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The Equation That Couldn't Be Solved 299

Posted by samzenpus from the math-is-hard dept.
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.
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The Equation That Couldn't Be Solved More

The Equation That Couldn't Be Solved

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  • As Barbie says (Score:2, Funny)

    by Anonymous Coward
    Math is hard!

  • The real geek equation...solved! (Score:4, Funny)

    by dada21 ( 163177 ) * <adam.dada@gmail.com> on Friday November 18, 2005 @01:08PM (#14064416) Homepage Journal
    Shower^2 + Shave + BrushTeethx32 + Get(Own(Apartment)) + not(sqr(Clothing)) = Women

    • Re:The real geek equation...solved! (Score:3, Funny)

      by Anonymous Coward
      Wait, all I have to do is get my own apartment and not wear square clothing?

      Since right now:
      Shower=0
      Shave=0
      BrushTeeth=0

      which resolves to:
      0^2 + 0 + 0x32 = 0

    • Re:The real geek equation...solved! (Score:2, Funny)

      by Anonymous Coward
      Ah, but as every young mathematician knows, women are evil -

      Women takes time and money.

      Women = time x money

      Time IS money

      Women = money x money = money ^ 2

      Money is the root of all evil

      money = sqrt(evil)
      => money^2 = evil

      since women = money^2

      women = evil
    • Found your problem:

      shower^0.5 + shave + brushteeth*32 + get(own(apartment)) + not(clothes^2) ==> P(woman|low standards) > 0

      and its corollary

      hot(face AND body) OR $*1.0e+06 ==> P(hotness(woman) > 8) > 0

  • favorite math quote (Score:5, Funny)

    by flynt ( 248848 ) on Friday November 18, 2005 @01:09PM (#14064428)
    To paraphrase my favorite math quote (which I believe a physicist said): There are only two kinds of math books, those you can't read past the first page, and those you can't read past the first sentence.

  • Galois (Score:5, Informative)

    by Otter ( 3800 ) on Friday November 18, 2005 @01:09PM (#14064431) Journal
    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.

    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.

    • Re:Galois (Score:4, Interesting)

      by Anonymous Coward on Friday November 18, 2005 @01:25PM (#14064611)
      It was common that those who would participate in a pistol duel to stay up all night--writing a will, writing down their knowledge for posterity, praying, etc. Without having researched the issue, I can say that his opponent was likely up all night, as well.

    • Re:Galois (Score:4, Funny)

      by Beryllium Sphere(tm) ( 193358 ) on Friday November 18, 2005 @03:27PM (#14065807) Journal
      He was also a political activist, which lends a wonderful double meaning to "the quintic equation cannot be solved by radicals".

      As Tom Lehrer said, "It's people like that who make you realize how little you've accomplished".

    • Re:Galois (Score:3, Funny)

      by ozbird ( 127571 )
      "Guns don't kill people; maths kills people."
    • Here's something [galois-group.net] 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
    • That's a myth. He was working on that book for months beforehand and didn't actually stay up all night writing down his math.

  • (-(p/4))^1/4BR((((-5/p)^5/4)q)/4) (Score:3, Interesting)

    by digitaldc ( 879047 ) * on Friday November 18, 2005 @01:12PM (#14064460)
    "...write down the roots of the quintic in terms of square roots, cube roots, and the Bring radical, which is therefore an algebraic solution in terms of algebraic functions of a single variable..."
    see: http://en.wikipedia.org/wiki/Quintic_equation [wikipedia.org]

    It's really that simple.

  • Curious and interesting numbers (Score:3, Informative)

    by Skiron ( 735617 ) on Friday November 18, 2005 @01:13PM (#14064473)
    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.

    • Re:Curious and interesting numbers (Score:5, Informative)

      by TheoMurpse ( 729043 ) on Friday November 18, 2005 @02:53PM (#14065495) Homepage
      mathematics is PLURAL

      No, "mathematics" is a singular noun that just happens to end in 's' in the same vein as "his", "pus" and "psoriasis."

