
Mathematics all-in-one cheat-sheet (2013) [pdf] - rodmena
https://ourway.keybase.pub/mathematics_cheat_sheet.pdf
======
alister
I think the author's page (Alex Spartalis) deserves to be mentioned:

[https://www.alexspartalis.com/cheat-
sheet.html](https://www.alexspartalis.com/cheat-sheet.html)

He has the same version (v2.10) there and mentions that, "The Web version does
not include the distribution functions due to file size restrictions. Email me
if you would like a copy of these." That explains why pages 213-330 are
missing. Someone should offer to upload the full copy to keybase.pub (or
someplace) since his personal site can't handle the load.

People sometimes do tremendous work creating a program/book/artwork, and
_want_ the world to see it, but don't get around to really share it or promote
it.

~~~
lucb1e
I sent a message using the contact form. Thanks for mentioning this.

~~~
iagovar
Just use torrent! It's easy, and potentially free!

~~~
lucb1e
Distributing new versions is an issue though, as there is no way to tell
anyone who is currently seeding the old torrent. But it's not a terrible idea
if it turns out there is really a lot of bandwidth (I don't expect that), then
I could use my server for seeding at least.

~~~
CamperBob2
Hmm, that sounds like an interesting problem in itself, a torrent client that
handles versioning and updates behind the scenes.

~~~
lorenzorhoades
I mean, thats essentially the apple app store right? Just with torrents and
documents instead of apps? So like a freely distributed google docs, but for
everything. That movie you downloaded a week ago got a better version, the
sci-hub article you downloaded last week has commentary from the author or has
been disputed, That microsoft office (excel, word, onenote) you've been
working on with a group has a long revision history and automatic version
control. You could even download the 'latest version' at 'run-time'; when you
double click to open the document. There's a company, or atleast the very
start of one in these ideas.

------
supernova87a
Wow, I have to say I feel somewhat bad for the author who spent so much time
(months?) compiling this semi-comprehensive reference work. There's so much
information, but at the same time, so little useful information to any
particular reader. It's so broad as to be a hindrance to using it in any sort
of daily reference.

Who would use this? Wouldn't you probably resort to a reference more specific
to your field?

Is this the product of someone's superficial fascination with mathematical
equations combined with OCD to copy down everything ever read, gone awry? Or
is this like a strange version of a Noah Webster?

~~~
asdfman123
Well, I find it extremely useful.

This is my new go-to item when someone asks me, "What would you bring with you
if you could go back in time to the year 1600?"

~~~
lr4444lr
I thought I was the only one who pondered that! I think we'd have even more
fun times in 1st century CE Alexandria.

~~~
CamperBob2
They'd burn you at the stake.

------
rplnt
Cheat-sheet we were allowed to use during math exams at university:

Theoretical Computer Science Cheat Sheet
[https://www.tug.org/texshowcase/cheat.pdf](https://www.tug.org/texshowcase/cheat.pdf)

It's 10 pages, so 3 papers with one page free for something that might be
missing.

~~~
archi42
I often used this when solving assignments and can recommend it to any
computer science student.

------
kejaed
Making your own cheat sheet is half the battle when you are trying to learn
the topic. I remember in my 4th year Stochastic Processes class [0] (back in
2005, wow) we could bring in a double sided page of notes. I wrote my sheet up
in LaTeX [1] and shared it with the class, of course this made it better by
having others contribute. Some didn’t understand why I would “help others out”
by giving it away, but I never understood that, like a page of formulae are
going to get you an A (also why not help others out?).

Of course after spending so much time developing the formula sheet and working
through the example problems (always the most important thing you can do), I
think I looked at the sheet twice in 3 hours during the exam.

Also, Dr. Glen Takahara, this class and your instruction are one of my fondest
memories at Queen’s!

[0]
[http://kje.ca/school/stat455/cheat2.pdf](http://kje.ca/school/stat455/cheat2.pdf)

[1] [https://mast.queensu.ca/~stat455/](https://mast.queensu.ca/~stat455/)

------
te_platt
Nice touch: 4.15 FERMAT’S LAST THEOREM: ... General case when n>2 was proved
by Andrew Wiles (1994). The proof is too long to be written here. See:
[http://www.cs.berkeley.edu/~anindya/fermat.pdf](http://www.cs.berkeley.edu/~anindya/fermat.pdf)

~~~
throwawaymath
I actually can't tell if that was an intentional "margin" joke, or if the
author is just bluntly stating a fact about Wiles' proof...

