
Ask HN: Best Mathematics book for complete noobie? - babyboy808
Here's the story, I have never been very good at maths in school and to be honest, any other subjects really. I could get by, but never pushed myself. School was never the best environment for me to learn, but when I'm studying at home, I like it.<p>So over the past few weeks I have been thinking about trying to learn mathematics again at my own pace (starting at the basics), what books would you recommend?
======
mahmud
"An Introduction to Mathematical Reasoning" by Peter J Eccels. Teaches the
_vocabulary_ of mathematics, just the basics you need to think like a
mathematician, not a mathematics _user_ like most science texts.

"How To Solve It" George Polya. Heurists and problem solving skills, by a
great mathematician.

Do a google search, specially in the sci.math newsgroup. Again, read books by
mathematicians _for_ mathematicians; they're often far more enjoyable and
actually far more straightforward (I was often confused by the examples in my
school work; I didn't care for "vehicle moving at speed X" or "object falling
at from height Y". We all have a different internal _visual_ mind and I tended
to think in abstract patterns, usually colors, lines or nested bodies, without
real physical objects distracting me.)

~~~
babyboy808
Fantastic reply, thank you.

An Introduction to Mathematical Reasoning loks good, just going to read some
reviews on Amazon first.

Would 'How To Solve It' be aimed at someone like myself?

~~~
billswift
How to Solve It seems to be aimed at math teachers, but it is useful for
almost anyone wanting to learn more math since it gives ideas about how to
think about maths.

------
michael_dorfman
What kind of math do you want to learn?

There's a lot of great resources out there, but you need to be more specific.

For example: I really enjoyed Gilbert Strang's course on Linear Algebra,
available as a series of video lectures on MIT OCW
([http://ocw.mit.edu/OcwWeb/Mathematics/18-06Spring-2005/Cours...](http://ocw.mit.edu/OcwWeb/Mathematics/18-06Spring-2005/CourseHome/)).

If you are interested in Discrete mathematics, Knuth's "Concrete Mathematics"
is a great book--but it might not count as "basic" enough for your purposes,
depending on your background.

If by "basics" you mean "the stuff you should have learned in high school or
as an undergrad", the Standard Deviants videos are fun:
<http://www.sdlearn.com/default.asp>

~~~
jimbokun
Those Strang videos demonstrate that there are benefits to lecturing in front
of an empty chalkboard (or several, in his case), chalk in hand, vs.
displaying slides from a computer. Watching how he steps back and thinks out
loud about what he just wrote and what to do next, double checking his work,
making and correcting the occasional error, all greatly enhance the pedagogic
impact.

------
brg
Here is some general advice.

First have some kind of goal. Do you want to be able to determine the orbit of
an planet, evaluate the complexity of algorithms and computing models, study
human interactions as walks on graphs, or use statistics to model and predict
complex systems. Decide on this first, don't wonder around mathematics
aimlessly.

Secondly, work the books. Maths can not be learned by observation, and reading
proof after proof is simply observation. Memorize proofs, work from your
current point back to first principles, and do all the problems you can. Of
course there will be times when you simply can not find a means to start on a
problem, and at that time find help or try and come back to the problem later.

------
rickdangerous1
I'm in the similar situation as you (OP). My high school education was
interrupted quite badly and 13 years after graduating I lack confidence in my
comp sci endeavors because my maths sucks so bad. I'd be interested to hear of
any hackers who have missed education milestones (like high school maths) but
gone and successfully filled in the gaps. The reason I'm asking is because I'm
kind feeling that things like maths knowledge is layered on year after year
and if you lack the foundation its really a huge amount of work to repair each
successive layer.

Passing on some wisdoms to the young hackers around here... I wish someone had
grabbed me by the face in highschool and told how important all these layers
of skills/knowledge would be for getting the kind of jobs I want now. <I come
from a blue collar background - by the time I realised how important education
was (age 22) it was too late to do much about it>

~~~
bgutierrez
I left school when I was 16 to do homeschool, which turned out to be nothing
more than my parents buying me books when I asked for them and nothing more.

