Tscott
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I figured since there's an active Math Jokes thread I'd give this a shot. I'm interested in finding some more interesting and accessible books on various math topics, or logic or just thinking in general. Also, I would be especially interested in any good books related to teaching math as I may be moving into this field into the near future.

First, some great books I've already read and would recommend to anyone else interested:

Godel, Escher and Bach: An Eternal Golden Braid - Douglas R. Hofstader
One of my favorite books ever. A fascinating look at many subjects relating to intelligence (artificial or otherwise), self-awareness and of course Godel's inconsistency proof.

Linked: How Everything Is Connected to Everything Else and What It Means - Albert-Laszlo Barabasi
Great introduction to a new field of mathematics, the study of complex networks. Insight into how the Internet, businesses, fads, and even diseases like AIDS are networked and connected.

Everything and More: A Compact History of Infinity - David Foster Wallace
The author of Infinite Jest himself takes on infinity and infinitesimals and the mathematicians that discovered the basic ideas behind them that lead to today's calculus. Very readable and insightful.

The Nothing That Is: A Natural History of Zero - Robert Kaplan
The flip side of the previous book, this one looks at zero and when and why early cultures accepted it (or not) in their concept of math.

Fermat's Enigma : The Epic Quest to Solve the World's Greatest Mathematical Problem - Simon Singh
If you've had a math class odds are you've heard "Fermat's Last Therom" at least mentioned. Gone unproved for centurys, despite Fermat's famous claim that "I have discovered a truly marvelous demonstration of this proposition which this margin is too narrow to contain." Finally proved in the last decade, this book gives an interesting history of math from ancient Greece to today in the context of this simple yet elusive problem and somehow never gets too dense for even a layman reader.

A couple I'm maybe thinking about getting are "Surely You're Joking, Mr Feynman" memoirs of the Nobel Prize winning physicist Richard P. Feynman or "Demon-Haunted World" by Carl Sagan, a skeptical look at pseudosciences in the same vein as James Randi and Penn & Teller's 'B.S.' Showtime series. So any suggestions of other books of interest in these areas?

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Oneiromancer
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I can't really offer anything new, but "Surely you're joking Mr. Feynman" is a great book. I highly recommend it. Made my ex wish that all physicists could be like him.

Game on,

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ErikH2000
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A couple I'm maybe thinking about getting are "Surely You're Joking, Mr Feynman" memoirs of the Nobel Prize winning physicist Richard P. Feynman

I really love this book. If I am feeling tired and world-weary, it is a wonderful thing to pick up and read. Not that this is a self-help book or I particularly go in for that kind of thing, but Richard Feynman is an excellent role model to learn from. As far as math or science content, there is very little about that except from a metaphysical ("Why is science a good thing? What kind of thinking is scientific?") or historical context (Feynman's impressions of guys like Oppenheimer, his work at Los Alamos on nuclear fission, his investigation of the Challenger explosion). It is an entertaining and enlightening book and not because Feynman was some great writer, but because he was an amazing person that interesting things happened to.

My apologies for going on at length over something that wasn't particularly asked about. I just can't recommend this book enough.

-Erik

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Malarame
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My 12th grade math teacher made us all buy and read An Imaginary Tale, the Story of i. I don't remember who it was by, but it was a great book about the history of one of the most unique and interesting numbers.

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Tscott
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Thanks to everyone so far. I think I'll move Mr. Feynman's book to my definite buy list. I was soooo close to buying that the last time I was at a book store but I didn't- probably had something to do with the fact that I was supposed to be looking for Christmas gift ideas for people other than me.

"An Imaginary Tale" looks interesting and would seem to be a logical progression from the books on infinity and zero. I see they've got it at Amazon where they offer a link to buy it together with "e: The Story of a Number".

I'm still open to any other ideas anyone else may have.

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The Colossal Book of Mathematics by Martin Gardner
is a collection of the best of the author's Scientific American articles. Definitely worth checking out some time.

