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Pure Math Preparation (the ultimate high school / pre-university book thread)

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Main Question or Discussion Point

This thread is intended to give high school students the necessary information to cope with university level mathematics. I find the “who wants to be a mathematician” thread too convoluted to be of use when looking for books, and that it serves college students more than k-12. Nonetheless, it is a good read and contains a wealth of information for those with time and patience. I find a lot of people asking for high school resources, so hopefully this thread will serve as an oasis of sorts. Most importantly, I hope this advice exposes students to the beauty of math which has been all too absent in the modern curriculum. May it bring hours of joy and discovery.

Before I begin, I will first mention a little about mathematics education and how it has changed over the last few years. Traditionally, math has mostly been taught as a set of disjoint algorithms. Students were taught to memorize a bunch of procedures and to apply them to certain problems. It was criticized on the basis that students had weak conceptual understanding of material and relied heavily on rote memory. Traditional math was replaced by “new math” for a brief period in the 60s, partly because US was falling behind with Russia in the sciences. Books were more proof based and followed an axiomatic approach. Most modern math professors raved the new approach, and books from that era are considered ‘classics’. This new approach was quickly abandoned however, due to the fact that theory often obscured the main point and students had weak intuition of the material they knew. Also, it was deemed highly difficult for all but the ablest students. It was replaced by ‘reform mathematics’ in the 1980s, which sought out to connect mathematics to the real world and focus on applications and problem solving. This is largely what we are settled with today, mixed with the traditional approach. It is criticized as being dumbed-down math, and is part of the reason students experience difficulty transitioning to university mathematics.

In my opinion, I think students need both “new math” and “reform math”. No matter what math professors will want you to believe, mathematics has its roots in applications and solving problems; they create new branches of math and provide motivation for most subjects. Despite what engineers and scientists will tell you, a rigorous approach to math is absolutely essential as intuition is often wrong and limited; theory is more than “useless math” or an “art form”, it validates the tools of science and provides insight that intuition alone never could. Conceptual understanding is rarely needed by non-mathematicians, although at high school level I think everyone should have a strong background in elementary mathematics. Most scientists and engineers will succeed with reform math, as it was catered to their needs.

In what follows, the books are listed in the correct order of progression. In each grade, two subjects should be tackled simultaneously instead of one. Ie. Work through algebra and geometry at the same time, instead of first doing all of algebra and then all of geometry. Books are designated by two categories: skill and theory. Skill consists of books that are largely algorithm based, where you apply skills and techniques to solve problems; they are your cookbooks with recipes on how to solve questions. The material is often presented with a geometric/intuitive approach, and is of most use to the sciences and applied mathematics. This is the approach you’re used to in high school, so you can ignore these sections if you’ve been to class. Theory will give you the nitty gritty details of the contents, with a rigorous backbone and an abstract approach. These are your slick, black and white, no colors/photos booklets. You will be taught to discover mathematical methods on your own, and get practice at proofs. The questions will usually be highly difficult, requiring a lot of patience and determination. Solutions are not usually provided; you will have to gain confidence and get them yourself. These books will mostly be useless and annoying to most scientists, and are geared more towards mathematicians or those who want to perfect their gears. Snobbish mathematicians will tell you this is the only way to learn true math. You can ignore them if you’re in physics, but if you are taking pure math in university you had better heed their word. Lastly a few words of advice: do as many problems as time permits. Do not get obsessive compulsive about not getting every single question in a book. A good effort on half the questions is much better than rushing through all with minimal effort. More is merrier, but don’t let one question bog you down too long. A good rule is if you can’t solve a question in one hour, then move on. If a section is too difficult at first, take a break from it and return later. Do a section every two days, as this will aid in remembering the material. Where exercises occur is when a section ends, for those books that are not numbered by sections. Different people have different abilities, so that some may advance faster or slower relative to others their age. Background and interests will also affect how you progress. Generally, try to spend at least 10 hours on math each week. To a secondary school student, this list can take anywhere from 2-4 years to finish. College students brushing up can probably complete this in less than 6 months. Everybody finds math difficult at first. With practice and patience, it gets easier with time.


