
Re: books or courses with mathematical proofs
Posted:
Jun 24, 2012 10:32 PM


On Saturday, June 23, 2012 6:37:59 AM UTC4, ahum wrote: > On Friday, June 22, 2012 10:10:21 PM UTC+2, porky_...@mydeja.com wrote: > > On Friday, June 22, 2012 3:04:07 PM UTC4, ahum wrote: > > > On Friday, June 22, 2012 7:27:22 PM UTC+2, amzoti wrote: > > > > On Thursday, June 21, 2012 11:44:55 PM UTC7, ahum wrote: > > > > > Hi > > > > > > > > > > At this moment I'm studying IT(programming)in eveningschool. I will obtain a limited degree (one could compare it to a bachelor, but it isn't). This could lead to a full bachelor degree at a university. > > > > > I would like to refresh my mathematical skills trough selfstudy, I would like to study my highschool maths but have trouble finding books or courses which contain the full proofs of the subjects. I have found many excellent books on algebra, calculus, geometry and others, but it is al applied, and no mathematical proofs whatsoever. And I really want these proofs, because I want to fully grasp and understand what I'm doing (which sounds obvious). > > > > > Can you help me find the courses or books I need? I'm of course prepared to pay the price for books needed to increase my knowledge and skills. > > > > > Thanks in advance > > > > > > > > > > Best Regards > > > > > > > > > > Pierre > > > > > > > > Is this what you are looking for? > > > > > > > > 1. How to Read and Do Proofs: An Introduction to Mathematical Thought Processes [Paperback] Daniel Solow (Author) > > > > 2. How to Solve It: A New Aspect of Mathematical Method (Princeton Science Library) [Paperback] G. Polya (Author) > > > > 3. The Nuts and Bolts of Proofs, Third Edition: An Introduction to Mathematical Proofs [Paperback] Antonella Cupillari (Author) > > > > 4. Mathematical Proofs: A Transition to Advanced Mathematics (2nd Edition) [Hardcover] Gary Chartrand (Author), Albert D. Polimeni (Author), Ping Zhang (Author) > > > > 5. A Book of Abstract Algebra: Second Edition (Dover Books on Mathematics) [Paperback] Charles C Pinter (Author) > > > > 6. How to Prove It: A Structured Approach [Paperback], Daniel J. Velleman > > > > > > > > If not, I think you need to find older titles that actually contain proofs as many recent books are about mechanics of getting answers due to many factors in society. > > > > > > > > You can search out older books and if you tell which areas of mathematics, we could make recommendations as you question is a bit illposed. > > > > > > > > HTH > > > > > > Hi, looks interesting indeed. > > > > > > On Friday, June 22, 2012 7:27:22 PM UTC+2, amzoti wrote: > > > > On Thursday, June 21, 2012 11:44:55 PM UTC7, ahum wrote: > > > > > Hi > > > > > > > > > > At this moment I'm studying IT(programming)in eveningschool. I will obtain a limited degree (one could compare it to a bachelor, but it isn't). This could lead to a full bachelor degree at a university. > > > > > I would like to refresh my mathematical skills trough selfstudy, I would like to study my highschool maths but have trouble finding books or courses which contain the full proofs of the subjects. I have found many excellent books on algebra, calculus, geometry and others, but it is al applied, and no mathematical proofs whatsoever. And I really want these proofs, because I want to fully grasp and understand what I'm doing (which sounds obvious). > > > > > Can you help me find the courses or books I need? I'm of course prepared to pay the price for books needed to increase my knowledge and skills. > > > > > Thanks in advance > > > > > > > > > > Best Regards > > > > > > > > > > Pierre > > > > > > > > Is this what you are looking for? > > > > > > > > 1. How to Read and Do Proofs: An Introduction to Mathematical Thought Processes [Paperback] Daniel Solow (Author) > > > > 2. How to Solve It: A New Aspect of Mathematical Method (Princeton Science Library) [Paperback] G. Polya (Author) > > > > 3. The Nuts and Bolts of Proofs, Third Edition: An Introduction to Mathematical Proofs [Paperback] Antonella Cupillari (Author) > > > > 4. Mathematical Proofs: A Transition to Advanced Mathematics (2nd Edition) [Hardcover] Gary Chartrand (Author), Albert D. Polimeni (Author), Ping Zhang (Author) > > > > 5. A Book of Abstract Algebra: Second