
Real Not Complex: Free Math Resources - phonebucket
https://realnotcomplex.com/
======
bumblingmumbler
I am almost always disappointed in such lists as they almost never contain the
books I have never seen before. I got lucky today.

The list has [0] Multivariable Calculus by Don Shimamoto. I saw it on Amazon
first(so I have seen it before, but not on lists such as the linked one), but
didn't want to pay $40 for this strikingly beautiful book. Turns out it's
free.

The real analysis section has [1] by Lafferriere, Lafferriere and Mau Nam.
It's short, sweet and very pretty. No bloat. Logically organized. For example,
the section "Applications Of The Completeness Axiom" contains everything you
need to know as pertains to the topic (within elementary analysis) in one
place. Such things usually get lost in the weeds in regular analysis books.
Another book I like there is [2] which is on Lebesgue Integral and Measure
Theory by Cheng.

[0] [https://drive.google.com/file/d/1IB-fSe-
KR6mbo89iFTJVSA0CrFf...](https://drive.google.com/file/d/1IB-fSe-
KR6mbo89iFTJVSA0CrFfIpE-D/view)

[1]
[https://pdxscholar.library.pdx.edu/cgi/viewcontent.cgi?artic...](https://pdxscholar.library.pdx.edu/cgi/viewcontent.cgi?article=1015&context=pdxopen)

[2] [https://www.gold-saucer.org/math/lebesgue/lebesgue.pdf](https://www.gold-
saucer.org/math/lebesgue/lebesgue.pdf)

~~~
thaumasiotes
> The real analysis section has [1] by Lafferriere, Lafferriere and Mau Nam.
> It's short, sweet and very pretty. No bloat. Logically organized. For
> example, the section "Applications Of The Completeness Axiom" contains
> everything you need to know as pertains to the topic

I guess this is a matter of taste. I went to look at this and it really
bothers me that, true to the name they give, the "completeness axiom" is
presented as an axiom. My analysis class used Strichartz' _The Way of
Analysis_ , which is certainly not "short and sweet" \-- the professor
described it to us as "chatty" \-- but it goes ahead and actually proves the
"completeness axiom".

Compare the completeness axiom

> Every nonempty subset A of ℝ that is bounded above has a least upper bound.
> That is, sup A exists and is a real number.

with Strichartz' theorem 3.1.1

> For every non-empty set E of real numbers that is bounded above, there
> exists a unique real number sup E such that

> 1\. sup E is an upper bound for E

> 2\. if y is any upper bound for E, then y ≥ sup E

Maybe Laferriere, Lafferiere and Mau Nam could use a little more "bloat". :/

~~~
nointer
In the first one, they already defined upper bound and least upper bound
beforehand, so the statement is more concise than the Strichartz's one. On the
other hand, the proof of the completeness axiom, or rather a detailed
construction of the reals, is a part I always enjoy in analysis textbooks.

~~~
thaumasiotes
Yes, I have no problem with the fact that the concept "sup A" is defined
before the statement of the axiom and then not repeated in it. (It's defined
before the statement of theorem 3.1.1 in Strichartz too, it's just also
repeated there.) I'm objecting to the fact that they present a theorem as if
it were an axiom.

~~~
jfarmer
One model's axiom is another model's theorem.

~~~
thaumasiotes
Sure, the proof in Strichartz is constructive, defining a real number that is
the supremum of the set.

LLN (I'm going to assume that the family name is Nguyen) _can 't_ do that,
because they have no model of the real numbers, so it isn't possible to say
that some construct is or isn't a real number.

So this is more a case of "what is a theorem if you have a model is only an
axiom if you don't".

If you changed the statement of the completeness axiom from "every nonempty
subset _A_ of the real numbers that is bounded above has a unique real least
upper bound _sup A_ " to "every nonempty subset _A_ of the _rational_ numbers
that is bounded above has a unique real least upper bound _r_ ", you'd have
the Dedekind cut construction of the real numbers. That's not generally
presented as an axiom either.

~~~
zodiac
There are sometimes good reasons to define a mathematical object under study
(vector space, real numbers, ...) by an axiomatic list of properties it
satisfies instead of constructing it from simpler objects. John Conway's ONAG
has an interesting mini-chapter about this.

------
Koshkin
On the other hand, your time is never free - to the point that no matter how
much you pay for a math book, you will end up paying incomparably more with
your time spent studying it.

~~~
phonebucket
> On the other hand, your time is never free - to the point that no matter how
> much you pay for a math book, you will end up paying incomparably more with
> your time spent studying it.

This is true.

However, I find that I tend to buy much more than I read. Not just because I
tend to hoarde these things, but also because I try to seek out sources that
click with me before I work through them.

~~~
throwaway_pdp09
I utterly agree with @Koshkin. the monetary cost of a book is virtually
irrelevant compared to the days or weeks time cost in reading it.

I take your 2nd para in a way, the cost of a book being so cheap when I do
read up on something I tend to buy 2, sometimes 3, books on the subject so
where I get stuck on one the other will see me through.

Still, I am discomforted by your book-buying habits; a book unread is a
valuable resource wasted :) But each to their own way.

~~~
jonsen
>a book unread is a valuable resource wasted

“Unread Books Are More Valuable to Our Lives than Read Ones”:

[https://www.brainpickings.org/2015/03/24/umberto-eco-
antilib...](https://www.brainpickings.org/2015/03/24/umberto-eco-antilibrary/)

~~~
throwaway_pdp09
I can't even tell if that heap of twattery is self-aware enough to be parody.

~~~
SaxonRobber
Framing this on my blog

------
photon_lines
If you guys like visual / intuitive guides, I'm trying to release my own and
you can find them using the links below:

\- Linear Algebra: [https://github.com/photonlines/Intuitive-Overview-of-
Linear-...](https://github.com/photonlines/Intuitive-Overview-of-Linear-
Algebra-
Fundamentals/blob/master/PDF/An%20Intuitive%20Overview%20of%20Linear%20Algebra%20Fundamentals.pdf)

\- Maxwell's Equations: [https://github.com/photonlines/Intuitive-Guide-to-
Maxwells-E...](https://github.com/photonlines/Intuitive-Guide-to-Maxwells-
Equations/blob/master/PDF/An%20Intuitive%20Guide%20to%20Maxwell's%20Equations.pdf)

------
manthideaal
A more recent version of the last link in the Statistics and machine learning
section is (1)

(1)
[https://github.com/percyliang/cs229t/blob/master/lectures/no...](https://github.com/percyliang/cs229t/blob/master/lectures/notes.pdf)

