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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.
Math is hard!
or complex numbers :)
Shower^2 + Shave + BrushTeethx32 + Get(Own(Apartment)) + not(sqr(Clothing)) = Women
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, 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...
Hahahahaha! Nice one!
Computer Go: Writing Software to Play the Ancient Game of Go
RATING = 8/10
"...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
It's really that simple.
He who knows best knows how little he knows. - Thomas Jefferson
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.
Cool, the first book with dupes already integrated!!
Windows is like decaf - it tastes like the real thing, but it won't get you through the day.
Athletic Scholarships to universities make as much sense as academic scholarships to sports teams.
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.
DevBlogs
I typed in the ISBN into Google. Google told me 0 - 7432 - 5820 - 7 = -13259 Simple.
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!
"Worth the few bucks", or maybe a trip to the library?
Q: What's purple and commutes?
A: An Abelian grape.
I mean, come on, how many times have you seen "x"
or "cos()" "sin()"
-everphilski-
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
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.
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.
Set n=2.4879391731181746675433584949641.
4^n + 5^n = 6^n seems to hold and x,y,z are non-zero integers.
Experienced readers will revisit a familiar topic.....
I'm an experienced reader, but that doe not mean that this book makes sense !
What happens if you ferment a bunch of Abelian grapes in a Klein bottle?
It's a small world and it smells funny; I'd buy another if it wasn't for the money; Take back what I paid (SoM)
x= (b(+/-)sqrt(b^2-4ac))/2a
Shouldn't that be "x= (-b(+/-)...)?
math is good and all but where are the books for students with dyscalculia and taking finite math?
n is generally assumed to be a natural number.
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.
Sometimes it's best to just let stupid people be stupid.
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:
- For each x in G, x*e=e*x=x.
- 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.Wikipedia reckons that Galois's last words were "Don't cry, Alfred! I need all my courage to die at twenty." I'm sure that I read some where they were "What a wast to die over some stupid bint"
Hah! Nice try whore, I know a referral link when I see one.
(Perhaps you slept through Laplace transforms?)
-everphilski-
You'll save more by shopping via the 'bot at AddAll.com. You can usually get the book + shipping for less than the "discounted" book price (alone) at Amazon.
If you don't want to do it, send me the Amazon money and I'll send you the book when I get it from one of the less expensive outlets [and I'll keep the difference].
I have yet to understand why everyone heads to Amazon or B&N like a dog in heat. Do your research there, then save yourself money elsewhere.
> 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.
"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).
quidquid latine dictum sit altum videtur.
Pi are round. Cake are square.
No folly is more costly than the folly of intolerant idealism. - Winston Churchill
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).
Work = FD
F = MA
Work = MAD
--
"Open source is good." - Steve Jobs
"Open source is evil." - Microsoft
http://pomonahistorical.org/12times.htm
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.
x,y,z are generally assumed to be real numbers, but that didn't stop Fermat.
"At last, less-experienced readers will find a warm entry-way. . ."
.into one of the most fascinating and advanced branches of mathematics"
Oh God, finally!
". .
Doh!
Concepts in group theory are actually used quite a bit in chemistry. Molecules can be assigned to groups based on their symmetry properties, and these groups can be used to predict the molecules' spectroscopic properties. This is extremely handy in inorganic chemistry when you want to determine (for example) how many different infrared absorptions a metal complex should have.
The lingo of group theory has firmly entered chemistry as well. Even organic chemists will routinely talk about molecules with a C2 axis, systems with D3h symmetry, etc.
The book to read on this topic is "Symmetry and Spectroscopy" by Harris and Bertolucci. It's published by Dover, so it's cheap.
You'll make an Algebraic Topologist whine.
//Information does not want to be free; it wants to breed.
..the equations of love?
"It is only through the mysterious equations of love that any logic can be found."
ex/f What's the big deal?
.. just integrate the terms of the Taylor series ;-)
I'm still trying to figure out what people mean by 'social skills' here.
"There are 10 types of people in this world - those who understand binary and those that don't."
/. sig... but it keeps coming back.
Fortunately this is no longer the most popular
Here's a less-known, related riddle, though:
If only you and dead people understand hex, how many people understand hex?
There's the simple numeric answer, of course... I prefer: "Well, if I taught just one more person, we'd all be deaf."
Heh, heh.
