# Re-doing mathematics from the ground up - a rather odd request

1. Jun 3, 2009

### aneesh.mulye

Earlier today, I just took the last examination paper for my examination in my third year in computer engineering. (One more year left to go.)

Over the past year, now that I no longer have to do mathematics as a subject forced down my throat - it was a compulsory subject until the end of the second year - I've come to appreciate it much more. I've come to the realisation that throughout my mathematical education, there have been serious and structural deficiencies. It is a wish to correct those deficiencies that impels me to ask for advice here. In fact, reading this forum was one of the key motivators for my becoming interested again in mathematics. Thanks are in order.

The next few paragraphs are going to be a bit of a personal mathematical history, so those less patient can skip to the end if they so desire.

In high school, for our first board examination (a standardised test given at the end of the tenth grade) we were forced to do Euclidean geometry. We were mostly expected to memorise proofs by rote, and reproduce them as needed on the examination paper. There was a set of "standard" proofs - quite a large set, by the way, requiring a lot of memorisation - out of which questions were asked. I like the other parts quite well - basic algebra, and so on - but this was enough to make me hate this part of the subject for a long time, and it is only in the last semester that I dealt with the last of the proof-phobia that that phase induced.

Later, in the 11th grade, we were taught the standard pre-calculus sequence - trigonometry, co-ordinate geometry, the needed algebra, and so on. In the 12th came actual differentiation and integration, along with an absolutely elementary introduction to differential equations.

During this phase, the focus was on a set of standard "techniques" one could use to solve the question which would come on the test. The problem was not that we had to learn the techniques. The problem was that we were provided with absolutely no context in which to understand all these things. They were all isolated little islands of information which we had to remember. There was no overall structure into which to fit these things.

Now, when I look back over the curriculum, I realise that the reason we covered the concepts of functions, injectivity, surjectivity, bijectivity, and their relationship with the invertibility of a function before learning about inverse trigonometric functions was because they were needed to understand why ITFs work the way they do. However, when we learnt them, we did not relate the two concepts. They were separate. The chapter with the introductory concept of relations and functions was treated as one thing, and the chapter on ITFs as another.

This sorry state of affairs continued throughout my education, up to and including my mathematics courses for my engineering degree. This is in large part to the "high-schoolish" attitude of many of my peers. It isn't really their fault, to be honest - they live with their parents and are not afforded much freedom, so the college considers the parents their clients instead of the students, and attempts to act as an organisation would in loco parentis. Even today, mathematics courses for engineering are taught with the same idea - memorise the techniques, plug in the numbers. It would again be fine if there were understanding and the memorisation was supposed to help it along, but that is not so. Almost all students memorise the techniques, and forget them within a few months of the examination. I do not wish to be harsh, but most would not be able to recall what they are supposed to have learnt if you ask them now. Again, I cannot blame them, because this is all that they know. The problem is particularly acute in computer engineering, as most of the mathematics we learn, we do not really need.

If my mathematics education has been so uniformly dull and drab, not to mention off-putting, why am I still interested in mathematics? Well, why does an eagle fly? My innate love the the subject has kept me going through these nightmarish years. I could not believe - and still do not - that the torture I was put through was all there was to mathematics. It was only after finding this forum that I found that I was right and that there was hope.

To come to the crux of the problem:

Right now, I'm in a rather odd position. I know calculus, and have done the standard engineering mathematics sequence, but I'm not satisfied with it. Specifically, I don't think I have mastered anything that I want to.

Therefore, I'm looking to "re-do" my entire mathematical education, beginning with pre-calculus, moving on through calculus and related algebra, and finally doing my engineering mathematics the way it was meant to be done. (The material before that that I need, I have mastered.)

What is it that I want to do over in this fashion?

From high school:

Trigonometry (I was notoriously weak in this, and I want to master it)
Algebra (Binomial Theorem, Sequences and Series, Matrices, Determinants)
Co-ordinate geometry (Lines, Circles, Conics, 3D co-ordinate geometry)

After this is done, I'll move on to revising basic calculus and university algebra. Specifically:

Calculus I & II
Linear Algebra
Fundamentals of Multivariable Calculus
Complex Calculus (AKA Complex Analysis in our courses)
Differential Equations
Vector Calculus and Scalar and Vector Field Theory
Fourier and Laplace Transforms
Partial Differential Equations

Out of these, the only thing I've not studied before as part of the curriculum is the bit about PDEs, and Linear Algebra (our study for our degree was not deep).

The reason I posted this thread is because I want some advice relating to my plan.

First of all, most of what I'll be doing will be revision. However, we've only covered all these things (I refer to everything beyond Linear Algebra in the list) superficially. I want to not only study them, but have mastery over them at the undergraduate level. So what books should I use? I can find good books for university-level material, but I seem to draw a blank for the earlier stuff (basic trigonometry, algebra, co-ordinate geometry, and other high-school level stuff). In high school, we did only what was necessary for calculus. But now, I want an extremely firm foundation - after all, I'm spending the time to re-build my mathematical understanding from the ground-up, I might as well do it right - so I'm aiming for mastery, not merely competence.

Secondly, I don't know what is the right sequence for learning this. The sequence for the high-school level material doesn't matter - I'm familiar enough with the context to learn it in any order. But I'm not sure about the later material. I've put the subjects in the order I hope will be most conducive to learning linearly. I hope someone more experienced can help out here.

I would be grateful for any feedback.

2. Jun 3, 2009

### cshum00

Very interesting post. In matter of fact i am very similar to you in the part of wanting to re-do the math learning and by the way I am a computer engineering major.

