems in computer graphics (where matrix methods come into play).

So is there a singular book or at least a place that I can go to reference these symbols to get their meanings?

Yes. In fact, if I could rephrase your question to be

Is there a single book I can read to learn an entire undergrad math education in one go, without requiring other books or resources like the Internet, as if I was stranded on a desert island with only this one book for company?

Answer is still "Yes". The book is Courant and Robbins' What is Mathematics? It's a book I'd recommend alike to high schoolers and continuing education students. It was written at a time when masterful exposition for the intelligent layman was highly respected, and it still reads extremely well.

Please excuse the anecdotal introduction.

I started out as a musician (you know, music college and all that). Much later in life I did a degree that was part mathematics - I'm proud to say I got a distinction.

Why am I telling you this? Because I sympathise.

Here is the real problem and you don't find it in many maths books. The current users of these symbols are standing on the shoulders of giants. When a mathematical idea was first thought of there wasn't a symbol. For example, there was no symbol for zero at one time. Some clever person saw a problem, formulated an answer and then they or a successor encapsulated it in a symbol or an equation.

The revelation came for me when I wanted to understand neural networks. I couldn't understand the terminology. It was only when I obtained a reference book with all the original papers that I got it. I found I could easily program, e.g a Hopfield network by reading what Hopfield himself wrote. That allowed me to read other much more technical papers that followed.

My Answer

Wherever possible, go back to the source. This is easy now we have the internet. Find the person who first invented the concept and symbol and see what they said. They were as naive as anyone else when they first started.

This works for the old guys but unfortunately nowadays, many writers think it makes them look clever if they write as obscurely as possible. Avoid them and find the people who know how to teach. An excellent example would be Richard Feynman who used diagrams and taught his ideas clearly to generations of students

I recently bought the `Numerical Recipes' book after it was recommended to me by a physicist friend. It would probably help if you could see a mathematical formula implemented as an algorithm in order to understand what it's all about.

http://www.amazon.com/Numerical-Recipes-3rd-Edition-Scientific/dp/0521880688

The best way to learn math is to learn math. The best way to learn math, as someone who knows programming, is still to learn math.

Forgive the pithy wording, but I think it captures something important: you shouldn't be learning mathematical symbols, you should be learning math. For instance, you saw $\int$, the "squiggly f or s thing." I could tell you that that is an integral sign, but that won't do you a lick of good unless you have the 2-3 semesters of calculus under your belt that you need to actually use the sign. Until then, they're nothing but shapes on a page.

I went through this process learning Tai Chi recently (disclaimer: that should probably read "I am still learning Tai Chi"). I approached Tai Chi from a completely scientific perspective. I know science. If you can give me hard definitions for everything in your martial art, I can work with it. I tried to apply science to Tai Chi.

Fat chance.

It worked... well.. okay, I don't even have a euphemism for it. It failed so completely and utterly that there was almost no reason to even mention that the attempt occurred. I needed to try a different approach.

So, instead, I chose to learn Tai Chi. I didn't try to define Tai Chi in scientific terms; I didn't try to bound Tai Chi with probabilities. I learned Tai Chi. I learned what yin and yang meant not from some exotic analogy to a scientific model involving oscillating variables, but from what yin is and yang is, and how they work together.

Then, along the way, I found easier ways to do things. I found that Faraday waves over here could do a reasonable job of approximating one type of motion, so I developed my own way of thinking about that motion which flowed fluidly from "mostly Faraday waves, with errors described using yin and yang" when I just need something rough to work with to "yin and yang" on their own when I needed to actually delve further into the real nature of the motion. I never sought to find approximations first. I always sought to do it right, and lean on the approximations when they helped make it easier to reach further into the art.

You should do the same with mathematics. It sounds like you never took calculus. Don't learn what an integral sign is: take calculus and learn integration. You'll realize, along the way, that you actually have to learn differentiation before hand... its really easier to learn it in that direction. If I gave you the symbols ($\int$ and $\frac{d}{dx}(f(x))$), you wouldn't even notice they are related... yet the back and forth between those two is as fundamental to calculus as variables are to algebra.

This will save you a lot of trouble. Then, along the way, you may find a reason a programmer will want to study an Abelian group. You'll find that you don't need calculus for that one, but if you don't have a foundation in abstract algebra (which is different from the algebra you lean in high school), the meanings behind those symbols they use will vanish. However, if you took the time to do just a little abstract algebra, and Abelian groups look useful, you may be able to dig into it, using your programming knowledge to hoist you past some of the difficult details along the way.

Then you really use your programmer side. But always learn the fundamentals first, the way the discipline wants you to learn them. Its always worth it.

There are different mathematics Handbooks that are pretty good, with that word usually in the title. Flipping through one of them will give you a feel for the symbols and their names, and a loose idea of what subjects and searchable terms go together with the symbols. This at least gives an entry point when you are looking for one, or a place to browse when curious.

Purely top-down and bottom-up approaches don't really work for mathematics. The top down approach is to take a mysterious symbol, find out its definition, then the definition of all the mysterious words and symbols in that definition, and continue backtracking until there is a whole pyramid of words and symbols that ultimately build on things that are simple. This quickly becomes ridiculous and impossible (unless you are just fascinated by the words and symbols). The opposite approach, bottom up, is to do a self-learning adult education version of a university mathematics degree. This may exhaust your patience and interest.

It is more sensible to follow your interests wherever they lead, picking up smaller chunks of material but connecting it to previously acquired chunks where possible. That, done with the tools the Internet provides, is one kind of "side door".