Is A Course on Pure Mathematics by G.H. Hardy Worth Reading?

[murshid_islam]

have anyone read "A Course on Pure Mathematics" by G. H. Hardy?
is it a good book?

 

[fourier jr]

well since it's still in print after 100 years i would say it's pretty good. since he constructs the real numbers it's more theoretical than today's calculus texts, but it doesn't have anything on functions of more than 1 variable or vector calculus. in that respect it's lacking, but for single-variable calculus, sequences, series, properties of cos, sin, log, etc it's still one of the best there is i think, a real classic. it's got brutal problems.

 

[matt grime]

Its contents merely reflects the era it was written in. How could any modern introduction to pure maths fail to mention groups either? In anycase, multivariable calc is either a direct generalization of one variable and hence uninteresting for its own sake in an introduction, or it is Stokes' theorem which these days, if ever, hardly counts as pure maths at all unless one considers differential manifolds which are beyond the scope af any introductory book.

 

[benorin]

Here, read it for yourself: http://kr.cs.ait.ac.th/~radok/math/mat11/starter.htm#A%20Course%20of%20Pure%20Mathematics .

 

[fourier jr]

Its contents merely reflects the era it was written in. How could any modern introduction to pure maths fail to mention groups either? In anycase, multivariable calc is either a direct generalization of one variable and hence uninteresting for its own sake in an introduction, or it is Stokes' theorem which these days, if ever, hardly counts as pure maths at all unless one considers differential manifolds which are beyond the scope af any introductory book.

as hardy writes in the preface,

"this book has been designed primarily for the use of first year students whose abilities reach or approach something like what is usually described as 'scholarship standard'"

that's roughly what it would be used for today; pure math in first year would be calc I & II (& maybe some matrix theory). i only thought it might be worth mentioning that there's nothing on calculus of several variables & vector analysis since most big modern books & courant's text all have that stuff in them.

 

[DeadWolfe]

Here, read it for yourself: http://kr.cs.ait.ac.th/~radok/math/mat11/starter.htm#A%20Course%20of%20Pure%20Mathematics .

Is that displaying properly for anybody else?

 

[benorin]

Try this one: A Course of Pure Mathematics by G.H. Hardy. I found it in the Distributed Digital Library of Mathematical Monographs Collection.

 

[Chrono]

Try this one: A Course of Pure Mathematics by G.H. Hardy. I found it in the Distributed Digital Library of Mathematical Monographs Collection.

You mean I didn't even need to buy it?

 

[JasonRox]

You mean I didn't even need to buy it?

I bet that's not a good feeling.

If it was cheap, I wouldn't care. I rather have something tangible.

 

[Chrono]

I rather have something tangible.

True. It's so much better to actually have the book than just an electronic version of it. Oh, well, I'm sure it was well worth the $15 I'm sure I paid for it.

 

[matt grime]

as hardy writes in the preface, (quote omitted)
that's roughly what it would be used for today; pure math in first year would be calc I & II (& maybe some matrix theory). i only thought it might be worth mentioning that there's nothing on calculus of several variables & vector analysis since most big modern books & courant's text all have that stuff in them.

I think I must point out that is certainly not what Hardy would consider scholarship standard if he were alive today. Currently Cambridge considers that at the end of the first year in mathematics that you ought to be conversant in group theory, real and complex analysis (holomorphic and meromorphic stuff, not that they call it that prefering analyticity instead for some historical nonsense), calculus methods such as stokes theorem, discrete mathematics such as posets and generating functions, set theory, geometry, probability. And if you're a good student then you should certainly understand linear algebra (not the basic basis and matrix version, which is taught to all) but proper jordan normal form/idempotet/unipotent stuff over arbitrary fields.

I think they have changed it since I was there and now they may delay some of those topics and teach metric spaces instead, I can't say I'm up to date with the current syllabus

In Hardy's day the emphasis was more on special functions (elliptic things I imagine) and real and complex analysis of one variable. After all, algebraic topology and geometry was yet to be invented.