Proofs from the Book by Martin Aigner and Gunter Ziegler
has a nice collection of elegant proofs. Perhaps a bit too mathy for the armchair mathematician, but good nonetheless.

Flatland by Edwin Abbott
is a neat book that delves into logic of the fourth dimensions and also contains a biting satire of Victorian society. And if you read that, you'll have to read

Flatterland: Like Flatland, Only More So by Ian Stewart
which introduces all sort of concepts from math and physics (if only you can deal with all the awful puns). If you still haven't had enough with 2D worlds, you can also check out

The Planiverse by A.K. Dewdney
which tells the story of a creature in a 2D society, so you can learn how to build a flat steam engine.

Winning Ways for your Mathematical Plays by Elwyn Berlekamp, John Conway, and Richard Guy
is the resource for those interested in combinatorial game theory. This was recently rereleased in four volumes. And another important book on this subject is

On Numbers and Games by John Conway
from which one might say the whole of combinatorial game theory arose.

Oh, I'm sure I know some more good ones, but I can't think of them right now...

[Edited by bibelot at Local Time:12-15-2004 at 12:25 AM]

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There is one really good book I would recommend if you have never read it:

What is the name of this book? by Raymond Smullyan
is a book about mathematical logic, but written in a style that anyone (and I mean anyone!) can read and understand it. It has tons of logic puzzles in it, and it ends with Gödel. But the best parts of the book are the mathematical jokes and the paradoxes.

Not sure whether the book is still being sold, but you can always consult your local library.

-- Tim

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ErikH2000 wrote:
I really love this book. If I am feeling tired and world-weary, it is a wonderful thing to pick up and read.

I'm with you here. I also enjoyed the "sequel", What do you care what other people think?

[Edited by mrimer at Local Time:12-15-2004 at 04:34 PM]

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I think the only popular mathematics book I have ever read was Singh's "Fermat's Last Theorem"; I enjoyed it, I seem to remember. If you want to go a bit more hardcore, I'm currently reading "Principles of Mathematical Analysis" by Rudin. Difficult, but very interesting.

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Tscott
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Tim wrote:
There is one really good book I would recommend if you have never read it:

What is the name of this book? by Raymond Smullyan
is a book about mathematical logic, but written in a style that anyone (and I mean anyone!) can read and understand it. It has tons of logic puzzles in it, and it ends with Gödel. But the best parts of the book are the mathematical jokes and the paradoxes.

Not sure whether the book is still being sold, but you can always consult your local library.

-- Tim

I used to work at my public library while in high school and I tore through Smullyan's books while there. It's been a long time but I recall a number of his puzzles dealt with asking questions of people who always lied or always told the truth. He had quite a few books, though sadly many of his older ones are out of print and have been for a while, including "What is the name of this book?". "This Book Needs No Title" looks to be available though. Another one called "The Riddle of Scheherazade" looks interesting too.

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Mattcrampy
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Wait, wait... they proved Fermet's Last Theorem?

Man, I've been away from maths a looooooong time.

Matt

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wackhead_uk
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Here's something weird that I confused people with a few days ago, which I happened to relise in a maths lesson:

0 x ∞ = -1

In a graph of y=ax^n, the gradient at any point is nax^n-1.

On the graph of y=3, nax^n-1 = 0. Therefore, the gradient of this line is 0.
On the graph of x=3, nax^n-1 = 3/0 = ∞ (infinity).

In two perpendicular lines, the gradients multiplied together = -1.
y=3 and x=3 are perpendicular…

Therefore 0 x ∞ = -1

agaricus5
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wackhead_uk wrote:
On the graph of x=3, nax^n-1 = 3/0 = ∞ (infinity).

I think that's the problem. You can rewrite "x = 3" as "0y = 3 - x"

The question now is, can you multiply 0 by 0?