I did my GED and dropped out of college, so I think that I can relate to how
you feel. There's a huge inferiority complex that comes from having less
education than others. Filling in the gaps is difficult but I've found that
whenever I fill something in, I've benefited pretty quickly. Lately I realized
that I sucked at systems programming concepts, so I've been reading about
that, which has been really helpful. I make a living as a programmer, but I
have to continually be studying to try and fill in gaps before I'm hurt by
lack of knowledge.

If someone can work through a discrete math textbook and do the exercises, I
think they could get a lot out of it. Calculus has some important ideas, but I
think the discrete math is much more relevant to every day programming. I took
a course last year at my community college that used Discrete Mathematics with
Applications, by Susanna S. Epp. I don't recall it requiring higher math than
algebra, but it did require sharp thinking. Concrete Mathematics: A Foundation
for Computer Science is probably a good textbook, but I haven't used it, so
can't say anything about it.

------
kenshi
I find the Khan Academy videos to be pretty helpful. They start with the
absolute basics and go on up. <http://www.khanacademy.org/>

~~~
babyboy808
Thank you

------
WilliamLP
"What is Mathematics" by Courant and Robbins is quite good and respected, but
it will challenge you: [http://www.amazon.com/Mathematics-Elementary-Approach-
Ideas-...](http://www.amazon.com/Mathematics-Elementary-Approach-Ideas-
Methods/dp/0195105192) It may be more advanced then what you're looking for
though.

------
ochiba
Have you seen this thread?

<http://news.ycombinator.com/item?id=665029>

~~~
tokenadult
That thread recommends many very few good books, but probably mostly books too
hard at first for the participant who has posted this new thread.

I'll recommend a couple of books from that thread:

<http://www.springer.com/physics/book/978-0-306-45036-5>

[http://www.amazon.com/Mathematics-Short-Introduction-
Timothy...](http://www.amazon.com/Mathematics-Short-Introduction-Timothy-
Gowers/dp/0192853619/)

I agree with the recommendation of An Introduction to Mathematical Reasoning
in this thread.

Another participant has already recommended my favorite for background
reading, Concepts of Modern Mathematics by Ian Stewart.

[http://www.amazon.com/Concepts-Modern-Mathematics-Ian-
Stewar...](http://www.amazon.com/Concepts-Modern-Mathematics-Ian-
Stewart/dp/0486284247/)

Get that right away.

Sawyer's A Mathematician's Delight is surely also good (I've read other books
by Sawyer).

<http://www.amazon.co.uk/gp/product/0486462404/>

Read those for background as you get my favorite overviews of mathematics:
Basic Mathematics by Serge Lang and Numbers and Geometry by Joseph Stillwell.

[http://www.amazon.com/Basic-Mathematics-Serge-
Lang/dp/038796...](http://www.amazon.com/Basic-Mathematics-Serge-
Lang/dp/0387967877/)

(Basic Mathematics is mostly high school level math, with a minimum of fuss
and bother, and good exercises.)

[http://www.amazon.com/Numbers-Geometry-John-
Stillwell/dp/038...](http://www.amazon.com/Numbers-Geometry-John-
Stillwell/dp/0387982892/)

(Numbers and Geometry is mostly undergraduate level math, with very good
explanations and excellent exercises.)

