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."