The assumption you've made is that you can, which yields this:

y = 0 - 0x = 0 = (-1 * 0 * x)

And so, dy/dx = (0 * -1 * 1 * x)^-1 = -1/0x = -1/0 = ∞

This will then give the result you mentioned:

0 * ∞ = 0 * (-1/0) = -1

Edit: To avoid being too off-topic, this reminds me of another book that's very well worth reading (especially if you love infinity, and want to grapple with some pretty advanced Set Theory:

"The Art of the Infinite" by the author of "The Nothing That Is", Robert Kaplan.

[Edited by agaricus5 at Local Time:12-19-2004 at 10:50 PM]

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Both of those Feynman books own. I read "Surely, you're joking, Mr. Feynman" like 30 times, And it still amazes me. The best stories in that book are "Who stole the door?" "Los Alamos from below." and "Safecracker meets Safecracker." a lot of other people already told you to buy the book, so yeah...

Another good book I have if you like to solve puzzles that involve a lot of thinking, get "1000 Playthinks." by Ivan Moscovich.

Tscott
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I took everyone's advice and bought 'Surely You're Joking...' I'm on page 66 already and laughed out loud at the who stole the door story as well as a few other places. I don't know how this book flew under my radar for so long, as this just the sort of thing I was looking for when I read so much while working in the library in high school (way back in 1988-89).

I'll keep coming back to this thread as I look for more material next year and beyond.

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The Penguin Dictionary of Curious and Interesting Numbers D.G. Wells (editor)

It's a dictionary of numbers and their interesting properties, arranged in numerical order (starting from -1 & i) It offers only the briefest of glances at some of the topics covered by many of the other recommendations here, but is still worth a look for an enthusiast, and is of course easy to dip into. I particularly enjoyed the entry for 39.

bibelot
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Not a book, per se, but a neat reference nonetheless:

Sloane's Online Encyclopedia of Integer Sequences
http://www.research.att.com/~njas/sequences/

has basically every integer sequence you'd ever want to know, so if you ever have a sequence and you don't know where it came from, there's a good chance it'll be here.

wackhead_uk
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I think I read somewhere that 12345678901 was a prime number...

just making conversation.

...

agaricus5
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wackhead_uk wrote:
I think I read somewhere that 12345678901 was a prime number...

Actually, it isn't.

12345678901 = 857 * 14405693

1234567891, 12345678901234567891 and 1234567891234567891234567891 are prime, however. (From The Penguin Dictionary of Curious and Interesting Numbers, D.G. Wells (editor)

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Malarame
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wackhead_uk wrote:
Here's something weird that I confused people with a few days ago, which I happened to relise in a maths lesson:

0 x ∞ = -1

In a graph of y=ax^n, the gradient at any point is nax^n-1.

On the graph of y=3, nax^n-1 = 0. Therefore, the gradient of this line is 0.
On the graph of x=3, nax^n-1 = 3/0 = ∞ (infinity).

In two perpendicular lines, the gradients multiplied together = -1.
y=3 and x=3 are perpendicular…

Therefore 0 x ∞ = -1

That's incorrect. The slope (gradiant) of x=3 isn't infinity, it's undefined. You can't divide a number by zero. And zero times infinity is also undefined, same as zero over zero and infinity over infinity, and so on.

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agaricus5
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Malarame wrote:

wackhead_uk wrote:
Here's something weird that I confused people with a few days ago, which I happened to relise in a maths lesson:

0 x ∞ = -1

In a graph of y=ax^n, the gradient at any point is nax^n-1.

On the graph of y=3, nax^n-1 = 0. Therefore, the gradient of this line is 0.
On the graph of x=3, nax^n-1 = 3/0 = ∞ (infinity).

In two perpendicular lines, the gradients multiplied together = -1.
y=3 and x=3 are perpendicular…

Therefore 0 x ∞ = -1

That's incorrect. The slope (gradiant) of x=3 isn't infinity, it's undefined. You can't divide a number by zero. And zero times infinity is also undefined, same as zero over zero and infinity over infinity, and so on.