EDIT: Nope, probably intentional. The author doesn't say something similar for
any other proof based on a quick CTRL+F :)

~~~
piqufoh
Fermat wrote that in the margin of his book Arithmetica that a proof existed,
but there wasn’t space in the margin to write it. It took Wiles 385 years to
find a proof, and it won’t fit in a margin.
[https://en.m.wikipedia.org/wiki/Fermat%27s_Last_Theorem](https://en.m.wikipedia.org/wiki/Fermat%27s_Last_Theorem)

That’s the allure of the theorem; that a simple unknown proof may exist.

~~~
barking
Yes, I remember when news of his proof broke, being disappointed at how
voluminous and obscure (to someone like me) it was. I'd been hoping for
something I might be able to get my head around. (Hard as it was, I don't
think Wiles spent 385 years coming up with it btw!)

~~~
nurettin
Whether there is an existing and verified proof or not, there is still a great
mystery to be solved by figuring out what Fermat actually meant by what he
thought as an elegant solution, whether it is an actual solution or not.

~~~
EGreg
Well, it could have been like Kempe's chains... they finally realized there is
a problem, and then it took like 100 years before Appel and Haken made what is
probably the first computer-aided proof. And who can say it's really a "proof"
if it doesn't explain "why" it's true.

------
trilinearnz
I like the idea of this, but I'm not sure that "cheat sheet" is the best term
for it, given that it weighs in at quite a few pages (191 physical).

~~~
Sharlin
The sort of ~100 page softcover reference book that we were permitted to use
in highschool math, physics, and chemistry exams is called around here,
literally translated, a book of tables. The term is probably a remnant from a
time before pocket calculators when it contained actual trig and log tables.
"Book of formulas" would probably be a more apt name nowadays. Not sure if
there's a common word for such a thing in English.

~~~
mcnamaratw
This kind of thing?
[https://en.wikipedia.org/wiki/Abramowitz_and_Stegun](https://en.wikipedia.org/wiki/Abramowitz_and_Stegun)

I'm not sure there's really an English word for it. That's strange.

~~~
cosinetau
Normally, I've heard others reference books like A&S as a "handbook."

It's even in the formal title.

~~~
mcnamaratw
Right. Only "handbook" is so broad. It could be about any subject.

------
daveFNbuck
I got a B in probability because I didn't write a proof of the central limit
theorem on the allowed cheat sheet for the final exam. So of course it's the
first thing I looked for on this one. It's not there.

~~~
lucb1e
A B is like 80% or so, judging from A being the best iirc? I'm not familiar
with alphanumeric grading systems.

~~~
barrkel
Things can get a bit customized at either end of the spectrum, but generally
each letter covers 15%, with A..D covering the span 100% to 40%.

So a B would be 70 to 85.

~~~
nutjob2
Shouldn't that sort of grading be on a bell curve? That would seem to be the
fairest.

~~~
kaitai
Many smaller (<60 students) math classes, in my experience, have bimodal
distributions of scores, so a bell curve by definition simply doesn't make
sense. In addition, there's been a move to standards-based or mastery-based
grading, that is, grading according to what you know of the material rather
than how you compare to your neighbor. This allows comparisons over time and
consistency with regard to subsequent classes -- if you have a C as a
prerequisite for the next class, then a C should indicate the same mastery of
material rather than the same relative position in the class.

------
pja
More “Cheat Book” than “Cheat Sheet” :)

Looks like it could do with typesetting in LaTeX - I think the author started
doing this here:
[http://mathscheats.weebly.com/](http://mathscheats.weebly.com/) but never
entirely completed it. Plenty there to chew on though.

