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The Equation That Couldn't Be Solved
Joe Kauzlarich writes "There's an ever-growing number of fun niche books seeping onto the mathematics bookshelves, that, while not essential, are almost always guaranteed to leave the reader with a fuller taste of the subject at hand and an appetite to learn more. Mario Livio's The Equation That Couldn't Be Solved is a modest semi-classic of pop-math literature, focusing on the central concepts of group theory, the subject that turned mathematics on its head a century and a half ago and has ever since been one of the delights of studying higher mathematics." Read on for the rest of Joe's review.
The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry
author
Mario Livio
pages
335
publisher
Simon & Schuster
rating
8/10
reviewer
Joe Kauzlarich
ISBN
0-7432-5820-7
summary
Popular math/science
If you've studied group theory, you've probably heard it called 'the language of symmetry' or referred to by some such vague, colorful non-description, while your professor and textbook direct you to just memorize the handful of basic axioms, definitions, and theorems that reveal little to the unknowing eye in the way of having much to do with symmetry. Livio concentrates on the more colorful aspects of symmetry, spending little time with black and white textbook theory. For this reason, the book makes ideal extra-curricular entertainment for those enrolled in a first-semester course on abstract algebra.
It seems that Mario Livio's technique in writing books is to choose an ostensibly simple topic and explore it from a broad array of angles. In his second and most popular work, The Golden Ratio, he chose to write about the number Phi. The book reads like the front page of Slashdot, skipping quickly from topic to topic, though sticking to the general theme, insuring that the reader must never get bored. The treatment he once gave to Phi, he now gives to symmetry. Livio explores the concept of symmetry as it manifests itself in biology, art, physics and (especially, of course) mathematics. Then he broaches the most important topic of the book, group theory, and ventures upon the two stunning tales of its conception, as the book's two central figures independently discover that a certain equation cannot be solved by means of regular algebra (which, at the time, referred to the sort of formulaic manipulation done by today's undergrad algebra and calculus students; now, the word 'algebra,' in professional circles, includes group theory and much more).
At last, less-experienced readers will find a warm entry-way into one of the most fascinating and advanced branches of mathematics, one which has, through time, permeated most other branches. Experienced readers will revisit a familiar topic in its historical and mathematical-cultural context, as well as gain an 'intuitive' picture of group theoretical symmetry, an aspect often omitted from first semester advanced algebra courses. All readers can be comforted that mathematical notation is hardly anywhere to be found in the book. Experts need not fear wasting money to relearn what they already know and beginners can pick up the math through its brief mostly-English-language descriptions and should feel more comfortable diving into a course on the subject.
What is this Equation That Couldn't Be Solved? The equation in question is the quintic equation-- a polynomial of degree five (i.e. ax^5+bx^4+...+ex+f=0). You've probably studied the quadratic equation-- ax^2+bx+c=0-- as well as the quadratic formula, used to solve this equation-- x= (b(+/-)sqrt(b^2-4ac))/2a. The quintic equation cannot be solved by means of a formula and it took hundreds of years and two very young men to discover this. And as happens in so many famous instances throughout the history of science, the answer to a seemingly innocent little problem becomes the key to a revolution in thought.
A 22-year-old Norwegian named Niels Henrick Abel (1802-1829) and a 20-year-old Frenchman named Evariste Galois (1811-1832), discovered the impossibility of solving the quintic almost simultaneously in the 1820's. Both died within years of their discovery and both went unnoticed and uncelebrated until after their death. The tragedies that preceded their deaths-- Abel died essentially out of poverty; Galois, poor and already half-mad, in a pistol duel-- have served as a valuable lesson to the mathematical community ever since: spot genius early and foster it. Who knows what would have become of these men had they lived through the prime of their talents, just as the great Gauss and his contemporaries were developing the foundations for what would become Modern mathematics? It was Abel and, particularly, Galois, who defined the language of symmetry. Both saw The Equation in a light that had never been seen before.
Mario Livio is a historian as much as he is a scientist and the detail and color he gives to the lives of these tragic figures is unforgettable. Not only was his research thorough, but he even visited the regions he describes, and his results on the mysteries surrounding the death of Galois offer conclusiveness and definitiveness that seem hardly to have been matched in this particular line of research. Additionally, Livio digs up fresh mathematical anecdotes throughout the book, being careful not to repeat those stories or 'factoids' that are repeated ad nauseum across the genre.
Group theory has become an essential requisite of such diverse areas of scientific research as was unimaginable at the time of its inception. The fundamental particles of nature are arranged in groups, making the subject a cornerstone of particle physics and all physical 'theories of everything.' Group theory is the simplest sort of 'mathematical abstraction' (actually, it is a step past set theory) in that numbers and equations play no part in its basic definitions. Once you learn it well, then rings and fields follow. Then comes the fascinating study of topology, and then there is little that can stop you from learning anything you want mathematically (okay, that's a stretch). Cryptography is a modern applied field which requires a good working knowledge of group theory. I'm sure there are many other examples of applied group theory if you can't be convinced of the beauty of the subject in and for itself. Physics enthusiasts will enjoy the later chapter on group theory in modern particle physics, which is meant to show how integral the subject is to understanding and communicating the very laws of our universe.