Edition (Dover Books on Mathematics) [Paperback] Charles C Pinter (Author) > > > > 6. How to Prove It: A Structured Approach [Paperback], Daniel J. Velleman > > > > > > > > If not, I think you need to find older titles that actually contain proofs as many recent books are about mechanics of getting answers due to many factors in society. > > > > > > > > You can search out older books and if you tell which areas of mathematics, we could make recommendations as you question is a bit illposed. > > > > > > > > HTH > > > > > > Well, these books look interesting anyway, the book about algebra is spoton! I apologize for being unclear, but having received the tip of the "AoPs" books, I think I'm able to reformulate my question: I would like to study the following subjects: counting, number theory, geometry, prealgebra, algebra, precalculus and calculus, including all the proofs that are part of these subjects. So I'm talking about the level being thaught to 1218 year olds (I guess this is highschool? Over here we talk about the humanities) I have access to books who treath about this subject, but they are lacking formal proofs. The important thing to me is the proofs, as I really want to grasp what I'm doing (which I didn't when being at the humanities). > > > Thank you so much > > > Pierre > > > > With respect to Calculus, you probably want to read something that's used as a textbook for so called "Rigorous Calculus" courses. Among the usual choices are Apostol and Spivak. > > > > On the other hand, if you're so interested in proofs, you may find it useful to take a look at some introductory books on Analysis. I recommend Kirkwood, Introduction to Analysis. And finally, you probably want to read something about logic and proofs. S. Lay, Analysis with an introduction to Proof, contains a good chapter on logic and proofs (as well as introduction to analysis, but I like Kirkwood much much better). > > > > I don't know what you mean by Algebra. What's called a "College Algebra", the stuff you've learned in high school, is normally presented without proofs. Any proof requires some theoretical machinery on very high level. If you mean "Modern Algebra" or "Abstract Algebra" (as it's also called), any textbook on that subject is sufficiently rigorous to contain most of the proofs. > > > > PPJr. > > Hi > > Thanks for the tips. I indeed meant college algebra. If was convinced we had seen proofs, but my memory must have failed me :) (it is now 20 years ago). > I remember a small proof why x^0 is 1, under what subject can that be put? These are things I want to begin with anyway  super basic level  but I want to know it. > The same counts for geometry and trigonometry. Can you suggest any books on that subject? > The modern algebra clearly is too difficult for me right now. > > Thank you very much > > Pierre
Well, in this particular case, if we take the standard definition of some nth power of some number and then want to expand it to the 0th power, making zero power of any number equals zero makes it consistent with the way we have defined the power of n natural and also the fact x/x = 1. I would say in this particular case this is *not* the proof but rather expanding the notion of power. So, no one will ask you something like "proof that x^0 = 1", but rather "show that the definition of 0th power is consistent with how we've already defined the natural power".
Once again, on the level of College Algebra, what you'll find is mostly bunch of facts given without proof e.g., the addition of natural numbers is associative and commutative, and then you might be asked something like "prove that you can add any three numbers in any order with the same result", and you just repeatedly apply the associativity and commutativity. And that's a good skill to have, and probably the first step toward mastering the skills for more difficult proofs. But you won't find those "more difficult proofs" in the standard course of College Algebra. In College Algebra the emphasis is on the ability to manipulate with algebraic expressions. To prove that (a+b)^2 = a^2 + 2ab + b^2, you just carry out the multiplication (a+b) times (a+b) and you must be able to do that without thinking too much. The emphasis is on breadth, not on depth.
PPJr.

