Found in 9 comments on Hacker News
ufo · 2014-01-17 · Original thread
I guess any introduction to real analysis should cover that.

Unfortunately, I don't know any online references out of the top of my mind.

tokenadult · 2013-09-22 · Original thread
What would be a good book series (preferably a classic one that's stood the test of time) on math (Algebra, various Calculus topics, Statistics, etc)?

I'd like to edit this some more during the edit window for this comment. To start, the books by Israel M. Gelfand, originally written for correspondence study.

An acclaimed calculus book is Calculus by Michael Spivak.

Also very good is the two-volume set by Tom Apostol.

Those are all lovely, interesting books. A good bridge to mathematics beyond those is Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach by Hubbard and Hubbard.

A very good book series on more advanced mathematics is the Princeton University Press series by Elias Stein.

Is this the kind of thing you are looking for? Maybe I can think of some more titles, and especially series, while I am still able to edit this comment.

tzs · 2013-01-11 · Original thread
The publishers will tell you that it is because it is a lot of work to produce new editions every couple of years to keep the textbook up to date.

I'm pretty sure they are lying, at least as far as that explaining the high prices. Here is why I believe this:

These are Apostol's excellent two volume undergraduate calculus textbook. Volume I is $231.25, and volume II is $189.49.

They were around $22 each when I bought my copies in 1977. I'm sure that NONE of the subsequent price increase is due to revisions and updates, because there have been no revisions or updates. Volume I is still the second edition (from 1967) and volume II is also still the second edition (from 1969). (There has been no need to revise or update them).

Based on inflation, these books should be about $80-90 per volume now (if we assume that $22 in 1977 was a reasonable price).

jasim · 2012-06-08 · Original thread
I've Silvanus sitting in my shelf, but am yet to look into it yet.

A couple of recommendations (not specific to just Calculus):

- What is Mathematics? (Courant

- Calculus (Apostle

- Mathematics from the Birth of Numbers ( This book was written by a Swedish surgeon without any background in Mathematics. He started working on this when his son started attending university. A recommended read.

- The Calculus Lifesaver (Adrian Banner). This book is supposed to be a guide for students to crack their exams. But I found the book surprisingly informative.

- Godel Escher Bach. I've read only the first couple of chapters. My interest in mathematics was rekindled to a great degree by Godel and the Incompleteness Theorem. (

- The concept alone makes me happy! Metamath is a collection of machine verifiable proofs. It uses ZFG to use prove complicated proofs by breaking it down to the most basic axioms. The fundamental idea is substitution - take a complicated proof, substitute it with valid expressions from a lower level and keep at it. It introduced me to ZFG and after wondering why 'Sets' were being taught repeatedly over the course of years when the only useful thing I found was Venn diagrams and calculating intersection and union counts, I finally understood that Set theory underpins Mathematical logic and vaguely how.

- The Philosophy of Mathematics. From the wiki: studies the philosophical assumptions, foundations, and implications of mathematics. It helped me understand how Mathematics is a science of abstractions. It finally validated the science as something that could be interesting and creative.

I think the Philosophy of Mathematics should be taught during undergraduate courses that has Maths. It helps the students understand the nature of mathematics (at least the debates about it), which is usually pretty fuzzy for everyone.

tokenadult · 2011-03-23 · Original thread
Cal Newport, the author of the submitted blog post, draws comments both here on HN and on his own blog pointing out that deep understanding of a subject doesn't necessarily equate to VISUAL thinking about a subject. There is a big literature on "learning styles" and some attempts by some schoolteachers to categorize children by what their preferred learning styles are. When I have taken learning style questionnaires, and when I have asked my wife (a piano performance major and private music teacher) about this, the answer on learning styles is "all of the above." I personally think, based on my observations of successful learners of a variety of subjects, that learning styles are themselves learnable, and a learner with a deep knowledge of a particular subject will know multiple representations of that subject. My wife has had many piano performance courses, and also music theory and ear training courses, and has learned visual representations of music both in the form of standard musical notation and in the form of "music mapping,"

which she has found very helpful.

As for mathematics, the subject I teach now, I have always cherished visual representations of mathematical concepts, for example those found in W. W. Sawyer's book Vision in Elementary Mathematics

But other mathematicians who taught higher mathematics, for example Serge Lang, recommended memorizing some patterns of multiplying polynomials by oral recitation, just like reciting a poem.

The acclaimed books on Calculus by Michael Spivak

and Tom Apostol

are acclaimed in large part because they use both well-chosen diagrams and meticulously rewritten words to deepen a student's acquaintance with calculus, related elementary calculus concepts to the more advanced concepts of real analysis.

Chinese-language textbooks about elementary mathematics for advanced learners, of which I have many at home, take care to introduce multiple representations of all mathematical concepts. The brilliant book Knowing and Teaching Elementary Mathematics: Teachers' Understanding of Fundamental Mathematics in China and the United States by Liping Ma

demonstrates with cogent examples just what a "profound understanding of fundamental mathematics" means, and how few American teachers have that understanding.

Elementary school teachers having a poor grasp of mathematics and thus not helping their pupils prepare for more advanced study of mathematics continues to be an ongoing problem in the United States.

In light of recent HN threads about Khan Academy,

I wonder what Khan Academy users who also have read the submitted blog post by Cal Newport think about how well students using Khan Academy as a learning tool can follow Newport's advice to gain insight into a subject. Is Khan Academy enough, or does it need to be supplemented with something else?

krambs · 2011-02-10 · Original thread
To this day I still pull out Apostol's Calculus textbooks. If you're looking to learn calculus really well, or just brush up, these are the ones.
snowbird122 · 2010-06-06 · Original thread
Thanks for the recommendation. $154 new from Amazon and it was written in 1966. Killer reviews though. Thanks again.
jpwagner · 2009-08-19 · Original thread
volume 1 of the best series out there...

tokenadult · 2009-05-07 · Original thread
I think it would be enlightening if you could provide the textbooks you buy.

Interpreting that as a request to name the textbooks I find useful, I'll do that here.

Elementary mathematics:

Primary Mathematics


Miquon Math

Secondary mathematics:

The Gelfand Correspondence Program series


Basic Mathematics by Serge Lang


The Art of Problem Solving expanded series

When a student has those materials well in hand, it is time to work on AMC and Olympiad style problem solving,

and also the best calculus textbooks, such as those by Spivak or Apostol.

Elementary reading:

By far the best initial reading text is

Let's Read: A Linguistic Approach

but there are many other good reading series, including

Primary Phonics


Teach Your Child to Read in Ten Minutes a Day

(I devote more time than that to reading instruction, typically, because I use multiple materials)

and quite a few others. There is more junk than good stuff among elementary reading materials, alas.

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