------
EdwardCoffin
I've had pretty good luck using the MAA recommendations list for libraries:
[http://mathdl.maa.org/mathDL/19/?pa=content&sa=viewDocum...](http://mathdl.maa.org/mathDL/19/?pa=content&sa=viewDocument&nodeId=3219)

I recommend reading Theodore Gray and Jerry Glynn's Brain Rot article for some
ideas on what math skills are worth intensive study and development and which
are less important: <http://www.theodoregray.com/BrainRot/>

There are several articles and blog postings on the topic of math self study
that I found interesting and you might find useful in determining what and how
to study: Developing your intuition for math:
[http://betterexplained.com/articles/developing-your-
intuitio...](http://betterexplained.com/articles/developing-your-intuition-
for-math/) Math every day (Steve Yegge):
<http://steve.yegge.googlepages.com/math-every-day> Math for programmers
(Steve Yegge): [http://steve-yegge.blogspot.com/2006/03/math-for-
programmers...](http://steve-yegge.blogspot.com/2006/03/math-for-
programmers.html) How to read mathematics (Shai Simonson and Fernando
Gouvea):<http://web.stonehill.edu/compsci/History_Math/math-read.htm>

------
le_dominator
If you go back and learn Algebra, Trig, Geometry, then I fully recommend the
Cliffs Study Solver series of textbooks because they are very cheap and very
thorough plus each day you do a chapter, you'll make cumulative progress.

You are introduced to a concept, given a set of practice problems to see the
concept in action, then given a problem set to solve on your own.

[http://www.amazon.com/Algebra-I-Cliffs-Study-
Solver/dp/07645...](http://www.amazon.com/Algebra-I-Cliffs-Study-
Solver/dp/0764537636/ref=sr_1_1?ie=UTF8&s=books&qid=1249996670&sr=1-1)

Each book is about 350 pages and you'll be up to speed in no time.

------
papaf
I came back to doing maths after a gap of over a decade. I found the student
survival guide very clear and useful. I imagine it would be excellent to
someone who is not good at maths.

I read the guide every evening while I was cooking (its not a difficult read)
and my maths improved greatly.

<http://www.netcomuk.co.uk/~jenolive/> [http://www.amazon.com/Maths-Students-
Survival-Self-Help-Engi...](http://www.amazon.com/Maths-Students-Survival-
Self-Help-Engineering/dp/0521017076)

------
zppx
That's depends on what you want to learn and for why, well, some people want
to understand the formalism of a theory, as theoretical computer science or
theoretical physics where others are only interested in applications, so I
will take a generalist approach in the topics, yes topics not books, that I
will advise you to learn. Unless the book is awful (and there are many out
there that are) it will makes no difference which book you pick, you generally
will not "read" a math book, the only case in which you will is when it's a
book for divulgation (as Polya's "How To Solve It"). For me the basics is:
Statistics (Descriptive and some Probability), Calculus, the idea of Limits,
Derivatives and Integrals (for Multiple Variables) and applications, Linear
Algebra and also some Applications (there are many), numerical Linear Algebra
is totally necessary if you want to apply it in the real world, a basics in
Differential Equations, some Numerical Analysis.

If you want to learn things closer to computer science then learn something of
Number Theory, some Enumerative Combinatorics and Graph Theory as well. The
list is extensive because I come from a mathematical background. If you learn
at least a bit of these topics them the next step will be apparent for you.

------
brown9-2
I know this doesn't answer your question since you asked about math in
general, but in case anyone ever starts a "Best Physics books for complete
noobie?" thread I'd like to go ahead and suggest Brian Greene's "Fabric of the
Cosmos": [http://www.amazon.com/Fabric-Cosmos-Space-Texture-
Reality/dp...](http://www.amazon.com/Fabric-Cosmos-Space-Texture-
Reality/dp/0375727205/ref=sr_1_1?ie=UTF8&s=books&qid=1249998147&sr=8-1)

~~~
jimbokun
Maybe more on topic than you suggest, as some Physics books double as pretty
good books about math.

[http://www.amazon.com/Road-Reality-Complete-Guide-
Universe/d...](http://www.amazon.com/Road-Reality-Complete-Guide-
Universe/dp/0679776311/ref=sr_1_1?ie=UTF8&s=books&qid=1250000533&sr=1-1)

OK, I haven't actually read it, but it _looked_ like a really good book for
learning a lot of interesting math when I thumbed through its contents at the
library. :)

~~~
CamperBob
I agree with hyperbovine -- Road To Reality is a downright terrible book for
the casually-interested nonprofessional, and I don't understand why it gets
recommended so frequently. Roger Penrose is a bright fellow and a good writer
but this book is not for the person who did OK in high school math and physics
and now wants to take it to the next level.

~~~
MaysonL
How would it be for a guy who aced second semester freshman physics and the
math GRE (40 years ago), and top 100 on the Putnam?

