Year 12s: what will be the first way you research your future options? >>

Help me choose a maths book!!

Scroll to see replies

Walter Rudin's Principles of Mathematical Analysis (Aka "Baby Rudin" :wink:

is a very well regarded textbook into the subject of Analysis, which makes up a significant part of any mathematics degree (particularly pure). If that interests you it might be worth giving it a read.

Thank you for the reply!! I did try to go through this book once but the second chapter (one on basic topology) was just too much. I will definitely retry. I have heard a lot of good things about that book.

Reply 21

username4136150

Original post by RichE

Would endorse all of A02's post.

OP - have you completed further maths? are you on a gap year?

I have completed further maths and yes I am on a gap year.

Reply 22

username4136150

Original post by A02

I'm probably not the best person to comment, since I'm not in the year above you.

Anyway, given your goals such as preparing for MAT, I don't think there is much point reading on things like Cauchy Schwarz or generating functions, since I don't think you'll be expected to know about them. (I'm not 100% sure of the syllabus for MAT, can someone else please correct me if I'm wrong)

Also, Euclidean geometry doesn't really appear much outside of olympiads, so unless you're interested in olympiads or geometry in general, it's not really worth dedicating significant time to it. Conversely, if you do want to dedicate lots of time to olympiad geometry because you enjoy it, Evan Chen's Euclidean Geometry in Mathematical Olympiads is very good.

If you want to improve problem solving, my opinion is that books don't make you better, doing problems makes you better. I've personally found doing lots of problems to be the main way to get better at doing problems. Of course books can help with motivation by providing a list of problems to do, but in general I don't think you improve any more by working through problems in a book compared to working through a list of problems online.

I've heard the Higher Arithmetic is very good and interesting, but I have not read it.

Thank you!! This means a lot to me. The only reason I wanted to do Olympiad stuff is because problem in the internet use concepts I have not already mastered.

Reply 23

username4136150

Original post by Muttley79

I would not read any - prepare for MAT first - try UKMT BMO type problems and STEP. Plenty of time to read when you've got your offer.

Thank you! I will try to stick with BMO1 and STEP.

Reply 24

username1732491

Original post by Quantum Horizon

Ah yes chapter 2 is rather compact (if you'll excuse the pun). The whole book contains so much mathematics that it would normally be split over 3 university modules, so don't feel down if it takes you a long time to take it in. I'm 3 years into a university course and I've only just about covered all the topics in this book in various analysis courses. Learning the first chapter enough to do the exercises would be a good use of your summer time.

Reply 25

username4136150

Original post by timgb

ah yes chapter 2 is rather compact (if you'll excuse the pun). The whole book contains so much mathematics that it would normally be split over 3 university modules, so don't feel down if it takes you a long time to take it in. I'm 3 years into a university course and i've only just about covered all the topics in this book in various analysis courses. Learning the first chapter enough to do the exercises would be a good use of your summer time.

prsom and thank you!!!

Reply 26

Gregorius

Original post by Quantum Horizon

I have got this huge reading list and certainly I won't be able to have a go at all of them. What should I focus on?

In my experience, students starting a mathematics undergraduate degree tend to have difficulties in three areas:

(a) Algebraic manipulation. What you’re expected to do with relative ease at university is considerably harder than what you’ve done at school.

(b) Rigour. You’re finally expected to really prove things at university! Indeed, the first years (spilling over into the second year) of many maths courses are recapitulations of the experience of nineteenth century mathematics, when it was discovered that many things taken for granted are actually much more subtle!

(c) Abstraction. What you’ve done so far at school largely comes under the banner of “mathematical methods” at university (as in, “mathematical methods for the physical sciences”). All of a sudden, you’re exposed to “real” pure mathematics: analysis, topology, abstract algebra, in which, when you’re starting out, it’s very easy to lose your anchor in reality.

Now, of course, individuals suffer from their own personal weighted mixture of these three (including, “none of the above” for a lucky few). You’ll have to decide for yourself where you place yourself!

How to prepare for them? Well, (a) is probably the most straightforward. Practice, practice, practice! I think the “official” STEP preparation course (together with Siklos’s book) is very good for this. Even if you’re preparing for the MAT rather than STEP, I’d recommend this.

(b) and (c) are much more difficult to prepare for, and are part of that developing “mathematical maturity” that you just have to go through. For (c) I quite like to introduce the need for abstraction through the history of mathematics; John Stillwell’s “Mathematics and its History” is superb (in fact, any John Stillwell book is worth reading). Then if you read a couple of short books (and I emphasize “short”; don’t start reading massive tomes) on, say, group theory and analysis, this should get you started on (b) as well.