While this is surely a bias on my part, I wasn't impressed with the amount of actual math described in the book. The very basics of group theory, as I mentioned, are elaborated upon-- the definition of a group, permutation groups, symmetry groups-- but Livio makes few attempts to make clear what group theorists study (mathematically-speaking) beyond these simple sorts of ideas. To his credit, he does explain Galois's proof quite clearly, considering the amount of time a student spends getting to it in textbooks. The book, as I've said, is foremost a look at symmetry, secondarily historical, and lastly, a math text. It is light reading, but-- take my word for it-- extremely entertaining and worth the few bucks. If you aren't much of a math geek, this book provides a great chance for you to get a glimpse at abstract algebra, which, IMHO, is one of the most fascinating branches of mathematics and, oddly, seems normally to be kept well-hidden from the eyes of non-math or non-physics majors."
You can purchase The Equation That Couldn't Be Solved from bn.com. Slashdot welcomes readers' book reviews -- to see your own review here, read the book review guidelines, then visit the submission page.
If you've studied group theory, you've probably heard it called 'the language of symmetry' or referred to by some such vague, colorful non-description, while your professor and textbook direct you to just memorize the handful of basic axioms, definitions, and theorems that reveal little to the unknowing eye in the way of having much to do with symmetry. Livio concentrates on the more colorful aspects of symmetry, spending little time with black and white textbook theory. For this reason, the book makes ideal extra-curricular entertainment for those enrolled in a first-semester course on abstract algebra.
It seems that Mario Livio's technique in writing books is to choose an ostensibly simple topic and explore it from a broad array of angles. In his second and most popular work, The Golden Ratio, he chose to write about the number Phi. The book reads like the front page of Slashdot, skipping quickly from topic to topic, though sticking to the general theme, insuring that the reader must never get bored. The treatment he once gave to Phi, he now gives to symmetry. Livio explores the concept of symmetry as it manifests itself in biology, art, physics and (especially, of course) mathematics. Then he broaches the most important topic of the book, group theory, and ventures upon the two stunning tales of its conception, as the book's two central figures independently discover that a certain equation cannot be solved by means of regular algebra (which, at the time, referred to the sort of formulaic manipulation done by today's undergrad algebra and calculus students; now, the word 'algebra,' in professional circles, includes group theory and much more).
At last, less-experienced readers will find a warm entry-way into one of the most fascinating and advanced branches of mathematics, one which has, through time, permeated most other branches. Experienced readers will revisit a familiar topic in its historical and mathematical-cultural context, as well as gain an 'intuitive' picture of group theoretical symmetry, an aspect often omitted from first semester advanced algebra courses. All readers can be comforted that mathematical notation is hardly anywhere to be found in the book. Experts need not fear wasting money to relearn what they already know and beginners can pick up the math through its brief mostly-English-language descriptions and should feel more comfortable diving into a course on the subject.
What is this Equation That Couldn't Be Solved? The equation in question is the quintic equation-- a polynomial of degree five (i.e. ax^5+bx^4+...+ex+f=0). You've probably studied the quadratic equation-- ax^2+bx+c=0-- as well as the quadratic formula, used to solve this equation-- x= (b(+/-)sqrt(b^2-4ac))/2a. The quintic equation cannot be solved by means of a formula and it took hundreds of years and two very young men to discover this. And as happens in so many famous instances throughout the history of science, the answer to a seemingly innocent little problem becomes the key to a revolution in thought.
A 22-year-old Norwegian named Niels Henrick Abel (1802-1829) and a 20-year-old Frenchman named Evariste Galois (1811-1832), discovered the impossibility of solving the quintic almost simultaneously in the 1820's. Both died within years of their discovery and both went unnoticed and uncelebrated until after their death. The tragedies that preceded their deaths-- Abel died essentially out of poverty; Galois, poor and already half-mad, in a pistol duel-- have served as a valuable lesson to the mathematical community ever since: spot genius early and foster it. Who knows what would have become of these men had they lived through the prime of their talents, just as the great Gauss and his contemporaries were developing the foundations for what would become Modern mathematics? It was Abel and, particularly, Galois, who defined the language of symmetry. Both saw The Equation in a light that had never been seen before.
Mario Livio is a historian as much as he is a scientist and the detail and color he gives to the lives of these tragic figures is unforgettable. Not only was his research thorough, but he even visited the regions he describes, and his results on the mysteries surrounding the death of Galois offer conclusiveness and definitiveness that seem hardly to have been matched in this particular line of research. Additionally, Livio digs up fresh mathematical anecdotes throughout the book, being careful not to repeat those stories or 'factoids' that are repeated ad nauseum across the genre.