K-8

Arithmetic
Addition, subtraction, multiplication, division, ratio, proportion, area, volume, decimals.

You should know how to multiply 2 digit by 2 digit numbers in your head. Learn to divide and do percents/decimals as well. Mostly memorization here, practice about 5 of these a day. (Ie. Do 4/7 and 67x92 in your head). If these take longer than 1min each… you need practice!

See Singapore math books for this. (ie. https://www.amazon.com/dp/9810184948/?tag=pfamazon01-20)



Grades 9-10

Algebra I
Arithmetic laws (distribute law, etc), variables, equations, number line, proportion, graphs, y=mx+b, substitution/elimination, exponents and their laws, quadratics, polynomials, fractions, division, basic number sequences.

Skill:
Elementary Algebra, Jacobs
https://www.amazon.com/dp/0716710471/?tag=pfamazon01-20

Theory:

Algebra, Gelfand
https://www.amazon.com/dp/0817636773/?tag=pfamazon01-20

Geometry I (aka Euclidean/plane geometry)

Introductory proofs, from lines to planes, triangles, circles, quadrilaterals, platonic solids, constructions.

Geometry, Jacobs (1st or 2nd edition = theory, later = dumbed down )
http://www.amazon.ca/Geometry-Harold-R-Jacobs/dp/071671745X/ref=sr_1_12?ie=UTF8&s=books&qid=1239868442&sr=1-12

An alternative, especially if you can't find an older edition of Jacobs, is the following:
Kiselev's Geometry, Kiselev
https://www.amazon.com/dp/0977985202/?tag=pfamazon01-20
https://www.amazon.com/dp/0977985210/?tag=pfamazon01-20



Grade 11

Algebra II (aka intermediate algebra + precalculus)
Functions, graphs, analytic geometry, logarithms, trigonometry, conic sections, inequalities, sequences, series, vectors, matrices, dot and cross products (use Stewart in calculus as a source for more on vectors and 3d space)

Skill:
Precalculus, Sullivan
https://www.amazon.com/dp/0132256886/?tag=pfamazon01-20

Theory:
Functions and Graphs, Gelfand
https://www.amazon.com/dp/0817635327/?tag=pfamazon01-20

The Method of Coordinates, Gelfand
(This book is optional; you will need trigonometry for it.)
https://www.amazon.com/dp/0817635335/?tag=pfamazon01-20

Lines and Curves, Gutenmacher
https://www.amazon.com/dp/0817641610/?tag=pfamazon01-20

Logarithms,
FREE: http://www.mathlogarithms.com/

Introduction to Inequalities, Bellman
https://www.amazon.com/dp/0883856034/?tag=pfamazon01-20

Trigonometry (aka Geometry II)

Triangles, right angle triangles, trig ratios, unit circle, ptolemy’s theorem, pythagoras’theorem, trigonometric functions, graphs, inverse trig functions.

Skill: included in algebra II curriculum, in Sullivan’s Precalculus book.

Theory:
Trigonometry, Gelfand
https://www.amazon.com/dp/0817639144/?tag=pfamazon01-20



Grade 12

Proofs & Logic
Symbolic logic, truth tables, negation, conditional statement, equivalent statements, quantifiers, sets, unions, intersections.

Skill:
How to Read and do Proofs, Solow
https://www.amazon.com/dp/0471406473/?tag=pfamazon01-20

Theory:
How to Prove it, Velleman
https://www.amazon.com/dp/0521675995/?tag=pfamazon01-20

Discreet Math
Combinations, permutations, binomial theorem, induction, number theory, primes.

Mathematics of Choice, Niven
https://www.amazon.com/dp/0883856158/?tag=pfamazon01-20

Numbers: Rational and Irrational, Niven
https://www.amazon.com/dp/0883856018/?tag=pfamazon01-20

Calculus
Limits, derivatives, integrals, taylors formula, analytic geometry. Partial derivatives, multiple integrals, vector calculus, vector analysis, change of variables & coordinate systems.