      "Math" is the American abbreviation for the singular noun. "Maths" is the UK abbreviation for the singular noun.
      • No, "mathematics" is a singular noun that just happens to end in 's'
        Not according to the Merriam-Webster Online Dictionary [m-w.com]. It's description is: "Function: noun plural but usually singular in construction".
      • I'm not sure about whether it's singular or plural in English, but there's an interesting (IMHO) story attached to the French word. When a group of French mathematicians under the group pseudonym Nicolas Bourbaki started writing a series of books in which a very large part of mathematics was set up in a very general and formal style, they called it `Éléments de mathématique' (`Elements of mathematic'). Although the French word for mathematics is `mathématiques' (plural), they used the

  • Cool! (Score:5, Funny)

    by dorkygeek ( 898295 ) on Friday November 18, 2005 @01:13PM (#14064477) Journal
    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.

    Cool, the first book with dupes already integrated!!

  • Um, how about this? : (a + b)^5 = a^5 + 5(a^4)b + 10(a^3)(b^2) + 10(a^2)(b^3) + 5a(b^4) + b^5.
    • No, no. Sure, *some* fifth-order polynomials are factorable to a set of reduced-order polynominals, but not all. What's being said here is that you can't take an arbitrary fifth-order polynomial, in the form ax^5 + bx^4 + cx^3 + dx^2 + ex + f, and have a formula to provide a solution. So there can be no 'quintic formula' along the same lines as the 'quadratic' formula, making polynomials of fifth-order or higher much harder to solve.

  • It's been solved (Score:2, Funny)

    by Anonymous Coward
    I typed in the ISBN into Google. Google told me 0 - 7432 - 5820 - 7 = -13259 Simple.

  • if you want to learn a bit about group theory (Score:3, Informative)

    by flynt ( 248848 ) on Friday November 18, 2005 @01:16PM (#14064511)
    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!

    • Re:if you want to learn a bit about group theory (Score:5, Funny)

      by jfengel ( 409917 ) on Friday November 18, 2005 @01:59PM (#14064964) Homepage Journal
      Well, you got it partly right: there are an infinite number of books on group theory, but they're countably infinite, because each is of finite length, so you can assign an integer value to each (say, the ASCII coding of the book). And they're a subset of the countably infinite set of all books.

      Of course, only some of them have actually been written and sold, and that's a small and finite number. The books on group theory which have yet to be written are all out there, but it's left as an exercise to the writer.

      (Eh, it was a good joke when I started writing it.)
      • Are you sure they're infinite? You can only add so much random material before the book will fall outside the set of books 'on' group theory. Sure, it's a large set of books, but infinite? I don't think so.

        • Well, anecdotally I can tell you that when my group theory prof was talking, one got the very strong impression that he was never going to shut up. Which leads me to think that there's an infinite number of things to say about group theory. Or at the very least, you can say the same things over and over again.

          (Just kidding; he was actually a fine lecturer.)

  • Worth the Few Bucks (Score:3, Insightful)

    by fossa ( 212602 ) <pat7@g[ ]net ['mx.' in gap]> on Friday November 18, 2005 @01:17PM (#14064521) Journal

    "Worth the few bucks", or maybe a trip to the library?

  • Group Theory Joke (Score:5, Funny)

    by keithmo ( 453716 ) on Friday November 18, 2005 @01:21PM (#14064570) Homepage

    Q: What's purple and commutes?

    A: An Abelian grape.

  • Quadratic Equation (Score:5, Informative)

    by sameerdesai ( 654894 ) on Friday November 18, 2005 @01:23PM (#14064588)
    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

  • Pop Math? (Score:5, Funny)

    by Anonymous Coward on Friday November 18, 2005 @01:25PM (#14064610)
    Yeah, right. Pop Math. My friends I are always discussing popular equations around the water cooler.

