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.

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."

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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.

As Barbie says

[Anonymous Coward]

Math is hard!

Re:As Barbie says

[$RANDOMLUSER (804576)]

The actual quote was "English is easy. Math are hard."

The real geek equation...solved!

[dada21 (163177)]

Shower^2 + Shave + BrushTeethx32 + Get(Own(Apartment)) + not(sqr(Clothing)) = Women

Re:The real geek equation...solved!

[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!

[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

favorite math quote

[flynt (248848)]

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.

Re:favorite math quote

[vossman77 (300689)]

see reply to previous quote [slashdot.org]

"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."

CN Yang, Nobel Prize in Physics, 1957

Galois

[Otter (3800)]

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

[Anonymous Coward]

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

[Beryllium Sphere(tm) (193358)]

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

[ozbird (127571)]

"Guns don't kill people; maths kills people."

Duel Staged; death by suicide more likely

[Intelligent Design (909311)]

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

(-(p/4))^1/4BR((((-5/p)^5/4)q)/4)

[digitaldc (879047)]

"...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.

Re:(-(p/4))^1/4BR((((-5/p)^5/4)q)/4)

[miskatonic alumnus (668722)]

How is this Bring Radical an algebraic function? It's defined as an analytic extension of an infinite series.

Re:(-(p/4))^1/4BR((((-5/p)^5/4)q)/4)

[zootm (850416)]

Computational complexity scares the living daylights out of everyone.

Curious and interesting numbers

[Skiron (735617)]

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

[TheoMurpse (729043)]

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.

Cool!

[dorkygeek (898295)]

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!!

Math is all about the dupes

[everphilski (877346)]

I mean, come on, how many times have you seen "x"

or "cos()" "sin()"

-everphilski-

Re:Math is all about the dupes

[dorkygeek (898295)]

That's the very reason I always fell asleep during math lectures. It's always the same, day in and day out. It's always about x and y.

Sometimes s.

[everphilski (877346)]

(Perhaps you slept through Laplace transforms?)

-everphilski-

How is that called?

[n0other (907866)]

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.

Re:How is that called?

[Hiro Antagonist (310179)]

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

[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

[flynt (248848)]

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

[jfengel (409917)]

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.)

Re:if you want to learn a bit about group theory

[jfengel (409917)]

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

[fossa (212602)]

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

Group Theory Joke

[keithmo (453716)]

Q: What's purple and commutes?

A: An Abelian grape.

Hmmm.

[jd (1658)]

What happens if you ferment a bunch of Abelian grapes in a Klein bottle?

Re:Hmmm.

[tehshen (794722)]

You get an Abelian Soup.

Re:Hmmm.

[IntelliTubbie (29947)]

What happens if you ferment a bunch of Abelian grapes in a Klein bottle?

I suppose you'd get very disoriented.

Cheers,
IT

The answer is trivial

[abb3w (696381)]

What happens if you ferment a bunch of Abelian grapes in a Klein bottle?

You'll make an Algebraic Topologist whine.

Re:Group Theory Joke

[Kid Brother of St. A (662151)]

Q: What's purple, commutes, and is worshipped by a small cult of followers?

A: A finitely venerated abelian grape.

Quadratic Equation

[sameerdesai (654894)]

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

Re:Quadratic Equation

[farker haiku (883529)]

Sure, just because you actually read what the poster wrote, you get the easy karma points.

Re:Quadratic Equation

[servognome (738846)]

x = (-b(+/-)sqrt(b^2-4ac))/2a

Is it me or is 1337 sp3ak getting even harder to understand :)

Pop Math?

[Anonymous Coward]

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.

Re:Pop Math?

[richieb (3277)]

Funny, but we were actually talking about hyperbolic space at lunch. And we're just a bunch of Java programmers ... ;-)

Solve this...

[Reality Master 101 (179095)]

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.

Re:Solve this...

[podperson (592944)]

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 ?

Basic definitions without equality?

[nightlylemma (917760)]

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.

Look ma, no equations!

[Leadhyena (808566)]

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

Well-hidden?

[slavemowgli (585321)]

"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?

[Omestes (471991)]

Since when was mathematics a natural science?

It's like saying predicate logic is a natural science.

A few clarifications...

[Manchot (847225)]

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).

Equation For Folding Paper in Half 12 times

[capitalj (461890)]

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.

Re:Solve x = x+1 over the reals

[Anonymous Coward]

you know you could just use x++

Mod down, same kaleidojewel spam as always

[Anonymous Coward]

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.

Re:Slashdot review rating equation solved

[SMS_Design (879582)]

Good answer, but you lose points. You forgot to simplify the fraction. 4/5.