------
killjoywashere
Ah, 10% of the first section of the CRC Handbook of Chemistry and Physics,
with amateur typography. Sort of like Greenspun's tenth rule, but for math.

~~~
digitalsushi
I've heard of mathematicians handwriting proofs in the margins of books. Like
an MVC, I can appreciate the data without appreciating the presentation.

------
nicklaf
While we're on the topic of this kind of thing:
[https://en.wikipedia.org/wiki/Synopsis_of_Pure_Mathematics](https://en.wikipedia.org/wiki/Synopsis_of_Pure_Mathematics)

~~~
yannis
This is an excellent book, but perhaps a bit dated despite the fact that
Mathematical Books have a long shelf-life. Factoid: The book is noteworthy
because it was a major source of information for the legendary and self-taught
mathematician Srinivasa Ramanujan who managed to obtain a library loaned copy
from a friend in 1903...

------
pencillr
It's awesome. Though I prefer my Bronshtein :D
[https://en.wikipedia.org/wiki/Bronshtein_and_Semendyayev](https://en.wikipedia.org/wiki/Bronshtein_and_Semendyayev)

~~~
zwieback
Interesting - I've had the German version from my university days (every ME
student had one) and it's spelled "Bronstein". The English spelling makes me a
little dizzy.

------
floatinthecloud
When I was studying mathematical physics, my classmates and I made similar
quick reference notes in latex. Based on a cursory look, it seems to be
missing some important complex analysis and abstract algebra/lie theory
results (although I haven't read what the author studied).

~~~
kd0amg
Searching for stuff I've used, this document is also quite light on
combinatorics, complexity theory, and order theory (and of stuff I haven't
used much, I don't see anything on category theory or topology). I don't think
330 pages is nearly enough to be truly all-in-one.

~~~
throwawaymath
Agreed. This document is reasonably broad but it's missing quite a lot of
undergrad mathematics, let alone graduate math.

If you literally wrote down sequences of definition, theorem, proof,
definition, theorem, proof, ... with no exposition or exercises whatsoever, I
think you might be able to include most undergraduate math in about 1000
pages. That would include calculus, real analysis, complex analysis, linear
algebra, abstract algebra, discrete math. Maybe elementary number theory,
topology and probability theory as well. That would be...horrible to learn
from, frankly. Imagine a ten volume set of Baby Rudins, with no exercises.

If you really doubled down on the cheat sheet angle and didn't include any
proofs - just the theorems, identities and inequalities - I think you could
get through all undergrad math in a few hundred pages. But you wouldn't have
any of the peripheral content the author included, like physics/economics.

~~~
floatinthecloud
'Riley, Hobson and Bence' was a standard reference I used throughout my
undergrad and graduate studies. Sure enough, it's about 1200 pages.

------
sunstone
You know you're in for a head banging when the Table of Contents is 25 pages
long! :o

~~~
kwoff
[https://archive.org/details/elementsquaterni00hamirich/page/...](https://archive.org/details/elementsquaterni00hamirich/page/n13)
Hamilton's "Elements of Quaternions"'s table of contents is about 60 pages
long and includes footnotes.

------
kakaz
It is interesting and fun! I have a loose suggestion - it could be published
together with LaTeX codes for any formula present - so it would be great speed
up for someone who wants to use various formula in his works. However the
choice of various areas is strange for me ( partially lack of some engineering
areas, partially lack of very basic physics things, partially lack of
mathematics) A lot of things is missing: Maxwell-Clerk equations. Einstein
equations. Schrödinger equation. Dirac equations. Notable: Laplace equation
and Maxwell-Clerk equations, wave equation!!! Harmonic oscillator equation.
Quantum physics, Heisenberg relation at least please! Hydrodynamic, notable no
Navier-Stokes equations. Special functions: Bessel, Jacobi, Lagrange,
Chevyshev polynomials, various equations related to it. No elliptic functions,
no Weierstrass function mentioned. No various number theory objects: Mobius
function, Minkowski function ?() not mentioned. Congruences nearly omitted (
Chinese remainder theorem maybe) Group theory, some simple results from
category theory, Shannon theory completely missing. No universal algebra. No
basic cryptography. Complete lack of various numbering systems ( binary, hex
at least)

------
halfelf
Find a bug in page 39: y' is Lagrange's notation while dx/dy is Leibniz's.

------
nn3
Seems very similar to a classic formula collection like
[https://dlmf.nist.gov/](https://dlmf.nist.gov/)

------
imedadel
_One day_ , I'll make an Anki version of this.