Turning to the sources you mention:

1) Art and Craft of Problem Solving followed by Problem Solving Strategies.

I’ve only skimmed through it, but I quite like the look of Zeitz’s “Art and Craft”. I’d probably stick at just the one book on problem solving, so you have time for the rest!

2) Some algebra books like Cauchy-Schwartz master Class, Polynomials (the one by Barbeau) and Functional Equations and How to Solve them.

At the moment, focus on developing your algebraic manipulation skills (through STEP, for example). There’s a lot of material in these books! Barbeau is quite hard (you solve the cubic and quartic in the exercises near the beginning, for example) and goes off into material that you won’t cover in your degree.

3) HOW TO COUNT: AN INTRODUCTION TO COMBINATORICS (need to learn graph theory and generating functions)

Looks a nice book; but would it be a better idea to come at combinatorics from the direction of probability theory, which you will probably hit earlier in your course? Grimmett & Welsh’s “Probability: an Introduction” is pretty good. Another very good and accessible book for combinatorics and discrete mathematics in general is “Concrete Mathematics” by Graham, Patashnik and Knuth.

4) Geometry Revisited.

A classic book, but you’re unlikely to do much geometry in this style at university. That having been said, it would be worth reading this before you started a (probably third year) course in algebraic geometry.

5) THE HIGHER ARITHMETIC :AN INTRODUCTION TO THE THEORY OF NUMBERS

Another classic, but I’d recommend something a bit more modern (with a glance at some of the recent applications of number theory) such as Stillwell’s “Elements of Number Theory”.

6) Spivak's Calculus

Americans seem to like huge long books on calculus, and this is probably the best of them. I’m very dubious about this style of book for the English system; so perhaps have it on hand for reference, but don’t attempt to read it cover to cover.

7) Towards Higher Mathematics: A Companion. (Really skeptic about this book)

Don’t know this one.

(Unanswered Question) Which area of Olympiad mathematics: algebra, combinatorics, number theory or geometry is more closely related to university mathematics or better preparation for university course?

I’d go with algebra and number theory. But be aware that Olympiad mathematics is almost a specialist area in itself!

Is it better to focus on the Olympiad Style books or University textbooks?

For this choice, university textbooks; but at your stage, I’d be very selective about those books!

Any book recommendations??

Tom Korner’s “Pleasures of Counting” is very good. I can go on recommending books until I’m blue in the face, but remember that “of making many books there is no end, and much study is a weariness of the flesh”! And @Muttley79 gives good advice; you have MAT to get through first!

(edited 5 years ago)

Reply 27

Doones

Original post by Quantum Horizon

7) Towards Higher Mathematics: A Companion. (Really skeptic about this book)

I haven't read it, but the author is very reliable, and especially given their background I'd say this book is relevent to your needs. I wouldn't be sceptical.

Posted from TSR Mobile

(edited 5 years ago)

Reply 28

RichE

Original post by Doonesbury

I haven't read it, but the author is very reliable, and especially given their background I'd say this book is relevent to your needs. I wouldn't be sceptical.

Posted from TSR Mobile

Thank you, but the book is perhaps aimed more at someone starting year 13 so there may well be better choices for the OP on his list of texts. :smile:

Reply 29

username4136150

Original post by RichE

Thank you, but the book is perhaps aimed more at someone starting year 13 so there may well be better choices for the OP on his list of texts. :smile:

I did not realise you were the author. I am really sorry if I sounded arrogant in any of my posts.

(edited 5 years ago)

Reply 30

RichE

Original post by Quantum Horizon

I did not realise you were the author. I am really sorry if I sounded arrogant in any of my posts.

Don't worry and I'm genuinely interested in all opinions/feedback.

Reply 31

S2M

Original post by Quantum Horizon

I did not realise you were the author. I am really sorry if I sounded arrogant in any of my posts.

Mind me asking what other subjects you did?

Reply 32

username4136150

Original post by Black Water

Mind me asking what other subjects you did?

Physics, Maths and Further Maths

Reply 33

S2M

Original post by Quantum Horizon

Physics, Maths and Further Maths

Nice.

I’m guessing you’re hoping to get A*A*A or A*A*A*?

Reply 34

Doones

Original post by RichE

Thank you, but the book is perhaps aimed more at someone starting year 13 so there may well be better choices for the OP on his list of texts. :smile:

Lol.