~~~
CamperBob
I'd guess you'd be in his target audience. Someone who's already comfortable
with complex causality and relationships between seemingly-random facts.

------
kingkawn
Honestly I think Schaum's is pretty good because it does a bit of explanation,
but you primarily learn through doing two dozen pages of problems per chapter.

------
jorgetown
Some years ago I took the same approach to start again from ground zero. I
found Polya to be good, but a Mathematician's Delight is better and more
accessible:
[http://www.amazon.co.uk/gp/product/0486462404?ie=UTF8&ta...](http://www.amazon.co.uk/gp/product/0486462404?ie=UTF8&tag=jorg-21&linkCode=as2&camp=1634&creative=19450&creativeASIN=0486462404)

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billswift
If you want to DO math, nothing beats working through a decent textbook. I
have worked through several since I'm approaching 50 and don't use math enough
to keep my skills up (and like any skill, you have to keep practicing to stay
decently competent in math). The best Precalculus textbook I have used is
Swokowski's "Algebra and Trigonometry with Analytic Geometry" which is clear,
concise, and has lots of problems.

If you want to learn ABOUT math, Davis and Hersch's "The Mathematical
Experience" is a fairly easy read about philosophy of math, how it is used,
and a bit about studying math.

Eric Temple Bell's "Mathematics: Queen and Servant of Science" is a bit dated
but an excellent history of math for someone interested in learning to do
advanced math; the author's a bit biased towards algebra over analysis, number
theory, and geometry, but not excessively so. Its biggest lack is only one
short chapter on probability and statistics. This is not a particularly easy
read since it covers things in some depth, but I think it is worth the effort.

------
jedi_stannis
I haven't picked it up yet, but I remember reading about "The Princeton
Companion to Mathematics" on here a while ago. It looks like a pretty complete
guide to all of modern mathematics and sounded like it was easy enough for a
beginner to get through while still being able to teach math experts some new
things.

Thanks for reminding me about it - I think I'm going to order my self a copy!

~~~
arghnoname
My own math background is through Calculus II right now, so I'm a beginner,
but I can't say the Companion (though it is excellent) is something I could
'get through' in the sense that people usually mean it. By necessity it is
terse, despite its massive size. I personally find it useful as a survey on
the field, but I wouldn't recommend it as a teaching guide in and of itself
for people who happen to be as dumb as me.

------
unignorant
For proof based mathematics, I found "How to Prove It:A Structured Approach"
helpful.

------
seanstickle
I heartily endorse reading the classic works of geometry as a way to both the
subject as well as a way of thinking about math, proof, and argumentation.

Start with Euclid's _Elements_, and then move onto Archimedes' short books on
levers and floating bodies, Apollonius's wonderful treatise on conic sections,
and Ptolemy's _Almagest_.

They are excellent for self-education, providing both geometrical knowledge in
itself, as well as extensive training in sound reasoning. Don't be fooled by
the antiquity of their origin: they teach more clearly than most modern day
texts, and their content is timeless.

~~~
anatoly
I completely disagree. Do not rely on ancient texts to give you understanding
of what mathematics is about. They're very difficult to read, they do not
teach well, and you'll know next to nothing about even basic notions of
mathematics, as understood now, after you're through with them.