 

[JasonRox]

I think I must point out that is certainly not what Hardy would consider scholarship standard if he were alive today. Currently Cambridge considers that at the end of the first year in mathematics that you ought to be conversant in group theory, real and complex analysis (holomorphic and meromorphic stuff, not that they call it that prefering analyticity instead for some historical nonsense), calculus methods such as stokes theorem, discrete mathematics such as posets and generating functions, set theory, geometry, probability. And if you're a good student then you should certainly understand linear algebra (not the basic basis and matrix version, which is taught to all) but proper jordan normal form/idempotet/unipotent stuff over arbitrary fields.

I think they have changed it since I was there and now they may delay some of those topics and teach metric spaces instead, I can't say I'm up to date with the current syllabus

In Hardy's day the emphasis was more on special functions (elliptic things I imagine) and real and complex analysis of one variable. After all, algebraic topology and geometry was yet to be invented.

You should be knowledgeable in all those things after one year at Cambridge? Graduate School?

From what I see for most undergraduate schools, you don't get into any of that stuff or atleast not over half of it.

I must be far far far far far behind, but I'm not surprised though. My school is lacking. I tried to find time to do more rigorous work, like Spivak instead of Stewart, but it's not practical when you work 40 hours a week.

Note: I won't be working more than 8 hours a week next term. I'll be sure to do more than the class requires, which shouldn't be too difficult.

 

[matt grime]

That is undergraduate mathematics, Jason, certainly not graduate level mathematics.

 

[JasonRox]

That is undergraduate mathematics, Jason, certainly not graduate level mathematics.

I understand that the text is not graduate level.

I'm talking about this.

Currently Cambridge considers that at the end of the first year in mathematics that you ought to be conversant in group theory, real and complex analysis (holomorphic and meromorphic stuff, not that they call it that prefering analyticity instead for some historical nonsense), calculus methods such as stokes theorem, discrete mathematics such as posets and generating functions, set theory, geometry, probability. And if you're a good student then you should certainly understand linear algebra (not the basic basis and matrix version, which is taught to all) but proper jordan normal form/idempotet/unipotent stuff over arbitrary fields.

We don't learn half the stuff you mentionned in first year.

I was thinking of transferring to a better school, but financially that isn't feasible. (I'd have to work 24 hours a week just to survive, which is only food and rent.)

I feel like a big idiot now. :grumpy:

What's the first step to catching up now?

I'll tell you what I can say I am conversant in right now...

Group Theory (only up to External Direct Products, which includes Cyclic Groups, Isomorphism, Cosets and Lagrange's Theorem). I know about Homomorphisms, but not it's properties and I only know because I looked at the introduction of the chapter. I will later when I get the proper text from Hernstein (spelling?) because I currently have the one by Gallian (hurting textbook).

Real Analysis, well I would say nothing. Only Basic Analysis. I have a course in Basic Analysis next term, so I am ahead in all cases.

Complex Analysis, nothing at all. I just took a glance of an introduction to it, and I understand the similiarites between vector calculus and complex functions. Holomorphic and Meromorphic, I have no clue.

Stoke's Theorem, yeah.

Poset's, yes, but never worked with them at all. We haven't even learned about binary relations, mappings, equivalence relations yet. This is something I did. I still managed to do some stuff while working insane amount of hours.

Set Theory, basically nothing. de Morgan's Duality doesn't count since that's taught on the first page. I know about countable, uncountable, infinite and finite sets, but that's because I taught myself that just recently while going into the Topology book I'm going through, and before while going through some Basic Analysis.

Geometry and Probability are fine I guess. Not going into probability, but I might take a class just so that I'm acquainted with it (but I'm probably better off reading a book by myself).

I know about Linear Algebra, but not aware about unipotent's. Everything else is good. Maybe the only one in the class who understands most of it. Everything from Vector Spaces, Linear Transformations to Simliarities. Eigenvalues and Eigenvectors too. I independently thought that if a matrix has n distinct eigenvalues, then they are linearly independent. I went to prove it, but then I saw the Theorem then it was done from there.

I know what Fields, and Rings are because of my own readings.

The only thing I got from school is some Group Theory, Basic Calculus, and Linear Algebra.

Everything else was taught myself.

I haven't even found a student generally interested in mathematics, or atleast higher mathematics. No one cares period.