..but I need some inbetween math. I know algrebra and geometry. Can anyone recommend any websites for learning the maths about those?
Coder's Stone: The programming language quick ref for iPad
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, rings, fields, etc.?
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.
Sorry, don't buy it. Sure, telling people what a Hilbert Space is, is quite hard. But try explaining to a layperson (is that a politically correct term ;-)) that an electron is actually, well, you know, like a wave and a particle at the same time isn't that easy either. Wasn't it Bohr himself who said that either you didn't understand quantum mechanics, or you were insane?
Or try to explain to a layperson how quarks react...
Even special relativity, mathematically a quite simple theory, is hard to explain to a layperson... So it all depends on what you want to achieve. If all you need to do is scratch the surface, well, for a layperson you could use something like the jpeg file format: explaining that a picture can be represented in a series of 'functions' etc...
Or compare it do a euclidian space...
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.
Now think about pairs of points in space. We can talk about there being a distance between them. Well the same is true in some infinite dimensional spaces, there's an idea of the distance between two points. And if you have an idea of distance you have an idea of how close a pair of points is.
Now think about a sequence of numbers like 1,1/2,1/4,1/8,... halving each time. Notice that successive pairs of points are getting closer and closer all the time. The distance between 1/2 and 1/4, say, is half that between 1 and 1/2. Now the fact that they appear to be getting closer and closer to their neighbors suggests that actually the sequence is getting closer and closer to something. In the example just given you can see that the sequence is getting closer and closer to zero. It never reaches zero but it does get as close as you choose. But there are spaces where it's possible for successive pairs to get closer and closer without there being an actual limit that they approach. They're kinda annoying and it's often useful to restrict your attention to cases where whenever you appear to be approaching a limit there really is a definite limit. Infinite dimensional spaces like this are known as 'Hilbert Spaces'.
Hilbert spaces are very useful in modern physics where the state of a physical system in quantum mechanics is often infinite dimensional. But they also have many applications outside of physics.
I could give some examples but I have a day job. And yes, I know I haven't said what I mean by 'space', but this a layman's introduction. Next week I'll give an introduction to functional analysis on Sobolev spaces for the layman...
Wolfram Research has some interesting explication on historical methods of solving the quintic: http://library.wolfram.com/examples/quintic/main.h tml
My point was that you don't have to explain all the details in physics to give a meaningful and useful approximation of what things are like. You might not be able to explain all the details, but it often will be enough - if you are just interested in learning what water molecules look like (for example) and why they can form like this, you don't have to know about dual wave/particle nature of elementary particles. Sure, it'll become important at some later point - you can't really understand aromatic structures in chemistry without this, for example -, but for a layperson, it's good enough.
The same thing is not true in mathematics: no matter how simple a problem looks (that is, no matter how easy it is to formulate), the solution can be arbitrarily complex. Why can there be no natural numbers a, b, c (greater than 0) such that a^3 + b^3 = c^3? It's a question that everyone can understand, yet the solution is too complex even for most mathematicians to understand (I have a background in mathematics, and I tried to read it once; I had to stop after less than two pages or so).
Another problem with mathematics is that most people don't know the basic building blocks of mathematics. I think most reasonably intelligent people have a general idea of what molecules are, for example, but if you start talking about functions, or spaces, or groups, you'll immediately lose most people.
quidquid latine dictum sit altum videtur.
I bet Charlie Eppes could solve it in no time!
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 their connotative or layman's sense are not involved.
Group therapy, for all the geeks who get off on numbers.
she is;)
A modest semi-classic? With a recomendation like that I think I'll put it at the top of my reading list.
Ooo man the floppy drive is broken. No wait. The computer is just upside down.
What is the regular definition of "dimension", though? The dimension of a vector space at least is not the most basic concept you'll come across in that field - it's not immediately clear why you couldn't have two bases with different cardinality, for example. Or, for that matter, it's not immediately clear why there has to be a basis at all (and in fact, you have to use the axiom of choice to prove that this is always the case; if you don't accept it, there are vector spaces without a basis and thus without a defined number of dimensions, strange as that may sound).
And the Hausdorff dimension isn't really better, either.
