The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry

For thousands of years mathematicians solved progressively more difficult algebraic equations, until they encountered the quintic equation, which resisted solution for three centuries. Working independently, two great prodigies ultimately proved that the quintic cannot be solved by a simple formula. These geniuses, a Norwegian named Niels Henrik Abel and a romantic Frenchman named Évariste Galois, both died tragically young. Their incredible labor, however, produced the origins of group theory.

The first extensive, popular account of the mathematics of symmetry and order, The Equation That Couldn't Be Solved is told not through abstract formulas but in a beautifully written and dramatic account of the lives and work of some of the greatest and most intriguing mathematicians in history.

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All Customer Reviews
Average Customer Review:
25 of 25 people found the following review helpful:

interesting book about symmetry not enough on the theory of equations, 2008-03-19
I became interested in this book for several reasons. The first is that I find Livio to be an entertaining writer. I read his book on phi and its relationship to beauty and found it interesting and enlightening. I have reviewed that book on amazon earlier. I met Livio in Princeton a little over a month ago when he gave a lecture on symmetry at the Princeton Plasma Physics Laboratory in one of a series of lectures intended for high school students. It was a fascinating presentation and he briefly discussed the book, mentioning how his research into the death of Galois led him to a new theory about how he died in the duel and who killed him. I found this very intriguing and I wanted to read about it.

As a college undergraduate I majored in mathematics and modern algebra was my favorite subject. The course I took on Galois theory was the most fascinating to me and I marveled over the fact that a teenage boy had developed a branch of group theory that answered questions that had stumped the greatest mathematicians for centuries.

So I bought the book and read it with very high expectations. I preface my remarks this way because I was somewhat disappointed in the book and my disappointment leads to my criticism here. But I don't want the critcism to detract from the fact that it is a well written and researched book and written in a style that like his other books makes it accessible to the general public and even the highly motivated high school students.

First of all the title leads you to believe that it is completely about the solving of the problem for which polynomials can be solved by radicals (i.e. equations that only involve basic arithmetical operations a roots, e.g. square cube roots etc,)and which ones cannotbe so solved. The book starts out discussing symmetry and to some extent how it is connected to group theory and polynomial equations. Livio takes us through the competitions that existed as mathematicians struggled to solve the general cubic and quartic equations. He even goes into the many attempts and "near proofs" of the solution to the fifth degree equation. Here this leads into a natural discussion of the lives of Abel and Galois, the two young mathematical geniuses who both died in their twenties but did the amazing work of showing that the general fifth degree polynomial could not be solved by radicals.

Livio slips into vague terminology, at times referring to the result as simply the equation that could not be solved as he does in the title and at other times he calls it "the equation that could not be solved by formula." Both expressions are vague and a little misleading. All polynomial equations with real coefficients have solutions in the complex plane. In fact there are n solutions to a nth degree polynomial, some may be real, some complex and some may occur as multiples but every nth degree polynomial with real coeefficients can be factored into the product of n first degree polynomials with each term of the form x-a where a is a complex number.

Galois in his writings refers to this question as one of solvability by radicals which is the precise accepted mathematical term that I think should have been used in this book.

Although another reviewer, Steve Koss, criticizes Livio for his discussion of the lives of Galois and Abel I can see justification for this. I was actually most interested in learning more about these two geniuses and the circumstances that led to their early deaths. For this I give Livio high marks. I was a little disappointed in the Galois story because I had been led to believe that Livio had a new convincing theory about the duel that led to Galois' demise at the age of only 20. But the theory was not completely new and not as convincing as I had expected.

With regard to Galois, Livio refers to his discovery as the creation of group theory. I did not know this and I am a little skeptical because in my studies of modern algebra I have never seen a reference to Galois as the founder of group theory. What I do know about Galois is that the theory he developed solved not only the impossibility of solving the general polynomials of fifth degree and higher but also the trisection of an angle with only a ruler and compass and several other questions that had stumped the Greek mathematicians many centuries earlier. These problems of the Greeks were mentioned to us when we were in high school but often the fact that they cannot be solved is avoided because the mathematics that proves it is too advanced for high school. Unfortunately it has held many a student to try to come up with a construction and some have been convincing even though they are flawed.

This theory, that concerns itself with special groups and fields that are called Galois groups and Galois fields is highly focussed on the aspects of modern algebra that address the problem of solvability of polynomials by radicals which in turn leads to the results about the Greek constructions as well. I was disappointed that Livio missed the opportunity to point this out. This theory is rightfully called Galois theory and it encompasses groups, fields and isomorphisms. It does not cover all aspects of group theory but it does branch out into other areas of modern algebra. I agree with Koss that the theory of solvability was shortchanged in the book.

