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Download Algebra epub

by Israel M. Gelfand,Alexander Shen

This book is about algebra. This is a very old science and its gems have lost their charm for us through everyday use. We have tried in this book to refresh them for you. The main part of the book is made up of problems. The best way to deal with them is: Solve the problem by yourself - compare your solution with the solution in the book (if it exists) - go to the next problem. However, if you have difficulties solving a problem (and some of them are quite difficult), you may read the hint or start to read the solution. If there is no solution in the book for some problem, you may skip it (it is not heavily used in the sequel) and return to it later. The book is divided into sections devoted to different topics. Some of them are very short, others are rather long. Of course, you know arithmetic pretty well. However, we shall go through it once more, starting with easy things. 2 Exchange of terms in addition Let's add 3 and 5: 3+5=8. And now change the order: 5+3=8. We get the same result. Adding three apples to five apples is the same as adding five apples to three - apples do not disappear and we get eight of them in both cases. 3 Exchange of terms in multiplication Multiplication has a similar property. But let us first agree on notation.
Download Algebra epub
ISBN: 0817636773
ISBN13: 978-0817636777
Category: Science
Subcategory: Mathematics
Author: Israel M. Gelfand,Alexander Shen
Language: English
Publisher: Birkhäuser (July 9, 2003)
Pages: 150 pages
ePUB size: 1206 kb
FB2 size: 1701 kb
Rating: 4.7
Votes: 888
Other Formats: lrf docx rtf doc

As a returning college undergrad studying a math-heavy field, I picked up Gelfand's Algebra hoping to rejuvenate my very rusty algebra skills. To be honest, I didn't even learn algebra that well when I first studied it over a decade ago, but I slid by as a stereotypical "math student" in grade school. I wanted to do it right this time, and I'd heard very good things about this book.

I want to start with the positives, because there are many. If you're someone put off by the modern way of teaching math that forgoes all explanations and intuition in favor of step-by-step guides and formulas, then you'll appreciate Gelfand's approach to the field of algebra. Each subject is treated in a learn-by-doing fashion, and nothing is spoon fed. Clever intuitive explanations are provided for many topics that most books just take for granted. If you work your way through the entire book and all of its problems, you'll come away with algebraic understanding and intuition that probably exceeds that of most undergraduate math majors at the college level. As I said, I wanted to learn algebra the "right way" this time around, and I definitely achieved my goal through this book.

That said, this book has some pretty major flaws to the point that I have trouble actually recommending it. In fact, I don't think I'd even recommend it to my past self.

First, of the hundreds of exercises provided throughout the book, it feels like less than half of them have solutions provided. The author tries to explain that any problem without a solution is okay to skip if you can't solve it, but that explanation just isn't good enough. The fact is that the meat of this book is its problems, and if you skipped every problem without a solution, you'd probably learn less algebra than you would from a standard textbook. Like I said, Gelfand's approach is very learn-by-doing, but he doesn't give you the proper tools to actually "do" everything you're supposed to "learn." In fact, many problems build on previous problems, so if you skip any of them you risk triggering a cascade of unsolvable exercises that will leave you frustrated and confused. It's not like these are routine calculations you're doing, either. Most of the problems require creative problem solving, significant trial and error, or out of the box thinking to solve. There's nothing inherently wrong with this (in fact I love this approach, as it builds vital mathematical intuition and "real" math skills), but without solutions there's no way to check the correctness of your answer. If you get stuck (and you WILL get stuck), there's nothing you can do. Simply put, the challenge level of some of these problems is such that the typical math student is bound to encounter several of them that they will never solve on their own, and there's no way for them to help themselves along. I've heard some people suggest that the lack of solutions is good because that way you're forced to work through the problem on your own, but as someone with the self control to give each exercise a nice long, honest attempt, I found it to be a major hindrance to my learning. After hours of searching, I managed to find a document online with solutions to the unsolved problems, and that's the only way I was able to work my way through the book.

