Part I

I don't believe that an introduction to
math has to be long, and here I will try
to give an introduction, and overview,
just as blog posts. That is this
introduction is much shorter than a book.

So I introduce the main topics.
Throughout, nearly always you can get a
lot more from a simple Google search or
Wikipedia article.

By far the most important topics are
calculus and linear algebra. My view is
that a 12 year old with good interest and
some okay or better teaching and some good
texts and access to Google and Wikipedia
can do well with both calculus and linear
algebra.

Both calculus and linear algebra have
enough so that a person can spend their
life in research, maybe in applications,
in advanced parts. This is especially the
case for calculus. E.g., calculus can
include calculus on manifords, partial
differential equations, differential
geometry, and numercial and algorithmic
topics for all of these.

What is described here could be covered in
a good undergraduate math major. Such a
student would have all or nearly all of a
good math background for graduate work in
math, physics, or other STEM fields.

None of this math is nearly new. So, for
good texts, I recommend ones that have
been regarded as among the best for at
least 20 years. For some texts, 60 years
is not too old. E.g., my favorite text in
linear algebra was first published in
1942. So, look for old texts. Usually
can buy used copies in good condition for
less than $10.

How to use the texts: (1) Get a stack of
blank paper, a sharp pencil, a big eraser,
a comfortable chair, a good light, and a
quiet room. (2) Read a section of the
text, try to understand all or nearly all
the section, when time is available try to
get a first cut intuitive understanding
good enough to explain to someone who
knows no math at all, and then do the
exercises. If your text proves theorems
and is short on good exercises, then prove
the theorems yourself as a good source of
exercises. (3) For the more important
topics, especially calculus and linear
algebra, get 3-4 well recommended texts,
pick one as your main text, and use the
others for alternative explanations and
more exercises. (4) Occasionally look
back at what you have learned and find a
good, intuitive view of what is going on.

Qualifications: I hold a Ph.D. in
pure/applied math from a world class
research university. While I've published
peer-reviewed orginal research in
pure/applied math, mathematical
statistics, and artificial intelligence,
my interests in math are mostly for
applications; I've applied math for US
national security, in business, and in
computer science research. I've been
programming for decades, and now I'm doing
a startup, a novel Web site, that has some
old pure and new applied math at its core
(that users won't see) and have written
and run the software for the Web site.
The software is almost entirely in
Microsoft's .NET and consists of 24,000
statements, in 100,000 lines of typing.
Currently I'm collecting initial data
before going live on the Internet.

Below are 17 numbered sections. Calculus
is in section 11, and linear algebra,
section 12. For sections 1-10, plane
geometry in section 8 deserves careful
study, e.g., as in a good high school
course, but for most students for the
other sections of 1-10 just the material
here may be enough.

(1) Sets

A set is a collection, aggregation, box
of, etc. of elements -- we're supposed
to already know what the elements are.

In math, sets are useful in much the same
way as little covered plastic containers
are in cooking, say to store some left
over peas, cookies, pizza sauce, etc.

So, a set is a way to identify, keep track
of what we are talking about.

Given a set A and some element x, the x is
either in the set A or it is not -- there
are no other alternatives. If element x
is in set A, then it is in set A exactly
once. e.g., never 2 or 3 times.

The elements in a set are not in any
particular order. E.g.,

{1, 3, -4} = {3, -4, 1}

Given two sets A and B, an element x might
be in both A and B, might be in many sets.

We also permit a special set, the empty
set that has no elements. The empty set
is curiously useful, plays a role roughly
like 0 does for numbers.

We can define sets with notation, e.g.,

A = {x| x > 7}

We read this as "A is the set of all
elements x such that x > 7."

(2) Foundations

Near 1900 a lot of math was known. A
nagging question was, what are the
foundations that let us be confident
that what we are talking about makes
sense?

B. Russell noticed we could write

A = {x| x is not a element of x}

Then A is an element of A if and only if
it is not an element of A. So, we have a
contradiction, angst, tummy aches, heart
burn, head aches, etc. This is the
Russell Paradox.

The problem was solved by first deciding
what the elements were and then making
sets out of just those elements. Net, we
rule out asking of set A is an element of
set A. Of course this work was all just
conceptual.

Also there was work to set up some
axioms (properties) to be assumed for
sets; this was axiomatic set theory.
Then there was an effort, successful, to
use axiomatic set theory to define
everything else in math. So, roughly, if
the axioms are solid, so is the rest of
math.

Some deep, difficult, and profound
questions were found and some answers were
found for some of these.

In the end, except for people specializing
in foundations, the approach taken for set
theory is Zermelo–Fraenkel set theory
usually assuming also the axiom of
choice. I know that that work is solid
way down in the basement, foundations,
of math, but not many people go down there
very often. I've been there for a good
tour and have no intention of going back.

Roughly, axiomatic set theory is to add
credibility to the math you already knew
in the 8th grade.

It may be that this material gets taught
to give a deeper understanding to people
who intend to teach in K-12. E.g., Newton
invented calculus, and I doubt that he
knew about the Russell Paradox or the
Axiom of Choice.

(3) Numbers

The most important concept in math is
numbers, and the most important numbers
are the real numbers. To understand
these numbers it is nearly always enough
to visualize them as just the points on a
line. That is essentially the same as
using a yard stick. No biggie.

We can talk about the set of real
numbers; maybe we let R denote the set of
real numbers.

We can also consider the whole or
natural numbers -- 1, 2, 3, .... We
might agree that N is the set of natural
numbers.

We might define the set N of natural
number by: 1 is in N and, for each n in
N, n+1 is also in N. We will use this in
proofs by mathematical induction.

By including 0 and the negatives
of the whole numbers we get the integers

... -3, -2, -1, 0, 1, 2, ...

Maybe we say that Z is the set of
integers.

The rational numbers are all the p/q
where p and q are integers and q is not
zero. We can say that Q is the set of
rational numbers.

If we let i be or act something like the
square root of -1 all the complex
numbers are all the x + iy for reals x and
y. We can let C denote the set of all the
complex numbers.

Of course, there is no square root of -1.
But we can regard the complex numbers as a
clever bookkeeping trick that at times is
curiously useful in defining the sine and
cosine in trigonometry, electrical circuit
theory, analysis of wave motion, quantum
mechanics, etc.

This use of R, N, Z, Q, and C is common,
but there is no rule that says we could
not use other notation.

The most important properties of these
numbers you learned by the 9th grade and
learned most of those in grade school. No
biggies.

There is one more property
that is crucial in
calculus:
The set of real numbers R is
complete.
There are several equivalent
definitions of complete.
Here is maybe the simplest
definition:
Given a subset S of R and some x
in R, if for all a in S
we have that a <= x, then
x is an upper bound of set S.

The completeness property is that
each non-empty subset S of R with an upper
bound has a least upper bound.

E.g., let

S = {x| x in R and x < 2}

Then 3 is an upper bound of S and 2 is the
least upper bound.

Perhaps a better way to
explain completeness is to
say, intuitively,
that
whenever we move
in steps to be as close
as we please to some point
on the line, there is a
real number there we are
getting close to.

A big point is that
the rational numbers
are not complete.
E.g., let

S = {a | a rational and a < square root of 2}

Then the square root of 2 would
be the least upper bound but is
not rational.
So, the rational numbers are not complete.

The concept of completeness,
especially viewing it as
converging to something,
is a major concept in
advanced
work in math analysis.

There is something of a
joke that "Calculus is
the elementary consequences
of the completeness property of the
real number system."