The latter part of the book is not really at all about polynomial equations. Rather, the author has taken the liberty of moving on into discussions of mathematical symmetries and there relationships to physics, non-euclidean geometry, human psychology and other fields. Like Steve Koss I found this part much less focussed and somewhat disorganized. Also, I think the book title is misleading. The book is about symmetry as a part of group theory and mathematics and other disciplines. The solvability of polynomials is only a part of it. I didn't enjoy the latter chapters nearly as much as the earlier ones.

Inspite of these shortcomings this book is well worth reading especially if you have an interest in abstract mathematics and life's connections to symmetry. What Abel and Galois accomplished as very young mathematicians is the most difficult thing to do in mathematical research. They showed that something that people thought was possible to prove was actually impossible. It is very difficult to accept when doing research that failure to prove something is not your own personal shortcoming but rather the fact that the endeavor is futile. I have often heard the expression "How can you prove a negative?" My most recent recollection was Roger Clemens saying this in front of a congressional committee investigating the use of performance enhancing drugs in baseball. Well, as we see here in mathematics it is possible to prove a negative!


3 of 5 people found the following review helpful:

Many mistakes in the book, 2008-02-10
Brahmagupta's beautiful solution of the quadratic is ascribed to Diophantus. Linear indeterminate equations were never solved by Diophatus either. That was done by Aryabhatta. That type of errors takes away from other good stuff. Aryabhatta who was the first to solve equations using general methods never mentioned. Author should have run it through an expert prior to publication.


5 of 5 people found the following review helpful:

A Fine book with a few permutations, 2007-09-08
If you are not a mathematician (and I am not), but have an interest in the subject, and a working knowledge of some elementary ideas, this is a terrific book. It has the easiest explanation of symmetry/Galois groups, etc., of any of the books I have tried on the topic -- oh sure, it rambles (as the severe critics here say) -- but try and find some other book on the subject that doesn't immediately drop you far beyond your depth. Livio has a knack for very, very clear explanations and great metaphors (permutations and probability are discussed in terms of finding a mate). I recommend it highly, especially if you can get it with one of Ian Stewart's books on the same topic.


3 of 5 people found the following review helpful:

"Don't cry, I need all my courage to die at twenty."...Galois, 2007-02-09
When I came across this book,I thumbed through it and the figures that jumped out at me were a collection of things,mainly about mathematics,puzzles and other things that interest me. I graduated in Electrical Engineering nearly 50 years ago,and have had a lifelong interest in Mathematical Recreations and Puzzles of all sorts. Granted most of the Mathematics I studied has long since left me mainly because of lack of use.However,the lore,beauty,mystery and fascination of Mathematics has remained. A lot of the Mathematics discussed in this book falls into what I think of as Theoretical rather than Applied Mathematics;and then there's that whole area of Recreational Mathematics.
I have read all the other reviews here,and basically agree with all of them.Taken together they do a good job of telling what the book is about and the Mathematicians who searched for those elusive solutions.In fact,there is so much that could be covered that it would take many volumes to even only scratch the surface.
I don't know if I really "know" much more about Group Theory and Symmetry than when I started ,but I still found it a fascinating read. Kind of like a 5-day tour of Europe-Been there,done that,but do I "know" Europe?
Like I said,other reviews have pretty well covered the book;so I won't repeat.
However; I would like to point out a couple of things.
In chapter 6,the 15-Puzzle is discussed. This is one of the all time greatest puzzles.It has interested me for years. If you would like to know more about it,I strongly recommend you read "The 15 Puzzle" by Jerry Slocum and Dic Sonnefeld.After you see this book ,you'll probably agree it is one of the world's most interestting puzzles;and what a history and legend it has. I posted a review of it here on Amazon on June 6,2006.
If you haven't noticed ,the information on this book has a section "Inside the Book" and in this section under "text stats" ,it shows this book has a Fog Index of 16.2. A search on the net will show how it is calculated. It takes a sample of text,and by looking at the lengths of sentences,number of multiple syllable words,paragraphs,and so forth comes up with a number that shows how difficult it is to comprehend. 16.2 is a fairly high level; and that combined with the theoretical math concepts;there is lttle wonder tht many would find this a fairly difficult book to read.Of course,I'm referring to the Mathematical concepts as opposed to the Biographical information.
The author must have done a tremendous amount of research in writing this book, and in the extensive Notes and References he provides a huge amount of information for the reader who wishes to pursue anything further


1 of 2 people found the following review helpful:

A lively read for a wide audience, 2007-01-07
Symmetry is the topic of Mario Livio's THE EQUATION THAT COULDN'T BE SOLVED: HOW MATHEMATICAL GENIUS DISCOVERED THE LANGUAGE OF SYMMETRY, and will make an involving read for those involved in either science or art. Mathematicians solved algebraic equations until they came to a stop with the quintic equation, which resisted solution until two mathematical geniuses independently discovered it couldn't be solved using the usual methods. This account of 'group theory' explains both the concept of symmetry and the evolution of its foundations, and makes for a lively read for a wide audience from physicists and science majors to students involved in the arts.

Diane C. Donovan
California Bookwatch

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