Skill:
Stewart, Calculus (ANY edition will do... older are better & cheaper)
https://www.amazon.com/dp/053439339X/?tag=pfamazon01-20

Your first brush of calculus should be with a book like Stewart. This is how calculus was first conceived by Newton and Leibniz. Modern books will rob you of that experience, which I think is wrong. It’s best to experience calculus as infitesimals at first, even if it means upgrading to epsilon-delta formalism later. The modern approach is too slow, and you will need this math in all your science courses from freshman year. There will be plenty of time for you at university to develop the theory of calculus, where they won’t cover this stuff. I don’t recommend Spivak, Courant, or Apostol as first exposure… save them for undergrad.

Geometry III (optional)
Transformations, isoperimetric theorems, modern circles and triangles.

Geometry Revisited, Coxeter
https://www.amazon.com/dp/0883856190/?tag=pfamazon01-20

Geometric Inequalities, Kazarinoff
https://www.amazon.com/dp/0883856042/?tag=pfamazon01-20

Geometric Transformation Series, Yaglom
https://www.amazon.com/dp/0883856085/?tag=pfamazon01-20



Beyond

These books don’t really belong to a curriculum, but are good to know if you intend to compete in Olympiads or want to have a general knowledge of math. Most should be accessible once you finish Grade 11 section.

Problem Books
Compete in Olympiads! These problem books are excellent preparation and sharpen your problem solving skills. Once all of the above is mastered, test your knowledge with these.

Problem Solving, Engel
https://www.amazon.com/dp/0387982191/?tag=pfamazon01-20

Cauchy-Schwartz Inequalities, Steele

https://www.amazon.com/dp/052154677X/?tag=pfamazon01-20

IMO Compedium, Djukic

https://www.amazon.com/dp/0387242996/?tag=pfamazon01-20

USSR Olympiad Book, Shkalrsky
https://www.amazon.com/dp/0486277097/?tag=pfamazon01-20

General Audience

Concepts of Modern Math, Stewart
https://www.amazon.com/dp/0486284247/?tag=pfamazon01-20

What is Mathematics?, Courant

https://www.amazon.com/dp/0195105192/?tag=pfamazon01-20

Originals
Classical books by originators. Not really recommended unless you have the time. Modern approach is a lot better.

Elements, Euclid
https://www.amazon.com/dp/1888009195/?tag=pfamazon01-20

Works, Archimedes
https://www.amazon.com/dp/0486420841/?tag=pfamazon01-20

Geometry, Descartes
https://www.amazon.com/dp/0486600688/?tag=pfamazon01-20

Principia, Newton
https://www.amazon.com/dp/0520088174/?tag=pfamazon01-20

Disquisitiones Arithmeticae, Gauss

https://www.amazon.com/dp/0300094736/?tag=pfamazon01-20


OTHER RESOURCES


Stanford’s Education for Gifted,
Curriculum similar to mine, but you need to apply (and pay) to use resources. I much prefer the books I listed to theirs.
http://epgy.stanford.edu/courses/math/secd.html

Who Wants to be a Mathematician?
A thread on these forums by a professional mathematician. Gives solid advice on how to cope with undergraduate mathematics, and more or less provides a scattered list of university level books.
https://www.physicsforums.com/showthread.php?t=122924

How to Become a Pure Mathematician

A list of standard undergraduate books for pure mathematics. This is mostly university level stuff, but you can look into Stage 1 for some free resources.
http://hk.mathphy.googlepages.com/puremath.htm#1calculus
 
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Answers and Replies

  • #2
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This is a great list but in my opinion there seems to be a large gap between the books that you mentioned (which are great books) and the Olympiad books. I am currently in high school and attempting to bridge that gap so let me make some more suggestions of books one might want to consider. Books on this list do not necessarily attempt to develop your understanding of the theory (which is important to understand), but rather develop your problem solving ability, which is also important. Things without links can be found on amazon.