    I love it when I can throw in a funny "pop math" reference.
    • Funny, but we were actually talking about hyperbolic space at lunch. And we're just a bunch of Java programmers ... ;-)

    • The only way to bring math into the pop culture and call it pop-math would be to thinker up a formula for getting girls... ...but unfortunately f^3+b^2+r=0 where f=flowers b=begging r=ring ...still equals ZERO.
  • One we tried to solve in high school:

    Integral x^x dx

    It seems like a found a solution for it (this was a long time ago), but I think I later on figured out it was wrong. I haven't thought about it in a long time, but I suspect it's not integrateable. Any opinions from math geeks? I'm actually kind of curious.

    • I solved it in high school, but not in closed form, so for a lot of people, that doesn't count.

      I saw a proof here on /. that it couldn't be solved in closed form, but I don't remember what it was. Something like x^x = e^(x ln(x)), and for e^(f(x)) to be integrable, f has to have certain properties.

      Digression: if you really think about it, functions like sin(x) and ln(x) are really not closed form either - they are infinite series. But they are infinite series that we have given names to, and which we can

      • Re:Solve this... (Score:4, Insightful)

        by podperson ( 592944 ) on Friday November 18, 2005 @02:30PM (#14065266) Homepage
        if you really think about it, functions like sin(x) and ln(x) are really not closed form either

        Think about it harder.

        You can express anything as an infinite series. E.g. 1 = 1/2 + 1/4 + 1/8 + ..., so I can't integrate S 1 dx ?
        • .. just integrate the terms of the Taylor series ;-)

        • Re:Solve this... (Score:2, Insightful)

          by Anonymous Coward
          Think about it harder.

          Think about it a little bit harder.

          Sin(x) and ln(x) are transcendental functions. Any function or value can be expressed in terms of an infinite series. Some functions and values can not be represented without an infinite series. Functions such as sin(x) and ln(x). These are not "closed form." They are functions that can only be expressed (generally) as infinite sums, which we gave given specific names to.
      • Yes, closed forms is whatever we give a name to.


        The problem of solving polynomial equations was not to find a closed form solution, but to find an expression in terms of a finite combination of radicals (roots).


        It turns out that even for cubics there is no "closed form" solution unless you allow taking radicals of complex numbers.

  • 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.

    I thought, and Algebra by Isaacs confirms, that a group is a set G with an associative binary operation * such that there exists e in G with properties:
    1. For each x in G, x*e=e*x=x.
    2. For each x in G, there is a y in G such that x*y=y*x=e.

    Can anyone give the definition that doesn't use equations? I didn't think so.

    • I'm not sure what that comment meant either. I can do set theory without thinking about group theory. I can't do group theory (can't even define a group) without using sets. So I don't know what the poster meant by that.
      • I meant what you said.. groups need sets but sets don't need groups. As far as my comment about not needing equations, I was of course totally wrong; I should of said something to the effect that, unlike algebra or calculus, equations aren't the central focus of study... well, that's not quite precise either.. oh well
    • Yes you can do this non contructively. Either define the catagory of groups in terms of either derived functors or as universal objects with respect to other catagories. Then the elements of catagory are by definition "groups".

      I'm not sure if you have ever seen catagory theory and universals but a good (2nd semester or assuming a good undergrad background) graduate algebra class would have gone through universals.
    • A group is a category over one object with invertible morphisms. Pbbbbbbbbbt.

      Seriously though... every logical statement is technically an "equation". Even the definition of "definition" (if you allow me to quine for a bit) is a substitution of a long sequence of symbols with a smaller one, and substitutions are what equations are all about. I would argue with the submitter that Group Theory is not the simplest sort of abstraction (Category Theory is) but his point is still there: numbers and equations in

  • > 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.

    That sentence deserves to be taken out and shot.

    You may have had an interesting point but I'll never know - I stopped reading.
    • I'll stand by that sentence. It was years ago when I first studied group theory-- it was taught in a very standard format-- and I remember always thinking how the talk of symmetry was more confusing than helpful. Only in symmetry groups did the concept seem useful or relevant. Think about it: the usual lay-conception of symmetry doesn't work on, say, a non-abelian group. If you lay out the usual table of symbols and products, it isn't symmetric about the i=j line. What kind of symmetry exists besides visua

  • Well-hidden? (Score:5, Interesting)

    by slavemowgli ( 585321 ) on Friday November 18, 2005 @01:57PM (#14064942) Homepage
    "kept well-hidden"? Sorry, but that at least is utter rubish. No part of mathematics is kept well-hidden by anyone really; it's just that

    1. the general public isn't really interested in mathematics (unlike physics, for example; most non-mathematicians I've met seem to have an instinctive averse reaction when you even say "mathematics")

    2. mathematics, in general, cannot be dumbed down simplified for laypeople the same way that other natural sciences can. Someone can have a general idea of what a black hole is even when they don't understand the physical theories behind it, but how do you explain to a layperson what a Hilbert space is?