Well, if you ask yourself "how many times will zero go into any number?", I'm sure you'll say that it will go into it as many times as you like, since taking zero from a number doesn't affect it. At "infinity" you could argue that an infinite number of 0s taken from 1 would still leave 1, but then you could still keep going, beyond "infinity". I personally believe that at the absolute infinity, taking all those zeros from 1 will finally reduce it to 0, since at that infinity, there can be no other infinities beyond it, so this process must "stop" at some point.

In the words of Robert Kaplan, from "the Art of the Infinite", I quote:

When it does - when the doors of our perception are finally cleansed, as William Blake promised - then everything will appear as it is: infinite.

But which infinity will we see?

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TripleM
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Something of the form 0 times infinity should really be called interdeterminate, because it could equal anything depending on the circumstances.. for example, multiply x by (pi/x). You always get pi. As x goes to infinity, "infinity * 0 = pi!". Its all about limits really.

Malarame
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TripleM wrote:
Something of the form 0 times infinity should really be called interdeterminate, because it could equal anything depending on the circumstances.. for example, multiply x by (pi/x). You always get pi. As x goes to infinity, "infinity * 0 = pi!". Its all about limits really.

I stand corrected. It is indeterminate, not undefined. Thanks for pointing that out.

agaricus5 wrote:
Well, if you ask yourself "how many times will zero go into any number?", I'm sure you'll say that it will go into it as many times as you like, since taking zero from a number doesn't affect it. At "infinity" you could argue that an infinite number of 0s taken from 1 would still leave 1, but then you could still keep going, beyond "infinity". I personally believe that at the absolute infinity, taking all those zeros from 1 will finally reduce it to 0, since at that infinity, there can be no other infinities beyond it, so this process must "stop" at some point.

You can ask yourself whatever you want, and you can argue all you want, but that doesn't change the fact that you can't divide things by zero. Anything divided by zero is undefined. You can take a limit of something over x as x approaches zero from one side or the other, and that will evaluate to positive or negative infinity, but anything divided by zero itself is undefined.

You also have to remember that there is no such thing as infinity. It's a concept, an idea; it's not an actual number or place. You can't manipulate infinity like you would any other number (students often make the mistake of saying something like "infinity minus infinity equals zero!" or "infinity divided by infinity equals one!").

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Malarame wrote:
You can't manipulate infinity like you would any other number (students often make the mistake of saying something like "infinity minus infinity equals zero!" or "infinity divided by infinity equals one!").

Well...I might argue with that last statement, but perhaps it's because physicists aren't always mathematically rigorous. I mean, x/x = 1 should be true for all values of x (except for exactly 0 I suppose), since they are the exact same number by definition. I would agree that saying x/y = 1 for x -> infinity and y -> infinity is incorrect though.

Game on,

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eytanz
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Well, that's the point - x/x = 1 for every value of x.

Infinity isn't a number, and hence can't be a value. So, the previous statement doesn't apply.



[Edited by eytanz at Local Time:12-22-2004 at 03:23 AM]

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I've heard that there's a System that includes real number system
and infinity too. But I almost forgot all about it.
Is there any web that talk about it?

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Strabo
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:

:

That's incorrect. The slope (gradiant) of x=3 isn't infinity, it's undefined. You can't divide a number by zero. And zero times infinity is also undefined, same as zero over zero and infinity over infinity, and so on.

Well, if you ask yourself "how many times will zero go into any number?", I'm sure you'll say that it will go into it as many times as you like, since taking zero from a number doesn't affect it. At "infinity" you could argue that an infinite number of 0s taken from 1 would still leave 1, but then you could still keep going, beyond "infinity". I personally believe that at the absolute infinity, taking all those zeros from 1 will finally reduce it to 0, since at that infinity, there can be no other infinities beyond it, so this process must "stop" at some point.

Nice try. There is no 'ultimate' infinity. The infinty you all know and love is the first infinity, aleph-0(null). This is the infinity of rational numbers.
There's also aleph-1, the infinty of irrational numbers. This is different from aleph-0, and the two cannot be placed in one-to-one correspondance with each other. Aleph-1 is 'bigger' in a sense, than aleph-0. Likewise we have aleph-2, and aleph-3...infact there is an neverending number of these infinities..