General

The Art of Problem Solving Volumes I and II are good starting points

How to Solve it by Polya

Larson and Zeitz have good books which are comparable in difficulty to Engel.

Mathematical Olympiad Treasures is good.

Number theory

This isn't treated until Engel's book in the list. While a standard number theory book won't teach you to creatively solve harder problems, it's necessary to be familiar with the concepts of number theory. I used Elementary Number Theory by Eynden, but other books by the same name (eg Jones) will do.

After reading a book like that, this is the awkward transition phase. You need to learn to solve tricky problems with what you've learned. I personally just hung out on Mathlinks (mentioned below) and did some of the http://www.artofproblemsolving.com/Wiki/index.php/Category:Number_Theory_Problems on their site.

Once you're ready for harder material, try 104 Problems in Number Theory and read these http://www.artofproblemsolving.com/Resources/Papers/SatoNT.pdf

Inequalities

After bellman, you have a good overview of what doing inequalities is about. http://www.eleves.ens.fr/home/kortchem/olympiades/Cours/Inegalites/tin2006.pdf are available online to get supplement Cauchy-Schwarz Master Class

Combinatorics

Proofs that Count shows many great examples of bijective proofs.

102 problems in combinatorics is good.

GeneratingFunctionology is challenging (intended for senior undergrads) but some chapters are useful.

Calculus

I don't think that Spivak is too much for a high schooler (but it is challenging), and would recommend it.

Trigonometry/Geometry

103 problems in Trigonometry

Challenging Problems in Geometry

Algebra

Polynomials by Barbeau

Challenging problems in Algebra

101 problems in algebra [out of print :(]

More Resources

Mathlinks (AKA Art of Problem Solving) is amazing. Highlights of the site include
-Lists of http://www.artofproblemsolving.com/Wiki/index.php/Math_textbooks (in fact much of what I said here is probably a repeat of these lists)
-The resource section
-http://www.artofproblemsolving.com/Resources/AoPS_R_Articles.php

http://myhome.personaldb.net/ideahitme/

http://contestmath.com/

A lot http://www.princeton.edu/~ploh/olympiad.shtml http://www.tjhsst.edu/~tmildorf/math/Adv.htm http://activities.tjhsst.edu/vmt/archive/0708/pages/arml/lectures/0708/

http://www.math.sc.edu/contest/problems.html)

Kalva's Site (offline but thank god for Archive.org)

How to write proofs

Recommended Mathematics Literature
 
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  • #3
thrill3rnit3
Gold Member
713
1
as we know combinatorics isn't really taught in high school, what's a good book i can pick up before i start doing 102 problems in combinatorics (by andreescu)?? I picked up the book a couple of weeks ago, and I couldn't do anything, I'm not going to lie.

would niven's mathematics of choice be good enough for starters? I borrowed it yesterday, planning to start reading tomorrow.
 
  • #4
445
3
@qntty

the thread was never intended as one stop source for competition books, i briefly sketched those as a last minute addition. there are hundreds if not thousands of resources for those competitions, and such books would require a thread of their own. your suggestions are welcome however. polynomials by barbeau is a good one i should have listed.

competitions shouldn't be taken too seriously, and they are not the best way to prepare for pure math at university. your time is better spent on set theory and learning to prove rather than solving 103 geometry problems. maybe do a bit of linear algebra on the side. regarding spivak, i don't doubt most students could handle it. but the time invested is better spent on a rapid flare of calculus, because many science courses and competitions will rely on this knowledge. i think when one has a strong footing in elementary math and geometry, one can appretiate calculus a lot more. but you are encouraged to continue with it, it is probably the best way to prepare for college math.

@thrill
niven should suffice, it is a comprehensive book on combinatorics at the highschool level. the problem with imo competitions is that they include extremely difficult problems where students may not have to proper background to attempt. i think with niven you should be able to do many of Andreescu's problems.
 
  • #5
thrill3rnit3
Gold Member
713
1
To be honest, I don't do so well with the IMO problems. Probably I can do a couple here and there, but most of them I just give up. Do I need to be able to do them to succeed in university math (as that will be the path I will be pursuing in the future).
 