Do read the classical texts for pleasure, to round out your education, or to
understand the history of mathematics. But don't study math from them, that'd
be a terrible mistake.

~~~
seanstickle
I'd be curious to know what you find difficult about them, and why you think
they do not teach well.

I find Euclid extraordinarily clear in his exposition of geometry. And I have
not yet found a better teacher on conics than Appolonius, although Descartes
comes very close with his analytic geometry.

Are there some defective constructions in the Conic Sections or in the
Elements that lead you to say that "you'll know next to nothing about even
basic notions of mathematics"?

It's not clear to me why you think that the ancient texts are lacking in
pedagogical power, except for perhaps a personal aesthetic preference.

------
travisjeffery
I'm pretty at Mathematics and doing that takes a lot of work. And the
Mathematics books for noobs are no good.

The better approach is to get a theoretical book, something like Spivak's
Calculus or Linear Algebra by Friedberg, Insel and Spence. And then from there
whenever you have difficulty with the material spread out laterally and you
really start to gradually grow an understanding of mathematics.

And then perhaps, one day you'll be up for Spivak's Calculus on Manifolds!

------
MaysonL
_The World of Mathematics_ edited by James Newman (a four-volume anthology)
may give you a feel for and an interest in exploring math further. It's a
great collection ranging from mathematical curiousities and puzzles to memoirs
of mathematicians to a very moving short story by Aldous Huxley.

Also, _One, Two, Three, Infinity_ by George Gamow (a great physicist) is a
great intro.

Second the _How To Solve It_ recommendations.

------
brg
If you have some decent background in reasoning and logic, I would suggest
Linear Algebra Done Right by Sheldon Axler. It provides a general introduction
in mathematical reasoning as well as providing a strong framework for maths
needed in many fields.

[http://www.amazon.com/Linear-Algebra-Right-Sheldon-
Axler/dp/...](http://www.amazon.com/Linear-Algebra-Right-Sheldon-
Axler/dp/0387982582/)

------
suprgeek
For a "popular" treatment of mathematics that does go into some mathematical
detail "Journey through Genius: The Great Theorems of Mathematics" by William
Dunham is difficult to beat. It is by far one of the best "Math" books I have
read that have kept me coming back to it. Also, try some of the books by John
Derbyshire along similar lines.

------
springcoil
I second the OCW reference. Some of the Calculus for Dummies, type books are
good. Its good to remember that Calculus and Linear Algebra don't have to be
that complicated. I also recommend scan the books before you buy them, I
wasted far too much money at college on txtbooks that I ended up despising

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thunk
It's not strictly math, and I can't recommend it from experience, but I've
always been curious what a beginner's reaction to _Structure and
Interpretation of Classical Mechanics_ might be. It's free here:

<http://mitpress.mit.edu/SICM/book.html>

------
jaja
"Scientific Notation and Other First Principles: Comprehensive Mathematics for
Lawyers and Politicians," by Jacob Herwitz. Penguin, 1992.

Great introductory text which starts from algebraic first principles and goes
through pretty much everything up until differential equations. Very thorough.

------
ottbot
I think Kreyszig's Advanced Engineering Mathematics is an excellent text in
applied math.

I'm not sure where you're at or what direction you want to study, but once
you're familiar with calculus concepts this text is a good place to go to deal
with ODE and analysis.

------
patternexon
Concepts of Modern Mathematics

[http://www.amazon.com/Concepts-Modern-Mathematics-Ian-
Stewar...](http://www.amazon.com/Concepts-Modern-Mathematics-Ian-
Stewart/dp/0486284247/ref=pd_sim_b_88)

------
intregus
There are a few math books in the "Head First" series (which I always love).
Those might serve more as refreshers for most people, but for a true math noob
(like me), they have really helped.

------
paulreiners
"Naive Set Theory" by Paul R. Halmos.

~~~
mahmud
Set Theory is the building blocks of mathematics, and Halmos is a first-rate
mathematician and teacher. Excellent recommendation Paul. I worked through
Halmos 8 months into my mathematics self-study and loved it.

------
avner


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ftse
'Zero: The Biography of a Dangerous Idea' by Charles Seif

