
Math Intuition Cheatsheet - jgrodziski
http://betterexplained.com/cheatsheet/
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bithive123
I am often saddened by how much easier it has been for me to acquire computer
science versus mathematical literacy. Embarrassingly, years after a poorly-
timed calculus course left me thinking I had to be able to prove the central
limit theorem in order to use calculus, the biggest barriers have turned out
to be things programming as a discipline has learned to avoid; encouraging
varied and/or terse notations, opaque variable naming schemes, and arbitrary
use of jargon where simpler terms would suffice.

A friend of mine who is a physicist once complained to me that every time he
had to install some scientific software package on his computer, he had to
deal with a litany of arbitrary things which seemed to have nothing to do with
his task. I countered with my experience learning math and joked that at least
programmers are willing to occasionally refresh our idioms and notations to
better reflect our mutual understanding.

~~~
kalid
(Kalid from BetterExplained here)

Thanks for the comment. There's this weird notion in Calculus education that
we need to start from first principles. Limits were invented a century after
Newton died, yet they're taught first. "Oh, students won't understand calculus
unless they can build it from first principles. I don't care if Newton worked
out gravitation with his understanding, it's not good enough."

My little candle in the darkness
([http://betterexplained.com/calculus/lesson-1](http://betterexplained.com/calculus/lesson-1))
is to start with intuitive notions (X-Raying a pattern into parts, Time-
Lapsing parts into a whole) and then gradually introducing the terms.
Eventually, if people are interested, we can get into the theory (which is
like getting into the Peano axioms of arithmetic, if you care to go that
deep.)

I think in math there's a tendency to express things in the lowest-level
"machine code" we can. We need more comments and pseudocode outlines :).

~~~
vanderZwan
Same thing with logarithms, which were invented to simplify multiplying two
huge numbers. Combined with log tables (think paper LUT for humans) that
became a simple matter of looking up two log conversions, adding them, then
looking up the inverse of the answer.

Is that what we learn in high school? Nope. So everyone is left wondering what
the hell they are good for (or were, before we had calculators) the first time
around.

(btw, as far as I can tell you don't have that bit of history on your site yet
either - might be an interesting addition?)

~~~
marcosdumay
Well, I learned that at school. It still didn't answer what the hell
logarithms are good for. Everyone was left wondering why we were learning
something that can be replaced by calculators.

Seeing some modern application of lagarithms would be great. Even just making
a log-log graph at some point would answer every question. But those are not
at the official curriculum.

~~~
kalid
It drives me batty because we learn the properties of logarithms before (if
ever) internalizing what they mean. Here's my intuition if it helps someone:

[http://betterexplained.com/articles/think-with-
exponents/](http://betterexplained.com/articles/think-with-exponents/)

Exponents let you plug in time, and get the amount of growth. e^3 ~ 20, which
means "3 periods of 100% continuous growth [100% is implied by e, 3 = 3* 1]
will grow us from 1 to 20".

Logarithms let us plug in the growth, and get the time it took to get there.

ln(20) ~ 3 means "It takes 3 units of time [growing at 100%, continuously] to
grow from 1 to 20".

Exponents take inputs and find the future state, logs take the future state
and work backwards to find the inputs that got us there.

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arianvanp
I started my 1st year in college doing both math and computer science majors.
I really love math but I just hate how (especially calculus and linear algebra
courses) they just don't work on growing any intuition at all. in contrast, in
my CS major we had a linear algebra class (ugh overlapping classes, huh) as
well and we used computer graphics as a case study to learn about the basic
concepts which was really fun and enlightening.

Eventually I just ragequit math because I didn't have the patience and time to
search for the intuition myself, which is a pitty. I'm really happy I followed
the classes about mathematical proofs, groups etc because I really enjoyed
those but others were just painful.

Anyhow recently I discovered this blog too up my math concerning computer
science is this: [http://jeremykun.com/](http://jeremykun.com/). It's a tad
more advanced than this but I'd really recommend it. His blog is freaking
amazing and always a joy to read. Math as a sidehobby it is! :)

~~~
j2kun
A lot of mathematics people feel this way about programming, too. Both sides
expect one to devote the time to build intuition to grok their subject. I
think it's easier to learn programming coming from the mathematics side,
because you're going from nobody holding your hand to a compiler giving you
error messages. You can't interactively test whether your intuition in
mathematics is sound (you can, but you have to play both the role of the
compiler and the tester).

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akrolsmir
I'd read Yudkowsky's popular "An Intuitive Explanation of Bayes' Theorem" as
well as a few others, but the article on this site made me feel like I
actually understood it for the first time. The examples were clever, short,
and simple, while still demonstrating how powerful Bayes' Theorem can be.

The concise explanations here seem more helpful than semester-long university
courses I've taken. I definitely look forward to exploring his other articles
on math and programming.