  • #6
290
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To be honest, I don't do so well with the IMO problems. Probably I can do a couple here and there, but most of them I just give up. Do I need to be able to do them to succeed in university math (as that will be the path I will be pursuing in the future).
competitions shouldn't be taken too seriously, and they are not the best way to prepare for pure math at university. your time is better spent on set theory and learning to prove rather than solving 103 geometry problems. maybe do a bit of linear algebra on the side.

While it may not be directly relevant to university mathematics, having some experience solving challenging problems can help. I think that it's not a bad activity for high school students to do after they've learned the theory. In fact, most people who do well on the competitions do study linear algebra, set theory, etc which are certainly not to be ignored and should take priority to preparing for competitions. I not saying to do it for the sole purpse of doing well on the competitions or to limit yourself to math that appears on the competitions but as long as you enjoy it, it worth doing but certainly not necessary for university math. As for learning to prove, problems on the USAMO are proof-based and require a developed proof writing ability. Someone else said it better on another forum.

What Olympiad-level math will do for you is to improve your perspective and to strengthen your ability to deal with problem-solving heuristics and abstraction. You may not directly be able to apply, for example, your knowledge of how to prove inequalities, but if you've gotten good enough at it you're aware of useful ideas like homogenization, induction, and so forth. Depending on what kind of Olympiad math you focus on, it can also lead directly or indirectly into a lot of interesting college-level math. Abstract algebra and its many descendants are the big example I can think of right now.

My answer is to do all of the math you can. Whatever you can find and whatever interests you.

Edit: Here, I'll give an example. A month or two ago I noticed that several AIME problems (and at least one USAMO problem) take the form "find the number of words over some alphabet excluding these words as a substring," and I thought it would be interesting to figure out how to solve them generally. By posting a thread here and after some investigation on Wikipedia, I was led to:

- the Knuth-Morris-Pratt and Aho-Corasick algorithms,
- the notion of a regular language,
- this blog post.

So a problem in combinatorics ended up being directly relevant to my other budding interest right now, which is theoretical computer science. And the net effect of this exploration - which I'll remind you started with AIME and USAMO problems - is that I'm going to take more theoretical computer science courses now.
And Prof Tao made a blog post on the subject

I greatly enjoyed my experiences with high school mathematics competitions (all the way back in the 1980s!). Like any other school sporting event, there is a certain level of excitement in participating with peers with similar interests and talents in a competitive activity. At the olympiad levels, there is also the opportunity to travel nationally and internationally, which is an experience I strongly recommend for all high-school students.

Mathematics competitions also demonstrate that mathematics is not just about grades and exams. But mathematical competitions are very different activities from mathematical learning or mathematical research; don’t expect the problems you get in, say, graduate study, to have the same cut-and-dried, neat flavour that an Olympiad problem does. (While individual steps in the solution might be able to be finished off quickly by someone with Olympiad training, the majority of the solution is likely to require instead the much more patient and lengthy process of reading the literature, applying known techniques, trying model problems or special cases, looking for counterexamples, and so forth.)

Also, the “classical” type of mathematics you learn while doing Olympiad problems (e.g. Euclidean geometry, elementary number theory, etc.) can seem dramatically different from the “modern” mathematics you learn in undergraduate and graduate school, though if you dig a little deeper you will see that the classical is still hidden within the foundation of the modern. For instance, classical theorems in Euclidean geometry provide excellent examples to inform modern algebraic or differential geometry, while classical number theory similarly informs modern algebra and number theory, and so forth. So be prepared for a significant change in mathematical perspective when one studies the modern aspects of the subject. (One exception to this is perhaps the field of combinatorics, which still has large areas which closely resemble its classical roots, though this is changing also.)

In summary: enjoy these competitions, but don’t neglect the more “boring” aspects of your mathematical education, as those turn out to be ultimately more useful.
 