~~~
taliesinb
Speaking of, can anyone commend some good Anki mathematics decks?

~~~
mathnmusic
There is just one link here: [https://learnawesome.org/learn-
awesome/mathematics.html](https://learnawesome.org/learn-
awesome/mathematics.html)

If people share links here, I'll send a PR to learnawesome to add those under
mathematics#flashcards section.

------
sdan
First time I've seen someone use Keybase.pub. Last time I checked they gave
out 250gb for free for everyone... this is a good use of that!

~~~
ac29
I wonder how long that can last before abuse kills the system. Some quotes
from their page about KBFS [0]:

"The 250GB free accounts will stay free"

"we'll never run an ad-supported business"

"we're not trying to make money"

[0] [https://keybase.io/docs/kbfs](https://keybase.io/docs/kbfs)

~~~
ashton314
How _is_ Keybase funded anyway? They say they’re hiring, so money is coming
from somewhere.

~~~
johnnyRose
Their Series A raised $10.8M back in 2015:
[https://keybase.io/blog/2015-07-15/keybase-raises-
series-a](https://keybase.io/blog/2015-07-15/keybase-raises-series-a)

------
TorKlingberg
I remember using something similar called "Mathematics Handbook for Science
and Engineering" in university. [https://www.amazon.co.uk/Mathematics-
Handbook-Science-Engine...](https://www.amazon.co.uk/Mathematics-Handbook-
Science-Engineering-Lennart/dp/3540211411)

I assume there are many others, and this is a free alternative.

------
amitport
nice work. I would have removed

most of 'PART 1: PHYSICAL CONSTANTS', 'PART 8: APPLIED FIELDS', 'PART 18:
ELECTRICAL', and some of 'PART 99: CONVERSIONS'.

"all-in-one" math seems enough :). Other stuff seems arbitrary and leaning
towards physics (which could have its own giant book).

~~~
Aromasin
I'm biased as I'm an EE, but I'd say that the Laplace transforms within the
Electrical section are firmly in the realm mathematics, so keep them. It just
so happens that they are only really useful in the domain of digital
signal/control processing, and I think to get the true form or radioactive
decay(?) though I may be mistaken. I'd put it next to information on Fourier
transforms. The circuit theory may be a bit unnecessary though I agree.

------
droithomme
It's a very fine compendium of notable relations.

At 212 pages long, it's certainly not a cheat-sheet.

If it didn't have obvious spelling errors I might be more confident the rest
has been transcribed accurately.

------
r-w
Things get kind of pixelated for me around page 123. Anyone else?

~~~
trymas
You are not alone. Seems like formula's are not vector anymore from page 121.

------
ranie93
Add this to the list of items (or perhaps the only item?) you'd take if you'd
time travel to the past

------
fmajid
I've owned paper copies of Murray Spiegel's Schaum Outline of Mathematics, and
even the government-funded Abramowitz-Stegun doorstop, this reminds me of
those somewhat, but the title is misleading as it has as much physics and
chemistry as mathematics.

------
daurnimator
Pages 168 to 262 seem to be missing?

~~~
SubiculumCode
Fermat’s Lost Theorem?

------
karmakaze
I hate how often these sprawly things are called cheetsheets:

    
    
      LOGIC SYMBOLS
      Symbol Symbol Name Meaning / definition Example
        &        and     x & y

------
rando171717
Slightly off-topic: from a security perspective, how safe/not safe is it to be
clicking open random pdf's? Wouldn't this be a possible avenue for malware?

