Dear Math Students (Part 2)

Dear Math Students (Part 2)

Posted at: 26 Nov 2020

Single and multivariable calculus, linear algebra, real analysis, complex analysis, and abstract algebra are the core subjects in undergraduate mathematics that are not only taught to students who major in mathematics but also to those who study physics, engineer, and chemistry. They are very important for anyone who wants to do math, physics, and engineering at a serious level. This article is a short piece of advice on how to learn them. Although some part of this article is designed specifically for people who want to do a degree in mathematics, I encourage those of you who need mathematics as well as those who love the beauty of mathematics to read it.

First, you should start with single and multivariable calculus and linear algebra. The reason to start with them is that they are not too difficult as the other three mentioned above, and you don’t need to have any special knowledge in math to learn them- your high school mathematics is more than enough. For single and multi-variable calculus, Essential Calculus (Early Transcend) by James Stewart covers basic concepts of the functions and limits to double and triple integrals, and integral in vector fields. Of course, it contains so many pages but that’s really what you should learn. Also, I think you will find it interesting and enjoyable when you start going through materials that you don’t know before as well as those materials that highly interconnected with physics. If you already know things that are taught in high school, I think it might be good that you start at chapter 7, which is about the application of integral, and then jump to chapter 10th, which prepares the ground for you to study multivariable calculus, and read upward to the end. You can look back at things in previous chapters when you forget something about it. And it is likely that you might find yourself want to read the earlier chapters while you are reading those materials that are completely new to you. It is okay to jump back and forth as I myself do that very often. For linear algebra, Introduction to Linear Algebra by Gilbert Strang is a good choice. With this book, you might need to start from the beginning, and I think at least you should read up to chapter 7. These two books contain many exercises that you can practice, but it would take you years to finish them if you want to do all the exercises. Luckily, you don’t need to solve all the problems. All you should do is to read the reading section and then do some exercises and move to the next point. If you want to know which exercises you should do, I can select those problems and make a list of them.

Once you are done with multivariable calculus or linear algebra, you should start learning abstract algebra or real analysis. As a starter to real analysis, you should read Calculus by Michael Spivak. This book contains all the basics and fundamental concepts in real analysis. And in my opinion, you should start at the beginning of chapter 3. You need to understand that almost everybody finds real analysis difficult to understand and boring (I think the boring part comes when they don’t understand it), so don’t feel bad about yourself if you find yourself reading a section many times but still don’t understand it. I myself struggle to learn it, and I spent my whole three-month vacation learning it until I can get to see its beauty. Real analysis is a very, very important subject in math, and it is also very beautiful when you get to learn more advanced topics. So, I strongly recommend you struggle with the basic concepts in it and keep pushing yourself through, and as you understand and get to see its underlying idea, you will be able to find the fun of learning it. For abstract algebra, A Book on Abstract Algebra by Charles C. Pinter is a good start. To read this book effectively, you should do most of the exercises given at the end of each chapter. In the reading part of the books, the author doesn’t give you many things that you should know as an undergraduate. He wants you to learn the subjects by developing them by yourself. Plus, he helps you to do that by designing exercises at the end of each chapter. After you finish real analysis and abstract algebra, your mathematical maturity might be enough for you to do complex analysis. Therefore, I recommend Introduction to Complex Analysis by H. A. Priestly because it covers the minimum amount of complex analysis, and it is wise to start with it when you only know the four subjects I mention above. 