Posted from TSR Mobile

Reply 35

DFranklin

Original post by Quantum Horizon

I have got this huge reading list and certainly I won't be able to have a go at all of them. What should I focus on?

1) Art and Craft of Problem Solving followed by Problem Solving Strategies.

2) Some algebra books like Cauchy-Schwartz master Class, Polynomials (the one by Barbeau) and Functional Equations and How to Solve them.

3) HOW TO COUNT: AN INTRODUCTION TO COMBINATORICS (need to learn graph theory and generating functions)

4) Geometry Revisited.

5) THE HIGHER ARITHMETIC :AN INTRODUCTION TO THE THEORY OF NUMBERS

6) Spivak's Calculus

7) Towards Higher Mathematics: A Companion.

Goal: Prepare for interviews, entrance exams (its only MAT right now) and prepare for the course itself (which is mathematics).Gregorious's reply is pretty definitive, but here are my own thoughts:

My feeling is books 1-4 are probably more suited to BMO-type prep, 5-6 probably go into too much depth, and you are perhaps a bit beyond 7.

Out of them, I think I'd put the first on the reading list (although it seems ferociously expensive), but being aware not to get too bogged down on areas that don't seem relevant.

For STEP/MAT preparation, I do think your best bet is the Siklos booklets and the STEP preparation course, and also past papers (plus asking for help on forums as needed).

If you feel you need books, in terms of "bang for the buck", I'd be inclined to look at "methods" books; they tend to cover a lot of topics "shallowly", which gives you a broader base without going into too much detail, and the approach is more familiar to an A-level student than some of the pure mathematics books. The justification here is really to get some exposure to a wider range of topics, and practice in actually *using* your A-level knowledge (that is, you might, for example, need to use a Taylor approximation as part of solving a bigger problem, as opposed to at A-level being asked "find the first 3 terms in the Taylor expansion of ... "). I'm not sure this is a *great* way of preparing for STEP/MAT (because it's not *that* similar a type of mathematics), but I think it will help improve your skill-set.

For university preparation, I think the topics people struggle with in most in their first year tend to be analysis and group theory. I would look for *accessible* books introducing these topics - my first analysis book was "Mathematical Analysis: A Straightforward Approach by Binmore" - although it's very lightweight by professional mathematics standards, it's easily enough to give you a good head-start. I'm a bit less sure of what to recommend on group theory (I suspect you're a bit forced to get a "full-on" group theory book and only do the first few chapters). In practical terms, you're not looking to cover university levels of material, more get familiar with ideas and approaches (and possibly realise that "I really hate group theory" - probably better to be aware it will be an issue now than when you're in the middle of the course).

TBH, I wouldn't worry about university preparation for a good few months yet though.

Reply 36

Gregorius

PRSOM

Original post by DFranklin

For university preparation, I think the topics people struggle with in most in their first year tend to be analysis and group theory. I would look for *accessible* books introducing these topics - my first analysis book was "Mathematical Analysis: A Straightforward Approach by Binmore" - although it's very lightweight by professional mathematics standards, it's easily enough to give you a good head-start. I'm a bit less sure of what to recommend on group theory (I suspect you're a bit forced to get a "full-on" group theory book and only do the first few chapters). In practical terms, you're not looking to cover university levels of material, more get familiar with ideas and approaches (and possibly realise that "I really hate group theory" - probably better to be aware it will be an issue now than when you're in the middle of the course).

Many *many* years ago, when I was at @Quantum Horizon's stage, there was a lovely series of books ("University Mathematical Texts" :wink:

from the publishers Oliver and Boyd that had the triple benefit of being short, cheap, and, in general, pretty good. I especially remember Ledermann's Group Theory and Phillips' Complex Variable volumes.

Trouble with textbooks these days is they're tremendously expensive; perhaps the equivalent modern source are all the pdf's of lecture notes that are freely available on-line...

Reply 37

DFranklin

Original post by Gregorius

Trouble with textbooks these days is they're tremendously expensive; perhaps the equivalent modern source are all the pdf's of lecture notes that are freely available on-line...

Without wanting to cast accusations, I'm not sure students these days worry about the cost of *buying* textbooks...

Although I do think there's still something to be said for hard copy when it comes to textbooks.

Reply 38

Kummer

Original post by Quantum Horizon

I think that's the hardest chapter in the book.

I think the best introduction to analysis is this book. It spends an unusual amount of time on the least upper bound property. It constructs the real numbers in so many different ways that there are multiple entries outlined in the introduction if you're not interested to digest all the material it has on real numbers.