~~~
syn0byte
[https://github.com/osnr/horrifying-pdf-
experiments](https://github.com/osnr/horrifying-pdf-experiments)

[https://www.alchemistowl.org/pocorgtfo/](https://www.alchemistowl.org/pocorgtfo/)

Yes. Not only is it possible but historically _already_ an avenue of attack.

------
FerretFred
Awesome! It only serves to remind me I know practically nothing about
Mathematics. "Good Primes" and "Happy Primes" .. who'd have thought it ...

------
mpoteat
To me this seems to be exclusively undergraduate mathematics that one might
learn in an engineering program. No mention of topology or modern algebra.

------
daodedickinson
It's organized like a debate file. Fascinating.

------
zw123456
Reminds me of the old Schaum's Outlines, I still have a couple of them on the
book shelf from college days, saved my bacon many times.

------
paxys
Uh, this is a book.

------
BlackFingolfin
OK, so, I also hate to be "that guy", but, despite the admirable effort put
into creating this by the author, I am sad to say I don't see why some people
are excited about this... Because unfortunately after just looking at a few
pages, I saw lots and lots of error and misleading or plain confusing
statements, basically on every single page I looked at closer, which makes me
distrust it. Granted, these vary in their severity, but still...

Examples:

\- p. 50, definition of a complex vector space, says " A complex vector space
consists of the same set of axioms as the real case, but elements within the
vector space are complex.". The vector space does not "consist" of the axiom,
it adheres to them. And it makes no sense to say that the "elements" (vectors)
are "complex". Rather, that scalars are allowed to be complex, not just real.

\- p. 50: definition of a subspace is a mess. To pick just one obvious problem
with it: What even are "axioms (a) and (b)"? Perhaps the three unlabeled
"axioms" at the top of the page are meant (being closed under addition,
additive inverses, and scalar multiplication)? But certainly the third one
(axion "(c)"?) needs to be verified, too (whereas the second, about additive
inverses, is redundant).

\- p. 51 the examples at the top of the page end with mentioning C[a,b], the
set of all continuous functions on an interval [a,b]. But then it claims that
this is actually _not_ an example, because "it has infinite dimensions". But
really this is a perfectly fine vector space (also, we haven't even defined
what a "dimension" is yet)

\- p. 51, definition of linear independence says "c_1=c_2=c_n=0" which is
missing a "=..." before the "=c_n"

\- p. 52 the "general vector" given as a column vector doesn't make sense in a
general vector space

\- p. 53: "Matricis"

\- p. 56 "Laprange's theorem" should be "Lagrange's theorem"

\- p. 126: "EXPONETIAL"

\- p. 167: "Matricies", "Prinicples", "opertaions",

\- p. 168: "Determinate"; the formulas for 2x2 and 3x3 matrices implicitly
assume a labeling of the entries which is never given, rendering this semi-
useless

\- one section is titled "MISELANIOUS"

And on page 38, the fields of real and complex numbers, R and C, are
introduced as being written with a "blackboard" font (\mathbb), while on page
51 this is not followed (instead, we get \mathcal{R} and plain C). Why even
introduce these conventions if they are then not followed?

Sure, many of these are minor and perhaps even "obvious" mistakes. But if you
know the matter well enough to spot all of these, maybe you don't need this
cheat sheet? Also, as a mathematician myself, my experience is that the number
of spelling mistakes in a research paper (or in anything written and submitted
by students) is a good first proxy for the quality of the text: if you can't
be bothered to even run a spell checker on your text, should I really trust
you to have verified all your computations and logical deductions carefully?
(And no, I don't apply this to, say, posts here on HackerNews: I hate it when
people shoot down a comment, or a twitter post, or whatever, just because it
contains typo. But writing a book is a bit of a different affair, isn't it?).