Of course, all these things are basic (or at least easily grokkable) for a mathematician, but for laypeople, they may not really mean anything. I remember giving a talk about fractals and the fractal dimension in my math class at school once, in the last year, but most people couldn't even accept the fact that a dimension might not be an integer. For them, a dimension was not something you could define, but rather something that existed, in the sense that we have three (or four, if you count time) dimensions we live in, and nothing else. I doubt they would've been comfortable with hearing about things like infinite-dimensional vectorspaces, either. And that *was* in a math class.
quidquid latine dictum sit altum videtur.
The point is that there is no universal formula of the required form for all quintics and above.
See here for more details! It's unfortunate that the urban legend made it into one of my group theory textbooks.
"sweet dreams are made of this..."
You know. You lost me at cardinality. Then I looked up the definition and realized that the cardinality is dimension. So Im guessing you are trying to say that it's not entirely clear that the dimension of a subspace can be given of the minimum number of vectors that span said subspace. Please tell me if I have butchered the definition because honestly I was just taught this yesterday. Also, I believe my teacher never showed us the proof.
Ooo man the floppy drive is broken. No wait. The computer is just upside down.
Handbook of Mathematics and Computational Science
by John W Harris and Horst Stocker
ISBN:0387947469
if anyone needs a complete reference to any formula, theoreme, theories and details in an extremely syntesys+example way, to get up to Galois' groups for quintic equations and much, much more
by far the most fascinating non-literature piece of paper I've ever had in my hands
The dimension of a subspace is the minimum number of vectors that span said subspace, yes (at least, that's one definition of several equivalent ones), but you have to show that that minimum is actually well-defined - for example, there is no a priori reason why I couldn't find one set of four linearly independent vectors that span said subspace and another set of five, where neither set spans the entire subspace anymore if I remove one of the vectors. In this example, it would not be clear whether the dimension of the subspace should be four or five.
:)
This really cannot happen, but it's important to realise that you need to prove this, as it doesn't immediately follow from the definition - you need Steinitz for this (or you could use the more general ultrafilter lemma, too).
quidquid latine dictum sit altum videtur.
Paolo Ruffini made extremely significant contributions to this problem before Abel was even born, only to be largely ignored by the leading mathematicians of the day:
m aticians/Ruffini.html
http://www-groups.dcs.st-and.ac.uk/~history/Mathe
0743258307=3*11*79*285101
I'll be your candy shop of infinite deliciousity if you'll be my discotheque of endless rump-shaking.
WTF is this crap? Do Physicists really use abstract algebra? I question the validity of the summary's statement... The person who wrote the summary seems to have a naieve view that physics people are the only ones who really use the "hard" maths... (And they would be sorely mistaken).
For example, abstract algebra is extremely important in many areas of Electrical Engineering and Computer Science. (Although both majors have sub-areas cover a fairly extreme range from "using no math at all", to the almost unspeakable branches...)
Still not convinced. True, in physics there are many things which the layperson can see for him/herself. ;-)
;-)
So you can tell something about it. But the same is true about mathematics.
You argue yourself that the mathematical problem mentioned above is easy to understand... The proof is not.
But you could easily give people an idea by substituting '1' and '2' for a and b; giving 1^3 + 2^3 = c^3 where c is an irrational number...
And that there are many more irrational numbers than natural numbers, and that it is in fact quite special that a^2 + b^2 = c^2 has so many solutions for a,b,c natural numbers... You could explain something about the number of degrees of freedom. And then conjecture something about solutions to a^3+b^3+c^3=d^3...
Proving IS something quite different... In physics, it is quite diffucult to PROVE to a layperson why 'general relativity theory' and 'quantum theory' is so difficult to combine. Or tell people about why dissipation and quantum mechanics are hard to combine, at least in a non-phenominological description. (It is possible, I know, my own PhD thesis is about that combination; it is just not straightforward).
About your 'building blocks' of physics: molecules are more like building blocks of chemistry than of physics. Physics is on every scale: from supernovae to quarks inside protons inside atoms inside molecules. No way a layperson would have a general idea about those.
In mathematics, natural numbers is one of the big areas of study. So in a sense natural numbers are part of the building blocks of mathematics...
Do you really believe natural numbers are difficult to understand for laypersons
If I haven't convinced you now, I'll probably not convince you in a further post. So no more postings from me on this subject
...and they are divided into three classes: those who learnt to count to three, and those who are not.
"..."Physics is to math what sex is to masturbation."..
Richard Faynman
The answer is: 42
It is not unsolvable, it would just take a super computer just over 7 millions years to compute.
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