Note: I gave a suggestion for the newly formed math club to be named Peano Players because the Peano Axioms are the most basic (that I know of), and that you technically use these basic axioms to play the game of mathematics. It was rejected as no one cared to know what the Peano Axioms are. No, I am not part of the math club... work. :(

I'm in second year first term right now, and going into my second term (fourth total).

I don't feel well-rounded in mathematics at all. I attempted to do Spivak's Calculus in first year, but time did not permit (I work like a slave).

What should I do?

Note: I met students you didn't even know how to do Proof by Induction in third year! I bet you any money I can ask some what the definition of a limit is and they will have no clue at all.

I will only be working 8 hours a week next term, so basically I will have lots of time. I will be living a dream. :)

PLEASE HELP!

 

[matt grime]

I know what you were referring to, and that list I gave is of undergraduate topics in the first year.

You can only do the maths you're offered. At least you're trying to look beyond the level they deem adequate at your university. In anycase, comparing is pointless, since different courses have different aims. I imagine most Americans would be moderatley outraged that at university in the UK you wouldn't have to do courses in different departments, ie mathematicians only do maths.

And no one actually cares about the Peano axioms, do they?

 

[JasonRox]

And no one actually cares about the Peano axioms, do they?

They didn't care to know about them basically.

Yeah, I have to take a Humanities and Social Science credit next term. I don't want to. :yuck:

 

[selfAdjoint]

Cambridge is the legendary home of the tripos, which still survives, doesn't it? That means the undergraduate math there is as tough as any in the world and much tougher than some, especially the watered down stuff offered in the USA.

 

[JasonRox]

Cambridge is the legendary home of the tripos, which still survives, doesn't it? That means the undergraduate math there is as tough as any in the world and much tougher than some, especially the watered down stuff offered in the USA.

So, what can I do?

Just keep teaching myself.

I'd love to find someone who enjoys it just as much as do, so you can talk about it all day... or atleast a little bit. I can't even do that yet.

Sometimes people ask for "advice" (answers) for some of the assignment questions. I just give them good advice, but that's never good enough. Sometimes all you need to do is apply a Theorem and you are done.

This is how it goes...

"Can you help with this?" - Other Student.

"Ok, which question?" - Me.

"Do you have your assignment with you?" - Other Student.

*Implying to see how I did it.

"No, I don't carry it around. Just show me, and I'll see what I can do for you." - Me.

"Alright then, it's this question." - Other Student.

*They show me the text or hand written version.

"Read Theorem #, and you should be good from there." - Me.

"Which part of it?" - Other Student.

"What do you mean which part? It makes a simple statement." - Me.

"This is so complicated." - Other Student.

"Ok, the Theorem says that if you have this and this, then this is true. (Something like that). Now, what does the question give you?" - Me.

"I don't know... I'm so lost." - Other Student.

*Then the student gives up on his/her attempt to get a solution, and decides to leave to "work" on it some more.

I consider myself a great helper if you read some of the material. When I was in business school, I tutored and I was popular for knowing just about everything you needed to know. I'd always be waiting in front of the exam room an hour early to talk about things with students, and answer questions. I wasn't always right, but I definitely knew what was going on. When I tutored, I helped students surpass their goals set for the class, so I'm guessing I did a good jop. I'm patient, but I become impatient if you don't even try.

Anyways, that's my life at school.

 

[asdf60]

Jesus christ. Even the top ivys don't cover so much stuff in the first year. In most places, including where I am, even the Honors track is just calc III and IV (with theory) in the first year. How the hell do they manage to cover so much?

 

[matt grime]

Because they start from a better basis at high school equivalent (assuming that you're talking about cambridge). I believe that the good Ivy schools have *potentially* better programs at the end of 4 years, but that would demand that the students take the correct options.

 

[Treadstone 71]

Don't get me started about Peano's axioms. I have 5 math courses this semester, and each and every one of them started with Peano's axioms. I had to hear that same speech 5 times in a single day.

 

[mathwonk]

i respect hardy's book but it is not one of my favorite calculus books. i have one tiny gripe against it as well, as he seems to have originated the incorrect impression many current authors have, that the chain rule cannot be proved by the simple minded argument of multiplying two difference quotients.

I.e. although there is a special case to consider with that proof, it is trivially disposed of.