Hence my point that I wouldn't want to suggest this to anybody as a reference.
:-(

~~~
raister
Page 33: PART 2: MATHEMTAICAL SYMBOLS

------
orangewindhoek
_misprint_ PART 2: MATHEMTAICAL SYMBOLS

------
andrepd
I really didn't want to be that guy, and props for the author for such
comprehensive reference work, but god, what an _appalling_ typography. Bad
tables with too many lines bitmap formulas with misadjusted size, poor
spacing, ugly font... Almost everything that could be wrong is wrong.

~~~
xwin
I came here to say the same thing. Looks like the author did not use a good
authoring tool to produce good math output.

I generally take notes like this with LaTeX and compile them to PDF but PDFs
can be large and if committing to a Git repo can lead to large .git
directories. The BasicTeX package for Mac (brew install basictex) has worked
well for me:
[http://www.tug.org/mactex/morepackages.html](http://www.tug.org/mactex/morepackages.html)

These days I also take math notes as plain HTML files containing Markdown and
LaTeX, then render with the TeXme one-liner here:
[https://github.com/susam/texme](https://github.com/susam/texme)

I guess what the OP has done here is take all the notes in Word or some other
WYSIWYG document editor not specialized for mathematical typography and
converted the whole document to PDF. This won't lead to good results. It is
absolutely necessary to use tools specialized for mathematical typography to
write and publish math.

By the way, is there anyway to create high quality math distributables that
are both self-contained and lightweight?

~~~
mkl
Why commit the output files? I generally put "*.pdf" in .gitignore.

~~~
snazz
Probably so that someone can download it without compiling it? GitHub Releases
or an equivalent would probably be better, though.

~~~
xwin
That's right. I would like others as well as myself to be easily able to
download the PDF files without having to compile it. I tried out GitHub
releases and it works okay but not great. I cannot track all the compiled PDFs
I have at a single place or move them easily from one system to another like I
could with GitHub repos. I am also trying to avoid getting locked into a
specific service like GitHub releases.

~~~
ktpsns
I have very good experiences with Gitlab CI: It just recompiles the PDF(s)
from the repository within a minimal docker container (containing a full Tex
Live repository) and pushes them online on a website (sFTP or similar). Also
Gitlab allows to host the build assets.

------
pgreenwood
Not LaTeX; didn't read.

~~~
dngray
I have to admit it's a pretty ugly manual.

Some of those formulas were really blurry, while some of the tables spanned
weirdly over pages.

The hosting issues could have been fixed as well if it was LaTeX as it would
have been neat in a git repository.

Gitlab also has continuous integration for LaTeX

* [https://www.vipinajayakumar.com/continuous-integration-of-la...](https://www.vipinajayakumar.com/continuous-integration-of-latex-projects-with-gitlab-pages.html)

* [https://sayantangkhan.github.io/latex-gitlab-ci.html](https://sayantangkhan.github.io/latex-gitlab-ci.html)

------
chris_wot
Don't see any logic equivalencies... I think they are kind of neat!

------
remarkEon
Woof. Flashbacks to Sister Mary's Calculus class in 12th grade.

------
karthie
Quiet handy and amazing thank you very much for sharing this over.

------
tychonoff
Nice work.

But on page 43 "quadratic" is misspelled.

------
amai
Why does the author not use LaTeX?

------
phragg
bet i could fit this on a 3.5in x 5in notecard...

------
jlv2
s/Cheat-sheet/Cheat-book/g

------
symplee
Last page says "Page 212 of 330"

Pages 213+ are probably where they discuss cardinality.

------
lightedman
Missing the cheats for division (how to rapidly tell if a number is divisible
by 3, 7, 2, 5, etc. for prime factorization.)

------
mhartl
Someday the cover will read e^iτ = 1 and all will be right with the world.

~~~
alanbernstein
Some believe the pi version is superior due to the inclusion of three
operators (+, *, exp), and five numbers (0, 1, i, e, pi) which are all
fundamental in some sense. The tau version omits the + and the 0.

~~~
gizmo686
e^iτ + 0 = 1

Fixed.

~~~
ngcc_hk
-1 not 1 ...

e^iπ+1=0 from

e^ix=sinx + icosx with x=π

~~~
gizmo686
Thats why we are using τ not π.

