Thank you very much!
=> I am 17 years old by now and I want to study mathematics at the
=> university next year. Since I like to go into research (especially
=> number theory) I thought i might start to read books and solve
=> problems in a serious manner.
For Number Theory, you might look at Niven, Zuckerman, and
Montgomery, An Introduction to the Theory of Numbers. Hardy and
Wright, The Theory of Numbers, is more serious but doesn't have
Gerry Myerson (ge...@mpce.mq.edi.ai) (i -> u for email)
Here is my nickel's worth:
Anything by Paul Halmos is worthwhile and joy to read. Rudin's
"Principles of Mathematical Analysis" is a good starting point for
calculus and more.
If you want to save money, you will find some good books on the
Internet. You can start at
or search through Google.
Beyond books, here is an advice from an old sage:
Don't specialize too early. In the next few years, try to learn as many
different fields of mathematics as you can. There any many deep and
unexpected connections between different areas of mathematics, and big
breakthroughs often come from applying an insight from one area to a
problem in another.
And connect to other mathematicians. Mathematics is not as solitary as
some people make it out to be.
> Stephan Karnos wrote:
> > Hello folks,
> > I am 17 years old by now and I want to study mathematics at the
> > university next year. Since I like to go into research (especially
> > number theory) I thought i might start to read books and solve
> > problems in a serious manner. Thus i read some books including Hardy's
> > course of pure maths and solved a lot of problems from this book. Now
> > I wonder what book might be useful for someone who wants to be a
> > professional mathematician. There must be some books for instance that
> > cover plenty of ground in calculus. I heard that people used to read
> > Jordan's course d'analyse. But i bet that this is no longer up to
> > date.
> > So what books would you recommend me??
> > Thank you very much!
> > Stephen
> Here is my nickel's worth:
> Anything by Paul Halmos is worthwhile and joy to read. Rudin's
> "Principles of Mathematical Analysis" is a good starting point for
> calculus and more.
Especially his "Naive Set Theory" for a good introduction to a foundational
subject that the usual axiomatic approach seems to make more difficult than
it has to be.
"Topics in Algebra" by Herstein as an introduction to abstract algebra.
> I am 17 years old by now and I want to study mathematics at the
> university next year. Since I like to go into research (especially
> number theory)
It is good to follow the advice Nemo gave, not to specialize to early,
you are only 17, and there is plenty of mathematics to learn to
realize what interests you most and what you are best at.
> I thought i might start to read books and solve
> problems in a serious manner. Thus i read some books including Hardy's
> course of pure maths and solved a lot of problems from this book.
Hardy's book is very good, but it is not the kind of book that is
suitable for beginners, at least not the students entering
To see what a school boy and high school student had to study around
the time Hardy wrote his book, just read Littlewood's "A
Some excerpts from the book:
"...Born June 9, 1885, I was in South Africa from 1892 to 1900; I left
the Cape University at the age of 14, and after 2 or 3 months went to
England to go to St.Paul's School, where I was taught for 3 years by
F.S.Macaulay. My first knowledge then was slight by modern standards;
the first 6 books of Euclid, a little algebra, trigonometry up to the
solutions of triangles. During my 3 years at St.Paul's I worked
intensively; seriously overworked indeed, partly because it was a
period of severe mental depression.
The tradition of teaching (derived ultimately from Cambridge) was to
study "lower" methods intensively before going on to "higher" ones;
thus analytical methods in geometry were taken late, and calculus very
late. And each book was more or less finished before we went on to the
The accepted sequence of books was: Smith's Algebra; Loney's
Trigonometry; Geometrical Conics (in a very stiff book of Macaulay's
own: metrical properties of the parabola, for instance, gave scope for
infinite virtuosity); Loney's Statics and Dynamics, without calculus;
C.Smith's Analytical Conics; Edward's Differential Calculus;
Williamson's Integral Calculus; Besant's Hydrostatics. These were
annotated by Macaulay and provided with revision papers at intervals.
Beyond this point the order could be varied to suit individual tastes.
My sequence, I think, was: Casey's Sequel to Euclid; Chrystal's
Algebra II; Salmon's Conics; Hobson's Trigonometry; Routh's Dynamics
of a Particle (a book of more than 400 pages and containing some
remarkably highbrow excursions towards the end); Routh's Rigid
Dynamics; Spherical Trigonometry (in every possible detail); Murray's
Differential Equations; Smith's Solid Geometry; Burnside and Panton's
Theory of Equations; Minchin's Statics (omitting elasticity, but
including attractions, with spherical harmonics, and of course and
exhaustive treatment of the attraction of ellipsoids).
I had read nearly all of this before Entrance Scholarship Examination
of December 1902."
Similar thing are as far I know is only present in India today, and
that is the preparation for IIT JEE, this is extremly difficult exam,
and each year hundreds of thousands apply for IIT but only about 2
thousand are allowed in.
> I wonder what book might be useful for someone who wants to be a
> professional mathematician. There must be some books for instance that
> cover plenty of ground in calculus. I heard that people used to read
> Jordan's course d'analyse. But i bet that this is no longer up to
It is of no use to you, first is is very "out of time", foundations of
mathematics then were not as developed as they are today.
There is maybe at your library some copies of Goursat's Course
D'Analyse in English.
> So what books would you recommend me??
> Thank you very much!
Find some book that you find reasonably hard, then read it from page
to page very slowly (forget about speed reading that's bullshit when
it comes to math)
Try to understand the idea in the book, and most important of all,
solve problems, problem solving is the mother of learning mathematics.