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  • #7
thrill3rnit3
Gold Member
713
1
Thanks for the response. I was just wondering because I'm not the typical math genius who can think of tricks or have the ingenuity to think of the solutions for Olympiad problems quickly. I'm not one of those kids who learned Calculus in 8th or 9th grade. But I don't want those facts to ruin my desire to learn math and do math as my career choice.

Anyways, I'm looking into picking up an abstract algebra book. Any suggestions for a good introductory book? I've just finished differential equations at a nearby community college. Would that be enough to start an abstract algebra book? Or are there more pre-requisites? I also finished linear algebra before doing differential equations.
 
  • #8
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Apostol is extremely good at motivating concepts and describing how they came about historically. It's not all e-d formalism like you suggest (although formalism does have its place, and the book is completely rigorous). I'm confused at why you think Stewart is better than Apostol for able students. If you can't handle Apostol, by all means try with an easier book, but you shouldn't be dissuading students who can handle it from pushing themselves.
 
  • #9
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I could never do (67)(92) in my head!
 
  • #10
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Just FOIL it. 60*90=5400
6*2=120
7*90=630
7*2=14

5400+750+14=6164

Anyways, most of the math majors I know are terrible at arithmetic. Problems are designed to involve only the simplest of operations.
 
  • #11
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ok, where do you learn tricks like THAT....like foiling two double digits? i have been through so much math and have never known that. very cool. i want more.
 
  • #12
jk
148
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You probably encountered it in algebra class in high school:
(60 + 7)*(90 + 2)
 
  • #13
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oh, yes, i know foil....but i meant other tricks like this to apply to all math. shortcuts, etc. foiling binomials is easy, i just never thought of it for basic 2digit multiplication. is there a way to do it with 3 digit numbers? ie (112)*(561)=

is there a book you know of that gives short cuts to doing math in your head?
 
  • #14
jcw99
Gold Member
15
0
the Trachtenburg Speed System of Basic Mathematics is a good one
 
  • #15
mathwonk
Science Advisor
Homework Helper
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To help navigate the huge thread "who wants to be a mathematician", posts 7, 8, and 22 are specifically aimed at pre-university level students, and 38, partially so.

A remark on "Originals" as mentioned in this thread: I like Euclid a lot for geometry and admit I consider originals vastly superior to modern versions, once they become accessible. Another original I like is the Elements of algebra by Euler. A high school student who reads that book will know how to solve not just quadratics but cubics, and better than many math grad students.
 
  • #16
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Howers, and everyone else, I would really like to thank all of you for putting together and responding (post to post) in this thread. I, being a slave of the public education system, have never really had the privilege to have a good mathematics class. I mean, even my AP Calculus class was pretty crappy-y, so I dropped out of that one really quickly...

Anyhoozledoozle, now that I have graduated high school, I'm going to attend a technical college, to attain a job in the HVAC-tech. trade, later on. I'm too broke to jump into the academic university industrial complex, and I'm having legal residential problems (my green card expired or something), so I practically can't get any generous scholarships quickly.

Nevertheless, I really enjoy math, and I suspect it's an art which I might want to work with, in the future.

So, I'll take your list into account, pay up some library fines, and track some books down.

Also, my step-father moved in with my mother, and left me two mathematics textbooks; because he's trying to get rid of random things/objects/items he no longer wanted and/or needed. I haven't taken a look at them yet, since I've been überbusy, but what do you all think of these books?

Bellman, Allan, et al. Algebra: Tools for a Changing World. Upper Saddle River, New Jersey: Prentice-Hall, 2001. Print.



Bluman, Allan G. Elementary Statistics: A Step by Step Approach. Dubuque, Iowa: Wm. C. Brown, 1995. Print.


I actually plan on reviewing everything I learned from day one, with a huge emphasis on proofs, of mathematics. So, that's why I'm really grateful for the initial list of recommended books. I'll start taking a look at that algebra book, today. From skimming through it, though, it seems very tailored towards middle-schoolers...

o_O

* * *​


Also, I was going to link the books I mentioned to their amazon/ebay postings but I'm apparently not allowed to do so because I haven't reached "10 posts..." bummer

:b
 

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