Group theory has become an essential requisite of such diverse areas of scientific research as was unimaginable at the time of its inception. The fundamental particles of nature are arranged in groups, making the subject a cornerstone of particle physics and all physical 'theories of everything.' Group theory is the simplest sort of 'mathematical abstraction' (actually, it is a step past set theory) in that numbers and equations play no part in its basic definitions. Once you learn it well, then rings and fields follow. Then comes the fascinating study of topology, and then there is little that can stop you from learning anything you want mathematically (okay, that's a stretch). Cryptography is a modern applied field which requires a good working knowledge of group theory. I'm sure there are many other examples of applied group theory if you can't be convinced of the beauty of the subject in and for itself. Physics enthusiasts will enjoy the later chapter on group theory in modern particle physics, which is meant to show how integral the subject is to understanding and communicating the very laws of our universe.
While this is surely a bias on my part, I wasn't impressed with the amount of actual math described in the book. The very basics of group theory, as I mentioned, are elaborated upon-- the definition of a group, permutation groups, symmetry groups-- but Livio makes few attempts to make clear what group theorists study (mathematically-speaking) beyond these simple sorts of ideas. To his credit, he does explain Galois's proof quite clearly, considering the amount of time a student spends getting to it in textbooks. The book, as I've said, is foremost a look at symmetry, secondarily historical, and lastly, a math text. It is light reading, but-- take my word for it-- extremely entertaining and worth the few bucks. If you aren't much of a math geek, this book provides a great chance for you to get a glimpse at abstract algebra, which, IMHO, is one of the most fascinating branches of mathematics and, oddly, seems normally to be kept well-hidden from the eyes of non-math or non-physics majors."
You can purchase The Equation That Couldn't Be Solved from bn.com. Slashdot welcomes readers' book reviews -- to see your own review here, read the book review guidelines, then visit the submission page.
Galois, IIRC, was the one who stayed up all night before the duel, frantically writing down every half-formed mathematical insight for posterity. Which probably didn't help his shooting. He was only 20, I think.
What I'm listening to now on Pandora...
If you like books about maths (as we say here in the UK - mathematics is PLURAL), check out 'The Penguin Dictionary of Curious and Interesting Numbers' by David Wells - ISBN 0-14-008029-5.
There are countless (obviously not really) books on group theory at all different levels. If you're not a math major and want to learn a bit about group theory (and rings, too) from a book that makes it interesting, historical, and gives motivation for the theory, check out Galian's "Contemporary Abstract Algebra". This book clearly isn't meant to prepare you for graduate level algebra, but that's not what many of us are going for of course. It introduces the theory with LOTS of examples, and even relates most of the theory to ways you can use it in practice to solve all sorts of different problems in "real life". Check it out!
FTFR: You've probably studied the quadratic equation-- ax^2+bx+c=0-- as well as the quadratic formula, used to solve this equation-- x= (b(+/-)sqrt(b^2-4ac))/2a
The roots of the equation are x = (-b(+/-)sqrt(b^2-4ac))/2a
Informative? Please... don't be fooled by AC's "Fermat's Last Theorem is teh cool too!" statement. It's the same Amazon-referral-whoring post (see the redirect) that he sticks in every book review. Pathetic.
How is this Bring Radical an algebraic function? It's defined as an analytic extension of an infinite series.
Here's something that might deserve a closer look: The duel and the events leading to it are blurred by time and the phantasies of novelists and what's worse biographers. We can rule out or at least it is highly improbable that the duel was a plot of the royalists to murder him. Though this version is a favorite legend lingering in many biographies. Most probably it was Galois himself who incited this interpretation. He wanted himself to appear as a victim of the government, which should enrage the masses to revolt. He dropped remarks pointing in this direction: At a meeting of the Friends of the People and in his last letters. The most likely reason is: He was weary of life, because of his unhappy love affair, his fruitless efforts for gaining recognition for his mathematical work, his financial and work situation and he felt finished up a blind alley in politics as well. So his duel was like a staged suicide. It is still not clear who the other dueller a supposed political friend was. One thing is clear, though it kills a favorite legend: He didn't lay down his mathematical theory in the night befor the duel. He pointed out the cornerstones of his scientific life in a long letter to his friend Chevalier, so that everything might be properly evaluated and not be lost.
Wolfram Research has some interesting explication on historical methods of solving the quintic: http://library.wolfram.com/examples/quintic/main.h tml
A group is a set of elements under a binary operation (that is associative) such that:
1) there is an element call the identity such that:
- it is commutative with all elements of the group
- under the binary operation, the identity with a non-identity element results in that same non-identity element
2) for every non-identity element there is another non-identity element that under the binary operation results in the identity that commutes with said non-identity element.
3) for any two elements in the group, under the binary operation of the group, the result is in the group.
I think that given this word definition of a group, no-one will argue that the equation form is not superior.