
Ask HN: Good articles/courses for understanding mathematical notation? - rory_isAdonk
Required at times for interviews or programming puzzles, knowledge I lack.
======
ivan_ah
One thing to note is that there is no universally agreed upon convention for
denoting math objects. Textbook and research paper start with a section on
notation to clarify the symbols they will use. Notation often varies between
fields, academic schools, and sometimes there are even differences between the
notation you would use when writing on a blackboard vs. the notation you would
use in print.

That being said, for the most basic concepts the notation is pretty consistent
so if you skim through one or two books you'll be able to get the feel for it.
Understanding the actual math—that will take longer.

As for references, here is a very comprehensive standard, ISO 80000-2 that
defines recommendations for many of the math symbols, with mentions of other
variations:
[https://people.engr.ncsu.edu/jwilson/files/mathsigns.pdf#pag...](https://people.engr.ncsu.edu/jwilson/files/mathsigns.pdf#page=10)

For something shorter (and less complete), you can also check the notation
appendices in my books:
[https://minireference.com/static/excerpts/noBSguide_v5_previ...](https://minireference.com/static/excerpts/noBSguide_v5_preview.pdf#page=136)
[https://minireference.com/static/excerpts/noBSguide2LA_previ...](https://minireference.com/static/excerpts/noBSguide2LA_preview.pdf#page=159)

~~~
swimfar
"One thing to note is that there is no universally agreed upon convention for
denoting math objects."

This is the thing that trips me up most often. It's especially a problem when
the author believes that their notation and variables are universal and
therefore don't need to be defined. I've spent a lot of time these past few
months trying to "reverse engineer" calculations done in research papers to
verify their notation.

Textbooks and theses generally do a better job of defining everything because
they aren't as concerned about document length.

~~~
downshun
I think the reason is that math notation is used in context. Which is what
explains the two different notations for calculus derivatives. They were
interested in different things! Similarly in quantum mechanics (not my area)
you may want to use bra-ket notation or matrix representations if you care
about implementation details.

~~~
downshun
Hi U

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dannygarcia
Here are a few notation resources I've found helpful when teaching myself
computer science:

\- Mathematics for Computer Science:
[https://courses.csail.mit.edu/6.042/spring17/mcs.pdf](https://courses.csail.mit.edu/6.042/spring17/mcs.pdf)

\- Calculus Made Easy:
[http://calculusmadeeasy.org](http://calculusmadeeasy.org)

Not directly related to your question but useful for interviews and
programming puzzles nonetheless:

\- Algorithms and Data Structures, The Basic Toolbox: [https://people.mpi-
inf.mpg.de/~mehlhorn/ftp/Mehlhorn-Sanders...](https://people.mpi-
inf.mpg.de/~mehlhorn/ftp/Mehlhorn-Sanders-Toolbox.pdf)

\- Basic Proof Techniques:
[https://www.cse.wustl.edu/~cytron/547Pages/f14/IntroToProofs...](https://www.cse.wustl.edu/~cytron/547Pages/f14/IntroToProofs_Final.pdf)

------
stiff
For interviews and programming puzzles you only need to know notation for
basic mathematical logic, basic set theory and the summation notation and
maybe some bits and pieces from number theory:

[https://en.wikipedia.org/wiki/Logical_connective#Common_logi...](https://en.wikipedia.org/wiki/Logical_connective#Common_logical_connectives)

[https://en.wikipedia.org/wiki/Quantifier_(logic)#Notation](https://en.wikipedia.org/wiki/Quantifier_\(logic\)#Notation)

[https://en.wikipedia.org/wiki/Set_theory#Basic_concepts_and_...](https://en.wikipedia.org/wiki/Set_theory#Basic_concepts_and_notation)

[https://en.wikipedia.org/wiki/Summation#Capital-
sigma_notati...](https://en.wikipedia.org/wiki/Summation#Capital-
sigma_notation)

[https://en.wikipedia.org/wiki/Modular_arithmetic#Congruence](https://en.wikipedia.org/wiki/Modular_arithmetic#Congruence)

If Wikipedia is too hard to follow, you can learn this from early chapters of
a discrete mathematics textbook.

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davismwfl
There is a good book for math notation that I like:

"Mathematical Notation: A Guide for Engineers and Scientists"

[https://www.amazon.com/Mathematical-Notation-Guide-
Engineers...](https://www.amazon.com/Mathematical-Notation-Guide-Engineers-
Scientists/dp/1466230525)

~~~
binarymax
An upvote isn’t enough for this book, so I need to comment that it’s the best
I’ve come across for my needs. When I was getting into the more mathematical
aspects of coding when I was getting started with machine learning 5 years
ago, this book was invaluable.

Having thought in code (with verbose variables and structure) for many years,
I needed a Rosetta Stone for the ambiguous symbology of mathematics - and this
is it!