Again, it is really, really important that you at least know linear algebra, multivariable calculus, abstract algebra, real analysis, and complex analysis. At first, you might find subjects like linear algebra or abstract algebra boring, and you do not see their significance. If you read the book by C. Pinter, the first triumphs of abstract algebra you see are the constructions of the integers out of the natural numbers, and then the constructions of the rational numbers out of the integers, and then the constructions of the complex numbers from the real numbers (and if you read the book by Spivak, at the end of the book, you’ll the see the constructions of the real number out of rational numbers). These constructions are fundamental to mathematics. They give us the firm ground to claim that the most basic mathematical objects that we work with actually exist (other objects in mathematics are built up from numbers in one way or another). It means we are not imagining things like there are reals numbers or complex numbers. (The construction of natural numbers is set-theoretic in nature, and it provokes philosophical questions. If you are interested in it, I suggest Naïve Set Theory by P. R. Halmos; and you should do it after you learn abstract algebra because only then can you fully appreciate the whole enterprise). And later on, you would see the proof that it is impossible to construct tri-sectors of an arbitrarily given angle by using a compass and ruler. I hope you are interested in this theorem since It tells you that the reason that we cannot trisect an arbitrary angle is not that we are not smart enough to do it, but it itself is impossible to do. Even god can’t do it should god exist. Back to learning abstract algebra and linear algebra, you should not ask about their utility when you first learn them, and all you should do is learning them and being interested in them for their own sake. Later on, when you learn enough of them, you’ll see they are very important and beautiful. And believe me: they are.

When it comes to learning math, I don’t think it is good to have many books and read a bit of each. You should choose at most three books appropriate to you and stick with them until you get the most out of them. You have to remind yourself from time to time to learn from them and resist the temptation to read other books. Do not wander around or you have no solid understanding of anything. If you don’t know much about mathematics, the books I’ve recommended above are what you should read. Read them in that order.

I advise many people to learn mathematics and suggest books that they can read, and I tell them to come back and ask me if they don’t understand something in their reading. At first, they enthusiastically come to get a soft copy of the books but then give up shortly after they read them a couple of times. I don’t think it is because the books I suggest are too advanced for them. However, I think they don’t put enough effort into it. People tend to think I can understand those books because I am clever, and the reason they don’t understand them is due to that they are not, but they don’t think that very often until I can understand a passage I keep reading it very carefully through and through for many times. Besides the help from other people, it is the effort and patience that make me understand things that I understand now. In his book How to Think Like a Mathematician, Kevin Houston writes “It’s up to you- Your action is likely to be the greatest determiner of the outcome of your studies. Consider the ancient proverb: The teacher can open the door for you, but you must enter by yourself.”

I have no hesitation to say that learning mathematics is very enjoyable. Yet, that’s not to say I always have a good time with it. I can get frustrated when I find myself not to understand a passage that I’ve scrutinized many times, but I think it is a very usual kind of thing- it happens to almost everybody. The other thing is that you might find solving mathematical problems difficult, and you don’t want to do it. Many times people come to me and ask if I can help them solve some problems without trying hard themselves first. I advise you not to ask anyone to solve any problem for you in the first place you see it. No matter how difficult and hopeless a problem seems to you, and you should try it first to see whether you can do anything with it. Maybe, you might want to start by asking yourself what the problem is about and if you understand the question of the problem, and then you can try to figure out how to solve it.  If you ask others for help the first time you see it, it means you give up. You have to understand that it is totally fine if you can’t solve the problem. When you are a student, the point of solving problems is not to get the answer, but to practice your thinking skills and your trying to apply things you’ve learned. You might get it wrong, but that’s also fine. Houston also says in his book that “Prepare to be wrong- You will often be told that you are wrong when doing mathematics. Don’t despair; mathematics is hard but the reward is great. Use it to spur yourself on”. It is good to discuss mathematics problems with your friends. You can exchange ideas and work together. I find it is helpful to collaborate with other people when it comes to learning math; but if you try to do that, make sure you spend most of your time with math but not with another chitchat. When you really think you can’t solve the problem, you should ask someone to help you and learn from them.

Finally, I am going to finish with a word from The Basic Method of Meditation by Ajahn Brahmavamso: It is a law of nature that without effort one does not make progress. Whether one is a layperson or a monk, without effort one gets nowhere, in mediation or in anything.

By Vanny Khon

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