Algebra

Gelfand Correspondence Program in Mathematics

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.

Genres: Mathematics, Science, Textbooks, Nonfiction, Reference, Algebra, Stem

150 pages, Paperback

First published December 31, 1992

About the author

Israel Moiseevich Gelfand, also written Israïl Moyseyovich Gel'fand, or Izrail M. Gelfand (Yiddish: ישראל געלפֿאַנד‎, Russian: Израиль Моисеевич Гельфанд ; 2 September [O.S. 20 August] 1913 – 5 October 2009) was a prominent Soviet mathematician. He made significant contributions to many branches of mathematics, including group theory, representation theory and functional analysis. The recipient of many awards, including the Order of Lenin and the Wolf Prize, he was a Fellow of the Royal Society and professor at Moscow State University and, after immigrating to the United States shortly before his 76th birthday, at Rutgers University.

Reviews

[Mohammad Tanviruzzaman · Rating 5 out of 5]

From Gelfand's wiki page, it looks like he was a prominent mathematician; reading this book, I can attest he had a flair for educating as well. The book is a bit unusual because much of it is dedicated to problems and the problems do not come at the very end of a chapter, they come frequently, between talks. Initially, I was a bit annoyed with the constant pop-up of problems, it felt like it was slowing me down. Since most other math books I read before, delegate the problems to the end of the chapter, I have been accustomed to being in two non-overlapping mindsets: one is to read and one is to solve. However, after a while, I got used to it and then if I did not see a problem in more than half a page, I would feel kind of lost. The problems gave me a concrete way to frequently check my understanding, which turned out to be awesome for the self-study that I was doing. I am now very interested in Gelfand's other books. The problems are invariably interesting and they are not ordered by hardness; rather, sometimes n-th problem would require much more time and effort than (n-1)th or (n+1)th problem--that kind of gave a sense of adventure. In another way the book is unusual because beginning algebra books do not usually include proofs, they only give computation problems; this book on the other hand had a fair share of both. I guess being a working mathematician, it was hard for Gelfand to not equip students with proof experience. Another way to view the book is that the authors took a traditional book, did not pay any attention to the talks that precede the exercises, then they threw away all uninteresting or repetitive problems, they then sprinkled some explanations, enough so the students are able to solve the following problems; at that point probably there were not enough problems to have a book, so the authors threw in a bunch of their own, including problems around proofs. If the authors included the topic-map that they had in mind while writing the book, the readers could refer to it time-to-time to orient about the big picture, which would have made this great book even better.

Here are some notes that I took while I was studying the book: https://github.com/tanv1r/math/blob/m...

[Phil Filippak · Rating 5 out of 5]

I'm sure this is one of the best introductory algebra books. I've been moving slowly through it only because of my lack of any prior experience in the field. In a bit more than one year I have covered nearly the whole school algebra curriculum, plus something extra. Explanations are beyond any measure. This book is concise, thorough, and provides a great deal of problems, either with and without solutions (those with solutions are also thorough and clearly explained). I'm happy to have read this book, and slightly proud for achieving ~95% completion rate on the problems.

[Maxim Perepelitsyn · Rating 5 out of 5]

Concise, packed with interesting and sometimes challenging problems. I would highly recommend it to anyone who is interested in Math or wants a refresher on high school algebra.

[Balázs · Rating 5 out of 5]

pg. 52
"Probably you are discouraged by this solution because it seems impossible to invent it. The authors share your feeling."

[Angelino Desmet · Rating 4 out of 5]

Very educative. Not rote learning, but understanding why math works. I grew up being taught math by memorizing and applying formulas. For some reason, regardless of the latter, I still had affinity to math, particularly probability. That was the only branch not destroyed by the soul crushing application of formulas; you still had to think. This book revamped that feeling towards algebra. It felt uncomfortable at first, not just thinking in terms of a formula. Instead it magnifies that math is about finding creative patterns, and how elegant, yet subtle, proving seemingly obvious facts can be.

This was recommended to me as suitable for self-learning. However I do not think it's ideal to learn this book alone, or I'm too stupid. Many problems come without a solution, which is quite annoying if you want verification and an explanation. Luckily there is a PDF with all the solutions floating around on the web, but it's not always thorough or without mistakes. Gelfand does mention that the reader needn't worry if a solution was not found, and that a lot of problems are pretty hard. Nevertheless, I prefer understanding all my mistakes and mathematical sinkholes. Aside from its flaws, this book definitely beats any math curriculum I've had in high school.

PS PM me if you'd like a copy of the PDF if you can't find it.

[Andyinverted · Rating 5 out of 5]

I found the technique of using the problems as a direct part of the instruction quite helpful. It was more challenging up front but it made later insights easier to understand. I was worried I would miss some vital information because of this technique but solutions are provided for all the problems the authors deem important - I never had a problem here.
The shortness of the book is a great feature in this age of gargantuan math texts in that it flows through a large number of ideas with a comparatively small investment in time. The small page count belies its depth - it was a great review of much of high school algebra, providing some insights entirely new to me.

[Logan · Rating 4 out of 5]

4 stars

I can honestly say that I've never seen elementary algebra presented in a manner quite like this. There are exercises scattered throughout the book to improve computation, but there's also an abundance of proof questions that are both challenging and deeply interesting. Despite the identity crisis the book seems to have (try explaining a derivative to an algebra I student!) and the rather confusing section on inequalities, everyone with even a little bit of room to improve in their elementary algebra skills can benefit.

[Paul R · Rating 5 out of 5]

Legendary very difficult problems that will improve your logically thinking and mathematical reasoning.

[braindead · Rating 5 out of 5]

Helps you develop intuition and derive every formula. It isn't the typical basic math book where formulas are presented and you just solve problems, leaving many questions unanswered.

Found it super helpful. After reading it, gained a real passion and love for mathematics. Thank you, Gelfand.

[Mykyta Kuzmenko · Rating 4 out of 5]

Отличное пособие с прицелом на матшколу.