Wierd. huh?

Anyway,I enjoy 'Curious And Interesting Numbers' as well. I like to read a number or two every so often.

Is it just me, or do we have a higher percentage of mathsy types in here than you'd normally expect? Does DROD attract mathy people?

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So there is such a thing as infinity plus one! And infinity plus 2! A whole new world of one-upsmanship can start with your younger sibling...

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Strabo wrote:
Nice try. There is no 'ultimate' infinity. The infinty you all know and love is the first infinity, aleph-0(null). This is the infinity of rational numbers.

Aleph-0 is the set of all the ordinal numbers from 1-ω, where ω stands for the number that comes after all the other ordinal numbers, that which we call "infinity". This is the first "transfinite" cardinal, which is the actual size of the whole set of numbers.

There's also aleph-1, the infinty of irrational numbers. This is different from aleph-0, and the two cannot be placed in one-to-one correspondance with each other. Aleph-1 is 'bigger' in a sense, than aleph-0. Likewise we have aleph-2, and aleph-3...infact there is an neverending number of these infinities..

Isn't aleph-1 actually the size of this infinite set of "infinities"?

[ω, ω+1, ω+2,... ω+ω,... ω^2,... ω^ω]

If we call this sequence "Ω" as a short form, then aleph-2, according to Cantor, can be defined as the size as the set of the numbers:

[Ω, Ω+1,... Ω+Ω,... Ω^2,... Ω^Ω]

Aleph-3 and further alephs from that point on supposedly follow this "sequence, leading to an infinity of infinities, the "endpoint" of which would be the absolute infinity, something which is probably impossible to comprehend in any way.

Is it just me, or do we have a higher percentage of mathsy types in here than you'd normally expect? Does DROD attract mathy people?

That's a good question.

You may wish to refer to this thread to find older threads that cover various topics about many aspects of DROD:

A Cultured Eighth

Specifically related to this question, these threads should interest you:

Math gurus go! Or: How to play DROD

DROD Tendency

I'd say that the major appeal of DROD is its highly logic-based syle of gameplay, which attracts people who enjoy logic puzzles, who are likely to be logical people. Many logical people are scientists or mathematicians, for whom logic is a necessity, so therefore, I guess this is the resultant effect that you're seeing.

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agaricus5 wrote:
Aleph-0 is the set of all the ordinal numbers from 1-ω, where ω stands for the number that comes after all the other ordinal numbers, that which we call "infinity".

I'm having trouble accepting wording of this definition as it makes it seem that ω is the "biggest number" when infinity is actually the absence of a "biggest number". This is where most people seem to get hung up when trying to understand this concept. I think it's better to say ω is the set of all finite ordinals {1,2,3,...}. (For more ordinal fun here a fun, formula filled page that looks at them.

Also, it's probably better to say that Aleph-0 is isomorphic to the set of all the ordinal numbers from 1-ω as there's an infinite number of sets that are of this order of infinity.

The natural numbers {1,2,3,...}
The whole numbers {0,1,2,3,...}
The integers {0,-1,1,-2,2,-3,3,...}
Positive even numbers {2,4,6,8,...}
Positive odd numbers {1,3,5,7,...}
The Primes {2,3,5,7,...}
All the fractions between 0 and 1 {1/2,1/3,2/3,1/4,3/4,1/5,2/5,...}
and so on.

Those (and more) can all be shown to have a one-to-one correspondance with the Aleph-0. The fractions between 0 and 1 is an especially good one to try and grasp as it gives an example of an infinite set where the numbers are bounded by a finite number, i.e. no number of this infinite set is ever bigger than 1- yet there is no "biggest fraction" of this set.

------
Oh and I just picked up The Millennium Problems by Keith Devlin, which looks at the 7 most important unsolved problems of our time. I've only read the first part on the the Riemann Hypothesis but it's pretty good so far (and I'm still reading the Feynman book). I'm thinking of picking up Devlin's The Language of Mathematics next.

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