Master Math: Step-by-Step Guide to Essential Topics - Pre-Algebra to Calculus

[Charlie Atles]

What topics of mathematics should I learn exactly step by step: pre-algebra, algebra, pre-calculus, calculus? Am I right? Can you describe the exact steps, topics of learning as in detail as possible, please?

 

[Mentallic]

I can't help you with the exact steps, but you at least need to learn geometry to supplement your algebra studies and then trigonometry after that. I'm probably also missing other topics too, but each topic is so broad that you're better off just getting a textbook and working through that.

 

[Charlie Atles]

ok, thanks!

Can you please tell me what college or university (in English) can help to obtain a math diploma only by distance? Is there any possibility to start since algebra, to go through the other topics of math and at the end to obtain a math diploma by distance (for sure I'm ready to pay for my education).

 

[Dishsoap]

Whoa now... now you're talking about something different. You just want to go up through calculus, or to get a degree in math? Those are two totally different things.

 

[QuantumCurt]

If you're talking about a whole math degree, then it depends on a lot of factors. All math majors will take algebra>geometry>college algebra>trigonometry>calculus, but these are basically high school level and freshman/sophomore level college courses. After that point it really varies a lot. It becomes far less sequential. Typical classes include both ordinary and partial differential equations, linear algebra, abstract algebra, real analysis, complex analysis, statistics and probability, number theory, topology...and many others.

I'm curious...if you haven't yet taken any of these classes, then how do you know that you want a degree in math?

 

[Charlie Atles]

Whoa now... now you're talking about something different. You just want to go up through calculus, or to get a degree in math? Those are two totally different things.

Dear, for sure I would like to obtain a degree in math. I just looking for the proper way to do that.

 

[Charlie Atles]

If you're talking about a whole math degree, then it depends on a lot of factors. All math majors will take algebra>geometry>college algebra>trigonometry>calculus, but these are basically high school level and freshman/sophomore level college courses. After that point it really varies a lot. It becomes far less sequential. Typical classes include both ordinary and partial differential equations, linear algebra, abstract algebra, real analysis, complex analysis, statistics and probability, number theory, topology...and many others.

I'm curious...if you haven't yet taken any of these classes, then how do you know that you want a degree in math?

For sure I have taken the classes at various times but now I'm going to do it again with a goal of getting a diploma.

 

[mathwonk]

This is a very general question and maybe has no reasonable answer, but I will make some suggestions anyway, which you may/should take with a grain of salt.

Learn arithmetic first, then study Euclid's Elements, Euler's Elements of Algebra, Allendoerfer and Oakley's Principles of Mathematics, Cruse and Granberg's Lectures on Freshman Calculus, Martin Braun's book on Ordinary differential equations, Frederick Greenleaf's book on complex variables, Mike Artin's Algebra, Massey's Algebraic Topology - an introduction, and Wendell Fleming's Calculus of several Variables. This leaves out statistics and probability except for very elementary discussion in Allendoerfer and Oakley.

 

[QuantumCurt]

Euclid's Elements is a very cool book that I see as a must-read for anyone interested in geometry, but I can't really recommend it as a good tool to actually learn geometry. Kiselev's two part treatment of geometry is a much more accessible text in my opinion, and still has a TON of rigor.

 

[mathwonk]

Thanks for your perspective. Perhaps you will suggest a list of books yourself. I admit my criterion for my titles is first of all the depth of content, i.e. what do they have to offer, if mastered. I tried to choose them also as accessible, but not to the point of sacrificing quality of mathematics. Having consulted it briefly, I am not myself a fan of Kiselev. I admit I came to admire Euclid only after a lifetime of over 50 years of study and research in geometry. It really does help I think to approach Euclid with Hartshorne's book in hand, Geometry: Euclid and beyond. My list is composed with a view to the logical structure of mathematics from the ground up, and in my opinion Euclid definitely belongs at the beginning of that structure. But it is only one mans' opinion.

To give one concrete example and reveal my own bias, Kiselev inverts the presentation of Pythagoras's theorem from Euclid, presenting the theory of proportions first and deducing Pythgoras. In Euclid the theory of area is presented first using elementary decompositions of figures and deducing Pythagora from geometric considerations then deducing the theory of proportions afterwards. True, the theory of proportions makes the theory of areas easier, and I at first thought this was the better way to do things, but later changed my view. In particular there apparently has never been any civilization in which proportion was discovered before area via decompositions, so that order of doing things is opposite to the intellectual history of the human race and presumably also to the way the mind works. In a nutshell, congruence, on which decompositions is based, is more fundamental than proportions. I myself favor presenting idea as they were actually discovered, as a better way to understand them deeply.

My most difficult choice was of an introductory calculus book. My favorite for content is Courant, but that seemed not too accessible for many people. Maybe I will suggest rather Calculus: the elements, by Michael Comenetz, as a book that really tries and hopefully succeeds at giving a good feel for the meaning of differentiation and integration, as well as rigorous proofs of basic theorems. He also tries to convey a feel for infinitesimals, a rare topic.

 

[mathwonk]

I regret I do not have a copy of Kiselev at hand for this commentary, having given mine away after becoming dislliusioned with it, but from searching online I offer another possible example that is especially important to me. It seems, that Kiselev, like many other people, defines a line as tangent to a circle if it meets that circle in only one point. (This information comes from a homework set on Kiselev and not from the actual book, but if correct, it is to me a significant flaw in Kiselev compared to Euclid.) This definition is not Euclid's and has almost no value in terms of understanding the concept of tangency. It applies only to quadratic curves, as opposed to Euclid's which applies to all curves at all except inflection points. Namely Euclid deines a line as tangent to a curve if it meets it without "cutting" (across) it. This is indeed the basic idea via intersection theory of a line meeting a curve with multiplicity two (or any even multiplicity).

Then Euclid proves the more usual result that a tangent line to a circle is perpendicular to the relevant radius, and more interesting, if given any other line through the same point, the circle eventually interposes itself between that line and the tangent line. If you think about it, this is the limit definition of a tangent line in terms of secant lines. From what I have read, Newton may have read Euclid just before giving his own version of this definition, and in any case Euclid's definition and theorem prepares the mind for the general modern definition. If you have Kiselev at hand, perhaps you can give us more data on this concept as treated there, in particular whether he includes this fundamental limiting version of tangency.

Another interesting result in Euclid, which is not always credited with containing any trigonometry, is the geometric version of the law of cosines, Props. II. 12 - 13. These are clearly presented as simple generalizations of the Pythagorean theorem, but most of my students have had no familiarity with them, or often do not realize that the analytic versions they have seen are just that.

I should admit that in teaching it, I also alter Euclid a little in Book II, introducing a bit of algebraic notation for the (to me) somewhat tedious geometric statements that things like A(B+C) = AB + AC, and I make explicit the idea of a geometric multiplication, in which the product of two segments is a rectangle.

But the fun of geometry is that there are many ways to try to present it, all of them interesting in their own way. After trying many of them over a lifetime of teaching I just prefer Euclid's now that I see how it is foundational to so many modern ideas. In Book V e.g. we see clearly for the first time in history the idea, now often credited to Dedekind, of real numbers (arbitrary ratios of segments) as things approximable arbitrarily closely by rationals.

FWIW my notes on reading thr first part of Euclid are here:

http://alpha.math.uga.edu/~roy/camp2011/10.pdf