Ever since i was young I have always been bad at anything i started. When i started to learn English i was the worst one among friends. I was never able to memorize the multiplication table. I started being able to memorize it after i learned that multiplication is multiple additions and then with such pattern i was able to memorize most of it (sometimes still requires mental calculation in order to recall multiplications of 7, 8 and 9). In fact, i spent most of my time trying to look for a patter to memorize it while i was told to memorize it by rhythm.

But other than that i loved math and always did a lot of numerical calculations which led me to better than my classmates during middle school. But until i came from the Dominican Republic to the United States, i never knew how much math i was lacking. The system on education was really different i had yet to realize how much basic stuff i did not know until i started doing college calculus. The algebra problems given in high school was nothing compared to the algebra used in calculus. I had to re-learn many of the basic stuff i never knew that was when it was used for.

Now in college, what have hindered most of my college studies and years is that i don't keep moving until i understand fully and try to prove it myself each new mathematical theorem is right. Therefore i would move really slowly. In fact i move slower than anybody else but once i understand, i get to fully understand it better than most people inside out. And sometimes i even get to realize stuff that i would have not otherwise.

In addition, one of my future goals is to create a mathematics encyclopedia where i could include everything about mathematics including the different methods and point of views for explaining every mathematical phenomena.

3. Jun 3, 2009

### glueball8

I'm in grade 12 and I too feel the need to "Re-doing mathematics from the ground up". I always feel my basic are really bad. Does everyone have this feeling?

4. Jun 3, 2009

### physicsnoob93

Yeah I do, but for physics :(

5. Jun 3, 2009

### blerg

My suggestions:

High School
I don't have much experience as far as high school textbooks go (I will let someone else chime in about their suggestions)
However, I highly recommend "What is Mathematics" by Courant and Robbins. It is a wonderful overview of much of high school mathematics, in addition to calculus. It presents all the interplay between various mathematical branches that you seem to want.

University Level
I have chosen books that are more rigorous than ones the average engineer would encounter. As a pure math major my selections are probably more on the rigorous side. I believe they all provide extremely adequate treatment of everything you wish to have.

Calculus I & II: "Calculus" by Spivak
Linear Algebra: "Linear Algebra" by Friedberg
Multivariable Calculus, Vector Calculus, ...: "Vector Calculus" by Colley
Complex Analysis: "Complex Variables and Applications" by Churchill and Brown
Diff Eq, Laplace and Fourier Transforms: "Elementary Differential Equations and Boundary Value Problems" by Boyce and DiPRima
PDEs: "Boundary Value Problems" by Powers

Considering that you have seen much of the material before, I think all of these books will be accessible to you, while providing you with the "fresh start" you desire.

6. Jun 3, 2009

### aneesh.mulye

First of all, thanks to blerg and the others for their replies. It seems that there are quite a few here who share this desire to "build it up again, foundations-first". Hopefully this thread can be useful to them, too.

I think a few clarifications are in order.

My fundamentals aren't bad, but they aren't good, either. They're just "sufficient". As an engineer, I'm tremendously uncomfortable with that. No civil engineer, for example, would design a bridge without at least a 100% factor of safety. How, then, can I be satisfied with "works well enough to get the job done" when it comes to something as important as this?

Secondly, and my apologies for not mentioning this in the opening post, I am from India. In case someone thinks that I am blaming my school or anybody else for this - I am most definitely not. Schools have very little autonomy here, as the amount of material to be covered is vast. Calculus I and II, for instance, are covered in around six to seven months in school. The teachers have to deal with impossible amounts of work. I am, in fact, immensely thankful to the teachers I had, because they tried their best. The problem persists at the university level, too. Our mathematics teacher for our three mathematics courses - all the mathematics I mentioned was crammed into three courses - was excellent, but the sheer amount of material he had to cover meant that he had no choice but to focus on technique to the exclusion of a richer understanding.

@ blerg:

I will definitely look up the books you have suggested. You have said that they are more rigorous than most. That is good to know - I've been looking for rigour. The mathematics we do as engineers is the "get the job done" type variety. It is good enough for that, but leaves me unsatisfied, for the reason I gave above. Using books meant for mathematicians will correct that problem - the help is much appreciated.

7. Jun 4, 2009

8. Jun 4, 2009

### Vid

Serge Lang - Basic Mathematics for anything highschool related.

After that, get books on linear algebra and analysis. Shilov has books on both subjects that are pretty excellent and pretty cheap dover reprints.

9. Jun 4, 2009

### glueball8

I was planning to go though 1/3 of Spivak in the summer. Would that help my basics?

10. Jun 4, 2009

### blerg

Basics as far as algebra, trig, euclidean geometry? Probably Not.
Basics as far as elementary logic and Calclulus I & II? Absolutely.

That being said, for everyone in this thread who wishes to strengthen their mathematics: be careful. Because you have seen most, if not all, of the material in a book like Spivak before, it is very easy to breeze through it saying "I already know this." Also as many of you are engineering/physics you may be tempted to skip over the proofs because they are "only interesting to mathematicians." I urge you not to do this. Read every proof. As you have seen much of the material before, try to prove the theorem before you even read the proof. Finally, and perhaps most importantly, DO THE PROBLEMS.

I do not mean to sound condescending, but in order to "relearn math" you will also have to learn how to properly self-study math.

11. Jun 6, 2009

### Seth_Damien

Speaking as someone who learned absolutely no algebra in school, I can recommend Harold Jacobs' Elementary Algebra (https://www.amazon.com/Elementary-A...=sr_1_1?ie=UTF8&s=books&qid=1244340030&sr=8-1) and Geometry (https://www.amazon.com/Geometry-Har...=sr_1_2?ie=UTF8&s=books&qid=1244340085&sr=8-2).

Both books are designed to make you think, and are excellent for self-study. They won't get out a stick and beat you if you skip sessions with them, but before you know it you'll be factoring binomial squares and solving angles in no time at all.

Last edited by a moderator: May 4, 2017