It’s tinier than you’d think, but is an absolutely incredible reference. An
absolute requirement for any engineers bookshelf.

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gspr
I'd argue that any attempt at understanding mathematical notation universally
will fail. Different fields and different sub-topics and different authors
have vastly different conventions, for good reason.

Sure, one can perhaps expect that something that uses an integral sign shares
some properties with ordinary integration of real functions, but to really
understand what the notation entails, one really has to study the underlying
material.

I feel that what you're asking for is kind of akin to wanting to read a novel
in a foreign language using only a dictionary of the 10% most commonly used
words of said language, with each entry resolving only to one meaning of the
word.

~~~
maxmcd
Surely the optimist would take this question to assume that someone is
interested in expanding their math knowledge and might not have an issue
getting exposed to more math.

I think your answer is akin to telling a french tourist that they shouldn't
try and learn basic conversational french because they could never hope to
understand the complexity of the complete language.

~~~
gspr
I understand your point. I just suspect, perhaps wrongly, that people asking
the question on HN may be inclined to base serious stuff on their tourist's
understanding of French. There's a place for tourist math, but it can be
dangerous when misused.

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alokrai
You internalise mathematical notation by using it to solve mathematical
problems and express mathematical ideas.

Two excellent resources are:

1\. Introduction to Mathematical Thinking (if you prefer moocs) -
[https://www.coursera.org/learn/mathematical-
thinking](https://www.coursera.org/learn/mathematical-thinking)?

2\. How to think Like a Mathematican - [https://www.amazon.co.uk/How-Think-
Like-Mathematician-Underg...](https://www.amazon.co.uk/How-Think-Like-
Mathematician-Undergraduate/dp/052171978X)

------
jgwil2
In addition to some great responses already on here, I would suggest picking
up a functional programming language as a way to bridge the gap between math
and the C-style syntax that most of us learned to program in. Haskell and
PureScript are good for this; many programs actually use even more mathy
aliases for common tokens (e.g. `∀` for `forall`).

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mhh__
What does understand mean? Notation is just that, notation.

I think that the single biggest advantage one can have (in programming that
does something "non-trivial" \- loaded term I know, rather than as a person)
is to have a firm grasp of the mathematical basis of their work. It's so much
easier to start something new when you can derive it yourself.

If you have the time, I recommend "Advanced Engineering mathematics" for the
gap between calculus to applications and other topics like Linear Algebra,
analysis, and graph theory.

If you just want a mapping of symbols to words try LaTeX documentation

A la
[https://oeis.org/wiki/List_of_LaTeX_mathematical_symbols](https://oeis.org/wiki/List_of_LaTeX_mathematical_symbols)

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vthommeret
To understand basic notation like summations and matrix multiplications, I
created Math to Code which is a quick tutorial to translate math into NumPy
code:

[https://mathtocode.com/](https://mathtocode.com/)

Previous HN discussion / it was on the front page earlier this week:

[https://news.ycombinator.com/item?id=23513438](https://news.ycombinator.com/item?id=23513438)

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Rerarom
There is no rule of mathematical notation except that things that are written
as an index (whether as a subscript or superscript or argument) are stuff the
given object depends upon. Everything else builds upon that rule and is
defined in some context.

Source: I am a mathematician

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kasbah
Not an article or course but this Math as code cheat sheet is pretty good:
[https://github.com/Jam3/math-as-code](https://github.com/Jam3/math-as-code)

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andreygrehov
In addition to other comments, I would also recommend "A Programmer's
Introduction to Mathematics" by Dr. Jeremy Kun [0]. The HN submission [1] may
have more interesting stuff around the topic.

[0] [https://pimbook.org/](https://pimbook.org/)

[1]
[https://news.ycombinator.com/item?id=18579076](https://news.ycombinator.com/item?id=18579076)

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agentultra
Notation varies depending on the author and subject area but a good resource
for "programmer/computer science" notation is to skim through _Concrete
Mathematics_ or the preliminaries to _The Art of Computer Programming_ \-- I
find this notation to be common.

In more specialized areas like type theory, first order logic, predicate
calculus, temporal logic, etc you have to pick it up as you go.

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hansvm
This won't solve all your problems, but it _can_ be a big help to know what to
search when you see a wall of symbols, and detexify.kirelabs.org is a decent
resource for that -- you can draw a single symbol and get the latex code that
would generate it.

(if you're typesetting math it's invaluable, not just decent)

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jimmaswell
What notation was used in an interview/puzzle that you weren't familiar with?

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cosinetau
There are a lot in Abromowitz and Stegun handbook, last section "Index of
Notation". It's not quite what you're asking for, but it's fairly
authoritative.

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geoelkh
related topic: do you usually prefer seeing specs/design doc in plain english
or mathematical notation?

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brutt
Look at formula source in Latex, so you will have keywords to google.