When talking to many former Soviet mathematicans about how they
learned calculus, and similar things at undergraduate level, they
almost all responded, by solving a lots of problems! That is the
You are welcome.
> I'll get some problem books from
> the library and work through it.
Before the fall of USSR, there were many good books published in East
Germany, especially by Teubner, Leipzig.
One good problem book used in former Yugoslavia was: (here in German)
Minorski, W. P.
Aufgabensammlung der höheren Mathematik
has also written some good books
Ostrowski, Alexander M.
Vorlesungen über Differential- und Integralrechnung
Aufgabensammlung zur Infinitesimalrechnung
One famous three voulme book translated to German is a classic in
mathematical analysis, it was written by Russian professor
Fichtengolts (I'm not sure about spelling)
Sominski, I.S.: Die Methode der vollständigen Induktion.
This book is great if you want to learn mathematical induction, it was
published in a series of books, written by eminent russian
Try find more books from these series, they were written especially
for young mathematicians in gymnasiums.
Look also for series:
Elemente der Mathematik vom höheren Standpunkt aus
one book from that series is
Lüneburg, Heinz "Vorlesungen über Zahlentheorie"
These book are small about 100 pages, but they are worth gold, because
they EXPLAIN the CONCEPT of the subject to you in a clear, elegant
manner, understandible to the beginner.
It is also good for you to study physics seriously, in case you maybe
find out that you cannot earn good living as a number theorist, when I
was 17 I was fascinated by mathematics and wanted to study math or
physics, but somehow my cousin convinced me to study electrical
engineering, and I'm happy to followed his advice, later I talked to
few math professors at university and they almost all said, sure it
fine and great to study math, but what is life like after the school?
Who is going to employ you?
And you are not worth much if you don't have a PhD, that's 5 more
years in school, doing some really hard research.
And it isn't easy to find a nice job as mathematician, unless you are
some kind of genius.
Trying to get employment at university isn't either easy, you have to
wait until some older professor retires, etc, etc.
If you choose enginnering/computer science, and you are good at
mathematics and physics, you are going to kick ass!
And there is also a lot of exciting mathematics in Engineering, for
example in Control Theory some of the newest results in differential
equations and functional analysis are used.
Talking of Hardy, control engineers work in Hardy spaces!
The situation (in Sweden) is following:
If you are PhD in some engineering field/CS, the industy will be
fighting about you, you will earn big bucks
A as math PhD (say in topology) you are probably twice as clever as
the PhD in engineering but nobody is figting over you, you are happy
if the university offers a job to you, but in this job there is no
time for research only plenty of teaching duties.
I personally think that mathematicans are smartest people, now I mean
really good mathematicans, problem solvers, they kick ass out of the
all other disciplines, technology, arts, business, bla. none of them
are smart as a good mathematicians are, but unforunatly they are not
paid properly. They are paid very bad.
That is how the reality looks like.
Now onto physics, every serious study of theoretical physics begins
Bible of theoretical physics is:
Vorlesungen über theoretische Physik by Arnold Sommerfeld
His lectures are in 6 volumes and they are a masterpiece each
Good problem book in undergraduate physics is I.E. Irodov's (Zadaci po
obscej fizike) (Exercises in general physics)
One good book (online) is by David Morin of Harvard.
The book is really good, be sure to look at the lectures also:
H. A. Priestley's "Introduction to Integration" doesn't do what is
says on the cover at all. It instead introduces some-one who is
comfortable with calculating integrals to Lebesgue's theory of
integration, in a very approachable way. In particular, it assumes no
knowledge of the theory of Riemann-Stiltjes integration. Lebesgue's
theory is often delayed very far in undergraduate courses and done in
a very abstract way, using measure. This book is much more immediate,
and tries to stress that Lebesgue integration is the natural way to do
So, to illustrate this, one might look at elliptic curves, say "Elliptic
Curves" (note the sub-title: "Function Theory, Geometry, Arithmetic") by
Henry McKean and Victor Moll, Cambridge University Press.
I agree with pretty much all that's been said so far. Two points bear
repeating -- yes, Principles of Mathematical Analysis is really
excellent, and you should read it. It takes you from high-school
mathematics to basic differential forms more quickly and clearly than
any other text I know. Another thing worth repeating is to avoid
Advice and recommendations would be easier if we knew more about you.
(For example, maybe you could tell us more about the types of problems
you've been solving.) For example, I think the type of reading best
done by one of the world's top-scorers in the Math Olympiads would not
necessarily be recommended to someone who's highly interested and
motivated but with no particular signs of brilliance.
There are (at least) 3 reasons for reading a math book: 1) To learn
some theory, 2) Motivational -- to get excited and determined about
progressing in math generally, or in a certain branch of it, 3)
Strengthening one's mathematical muscles -- becoming stronger at math
These categories do overlap somewhat, but I think I have marked out
three distinct functions a math book can have, and most are noticeably
stronger in one of the three categories than in the two others.
I note that most advice given to you has been for instructional books
(my category 1.) For category 2 (motivational), I'd like to recommend
Khinchin's Three Pearls of Number Theory. It bears a close reading,
working through the proofs carefully. It also contains a moving
statement about the meaning of mathematics.
For category 3 (working those mathematical muscles), try and solve as
many problems as you can from international math competitions --
Puttnam and Math Olympiads come to mind. Usually, the setters of
these problems are well-known mathematicians. Furthermore, many
who've thrived at these competitions have become very successful
research mathematicians. Using the internet, such problems should be
easy to come by.
Feel free to share your experiences following any advice (by me or
other posters) with this newsgroup. I think (almost) everyone on sci
math relishes helping others grow into better mathematicians.