Second, much of the wording in this book is ambiguous, and in some cases there are major errors. It's not always clear what an exercise is asking you to do, and the mistakes can really set you back and waste your time. The subject matter is difficult enough without throwing away hours of effort just because the book didn't give you clear instructions or explanations.

Third, there's just not enough reinforcement in this book. On this plus side, this means you're learning something new with each and every exercise, but the problem is that the near-zero repetition doesn't leave much room for the learning to sink in. I would have really appreciated a handful of reinforcement problems for the most essential concepts.

As a tutoring aid, this book would probably get 5 stars. In that situation, the tutor can have worked everything out ahead of time and guide the student through the many challenges in this book. But as a self-study tool, or even a classroom text, this classic tome has a few too many issues. It earns 3 stars for the high quality of Gelfand's approach and explanations, but the issues with the exercises keep it from a true recommendation. If this book were to be republished some day with corrections and a full set of solutions, I would have no problem giving it 5 stars.
This is not meant to be a comprehensive textbook that covers all topics found in Precalculus courses in the US high school system. Instead, its layout aims to give Western readers a feel of the Soviet mathematical pedagogy. The emphasis is on problem solving, as most insights can only be gained when students try to solve problems serially. By diligently follow simple problems in the fifth section "Multiplication and multplication algorithm", students may find out why each immediate sums inside a multiplication algorithm must shift left. The reason is simple: all we are doing is to follow the distributive law of multiplication. Take 188x98 for example, what we do when we set them up in the algorithm is to break 98 down to 90 and 8, thus (90x188)+(8x188). What fill in the gap caused by the left shift is just a zero. Believe it or not, nowadays, the multiplication algorithm is nothing foreign to most adults, but if you ask them why we have to shift each immediate sums to the left, they will be caught muted.

Students are also exposed to reading and writing proof this early in the book. Problem 42, which asks the readers to prove that a/b and c/d is a neighbour faction, is already challenging enough to become a question on many math forums.

Less known are the particular problems 31 and 32 which asks you to come up with a way to put the parentheses in the product of 5 and then 99 terms so that all variations are not left out. The solution prompts one to review the basic formulas of permutation and combination, which is not covered in this book at all.

Towards the end the book even introduces the AM-GM inequality, this is something very familiar with almost all students in the former Soviet Union countries, but not introduced directly in American high school textbook, according to my own limited knowledge.

The aim is thus to introducing mathematical reasoning and rigour to young students or even adults and much less on computation. This is the essence of learning mathematics.

Thus, 4 stars for a simple but in-depth presentation of one of the oldest branches of mathematics, 4 stars for the achievement of Soviet mathematics education within 4 decades (1950s-1990s).
Let me start out by saying that this book is really good content-wise. It only covers elementary topics, but does so in a way that you get to see into the mind of a mathematician and how they think about the subject. Some of the proofs and other topics in this book you will seldom find elsewhere, and that alone makes this valuable. The book also has a light-hearted tone to it, which is encouraging considering the difficulty level of this book. Now, onto the cons of this book. This book is not suitable for a person trying to either learn the subject for the first time or someone who is going back to it after many years of being away from math unless they are particularly bright. This is primarily for two reasons. Firstly, this book is far from being comprehensive. It covers stuff learned in the equivalent of algebra 1 if you live in the states and nothing more. This book honestly suffers from the opposite problem from modern day textbooks. Textbooks today try to cover a ton of topics in a shallow and meaningless way, while this book is, in my opinion, too narrow and too deep. There are very few routine or application type exercises, most of it is simply theoretical proofs or concept problems. Secondly, the difficulty level of this book is simply too high for someone learning it for the first time. I have a solid background in high-school level algebra, and even I found this book to be quite difficult to work through at times. This is not a bad thing necessarily, but I would not give this book to a child as a first exposure. Instead, this book would be great as a supplement to a more comprehensive text.