~~~
kalid
Glad it was helpful!

There's a strange tendency in technical explanations to expand, expand,
expand. A heuristic I have: Imagine you're writing a letter to yourself,
before you started the course. There's no reason to waste your own time, just
cover the key difficulties and how to overcome them, in simple language you
would have understood.

(Oh, this letter also costs $5 a word, so let's make them count!)

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fenomas
This is terrific. I'm mathy enough to know _e_ to several places, and use it
in programs occasionally, but if you'd asked me how it might be intuitively
derived I'd have been at a loss. And that's just for something inside my
comfort zone - the article on linear algebra is just as illuminating.

This looks like it will be my commute reading for the week. But more than that
it just makes me happy that it exists.

~~~
kalid
Thanks, glad to hear you're enjoying it. I realized that until I could explain
a concept intuitively, I didn't really understand it -- and it became a
mission to find some analogy that really clicked for me. Hope it gives you
some entertaining reading.

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gajomi
What an excellent collection of articles (well, I only ready a few, but I
would like to think my experience extrapolates to the rest)! But I don't think
"cheatsheet" is really the right word. Really this is a collection of short
expository writings on particular mathematical concepts. And very well done, I
think, to the point that calling it a "cheatsheet" (which I associate with a
haphazardly conjoined sequence of facts and tricks out of context) is to do it
a disservice. Or maybe the name choice has some marketing value?

~~~
kalid
Glad you enjoyed them so far :).

"Cheatsheet" might not be the best term for the summary page, perhaps "quick
reference". I picked cheatsheet because it's a little more approachable than
"reference", which implies something formal (and perhaps offputting).

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NoPiece
I really like this cheatsheet, it is very inspiring actually. You should
consider doing a version written for elementary school aged kids. You need to
get buy in while they are still young!

~~~
kalid
Thanks! I'd love to do more elementary school topics. I still encounter
revelations about things like basic counting (fencepost problem) and
arithmetic.

My meta goal is to raise people's standards for what it means to understand
something. ("If it didn't click, I should keep looking for a better
explanation." vs. "It didn't click, I'm no good at this.")

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jgrodziski
For me the best one from Kalid is the explanation of the fourier transform:
[http://betterexplained.com/articles/an-interactive-guide-
to-...](http://betterexplained.com/articles/an-interactive-guide-to-the-
fourier-transform/)

This article is simply amazing, moreover Kalid is an humble person and
references his inspirations from several people over the web. Thank you very
much Kalid for the time you dedicated to share all this wonderful resources.

~~~
kalid
Thanks so much, and for originally submitting the cheatsheet to HN. (I hope it
helps people untangle concepts that confused me for literally a decade.)

I really appreciate the encouragement and hope to keep cranking as long as I
can.

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adamwong246
Sure could have used this site a couple years ago when I was in school!

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ithinkso
I like the idea but I personally really enjoy and prefer 'math english'. At
first it may seems odd but after a while you get used to it and it's actually
easier, you just scan through epsilons, deltas knowing exactly what they mean.
Intuition has edge-case scenarios, double-meanings and almost-truths. I don't
know why, but I think intuition (pseudocode math as someone had named it)
gives a little bit of a 'illusion' of knowledge. You think you got it but are
you really?

~~~
kalid
Great point. I use the word "intuition" but feel it might be overloaded,
things with intuitive knowledge, psychics, going with your gut despite the
facts, etc.

The sense I mean is closer to "grokking" or "clicking", when symbols and
definitions actually mean something. You start thinking with the symbols, not
despite them. You actually look forward to checking your understanding with
real problems, because when it clicks the problems become easy :).

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omnibrain
Those explanations remind me of Feynman's explanation of elliptical orbits.

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RobinL
This September, I'm starting tutoring A level maths to underprivileged kids
for a charity after helping them with GCSEs last year. This looks like it will
be really helpful. Thanks!

~~~
kalid
That's awesome! I hope you can make use of the site. One goal for the site is
to provide "ammo" to teachers who are looking for other ways to present
difficult concepts.

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curiousDog
This is gold. Please do point to more resources like this.

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Sharma
Thanks for posting this! I just read a small portion of Integration and trust
me, it all make sense now!

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agumonkey
I love the content, just wishing for some of their views on explaining
electronic circuit design.

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factsfinder
Seems like there are a lot of things i need to learn in maths

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pawannitj
Are there other resources like this on Math?

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FlyingLawnmower
Great work. Please keep adding examples!

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Mistral
Super, but no induction!