    Coupled together, these things mean that the general public isn't really aware of what mathematicians even study or why it's important to them, but it's not the fault of mathematics (or mathematicians).

    • Re:Well-hidden? (Score:3, Insightful)

      by Omestes ( 471991 )
      Since when was mathematics a natural science?

      It's like saying predicate logic is a natural science.
    • I'll assume you have an idea of what we mean by 1D, 2D and 3D spaces. An n-dimensional space is one where you can have up to n directions that are at right angles to each other. Well a Hilbert Space is a kind of infinite dimensional space. This is a space where no matter how many directions you have already chosen, you can find another direction that is at right angles to all of them.

      Now think about pairs of points in space. We can talk about there being a distance between them. Well the same is true in s

      • Hilbert spaces don't have to be infinite-dimensional.

        Outside of that... yes, that is a pretty good description of Hilbert spaces, but you'd probably lose most non-mathematicians I know by the time you said that there could be more than three dimensions.
        • Ooops. Slight mistake. But finite-dimensional Hilbert spaces are so boring. I might have lost most people but I'm sure your average pop science/science fiction reader could just about bend their mind towards the concept of always being able to travel orthogonally to any given bunch of directions.
  • Pi are round. Cake are square.

  • A few clarifications... (Score:4, Interesting)

    by Manchot ( 847225 ) on Friday November 18, 2005 @02:09PM (#14065064)
    Just so people don't get the wrong idea, it's not just quintic polynomials which can't be solved with one formula: it's all polynomials of degree five and higher. Also, "can't be solved" is something of a misnomer: there exist five solutions to a degree five polynomial, and they can be expressed either as infinite series or in terms of some non-standard functions. It's just that they can't be solved in terms of addition, multiplication, and exponentiation (i.e., using +, *, and radicals).

  • 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.

    If Abel was 22 when he made his discovery, that means he did it in 1824. If Galois was 20 when he made his, that means he did it in 1831. What if Galois actually discovered something else, something so powerful it could transport him seven years into the past, something like, oh,

  • Equation For Folding Paper in Half 12 times (Score:5, Interesting)

    by capitalj ( 461890 ) on Friday November 18, 2005 @02:26PM (#14065223)
    http://pomonahistorical.org/12times.htm [pomonahistorical.org]

    Britney Gallivan has solved the Paper Folding Problem. This well known challenge was to fold paper in half more than seven or eight times, using paper of any size or shape.

    The task was commonalty known to be impossible. Over the years the problem has been discussed by many people, including mathematicians and has been demonstrated to be impossible on TV.

  • But could they solve.. (Score:2, Interesting)

    by Unski ( 821437 )
    ..the equations of love?

    "It is only through the mysterious equations of love that any logic can be found."
  • ..but I need some inbetween math. I know algrebra and geometry. Can anyone recommend any websites for learning the maths about those?
  • Is there a place where all the groups that have names are listed, in a sort of catalog form? I keep finding references to various groups like SU(3) and the alternating group A5, and the names themselves give no sense of the structure of the given group. Once and for all I'd like to be able to read brief definitions of all of these named groups so when I see one I know what the heck they're talking about.

    Furthermore, is there a catalog of various named algebraic objects, like groups, subgroups, monoids, ri

  • Pretty graphs (Score:2, Informative)

    by trilliwig ( 558395 )
    Wolfram Research has some interesting explication on historical methods of solving the quintic: http://library.wolfram.com/examples/quintic/main.h tml [wolfram.com]

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