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In a recent diary I introduced this series on math.  This is the first of the regular diaries.  We will explore some basic properties of prime numbers, including Euclid's proof that there is an infinite number of them.  It's a beautiful proof, and doesn't require any advanced math to understand.  In fact, you just need the four basic arithmetic operations: Addition, subtraction, multiplication, and division (you really don't even need subtraction).  

I'll also talk a little about arithmetic, number theory, and esthetics.

This diary series is intended to be understandable by people without a heavy math background.  Others who wish to post a diary in the series are welcom, but please ask me.  If you have an idea for a topic, please let me know. And if you want to help explain stuff in comments, feel free, but be nice.  There are no dumb questions.

So join me below the fold

The most basic mathematical objects are what's known as the counting numbers (aka the natural numbers, or, for those who like fancy words, the positive integers - some people use `natural numbers' to include 0, but we're talking about just the positive ones).  These are the numbers you count with (we'll discuss other numbers in later diaries): 1, 2, 3, ....
The ellipsis (...) is math notation for `and so on'.  Kids learn early that there is no biggest number - there are an infinite number of numbers. In a later diary, we'll talk about some interesting things about infinity, but, for now, let's just say that it means that, no matter how big a number you name, I can name a bigger one.  Some kids go through a period where they are fascinated by large numbers - when kids learn I like math, some start asking questions like "What's 1 billion minus 1 million?"  or "What's the biggest number with a name?"  

Mathematicians spend a lot of time thinking about these positive integers.  When they do this, they are usually engaged in number theory, which is sometimes called the `higher arithmetic'.  Some of this is very abstract and difficult, and we won't go there.  But number theory is the branch of math where it is easiest for novices (even kids) to ask questions that have everyone stumped.  It's also a branch of math where there are a lot of questions that have interesting but understandable answers.

The counting numbers can be divided into different groups in lots of ways: Some are odd, some are even.  Some are square (that is, numbers that are a number times itself 1, 4, 9, 16 and so on) and some are not.  One of the most important ways to categorize numbers is as primes or composites.  A prime number is one that can't be evenly divided by any number but itself.  A composite number is one that isn't prime.  So
2 is prime
3 is prime
4 is composite (2 x 2)
5 is prime
6 is composite (3 x 2)
And so on

Primes are important for lots of reasons, but one of the biggest is that every positive integer can be factored into primes in one and only one way.  So:
2 = 2*1
3 = 3*1
4 = 2*2
28 = 7*2*2
And so on.

The first primes are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37   

They seem to be getting less common.  Of the first 10 positive integers, 5 are prime. Of the next 10, 4 are prime.  Of the next 10 (in the 30s) only 2 are prime.  Of the first 100, 25% are prime; of the first 1,000, 16.8%; of the first 1,000,000, 7.8%.  Do they ever stop?
No.  They don't.  This has been known for a very long time, since Euclid in Greece.  He devised a proof.  It's a kind of proof that's very common in math, called reductio ad absurdum (reduce to absurdity).  In it, we first suppose that what we wish to prove is false.  Then we derive a contradiction (these proofs are also known as proofs by contradiction).  

Theorem: There is no largest prime number
Proof:  Suppose there is a largest prime number.  Call it X.  Now write down all the prime numbers smaller than X.  (In math, the primes are often labeled as p with a subscript).

P1, p2, p3 and so on, until you get to X, the supposed largest prime

Multiply all these numbers together. Add one.  So, suppose we thought the largest prime was 7 (it's ridiculous, but I am just illustrating a point), then write down the primes smaller than that or equal to that

1, 2, 3, 5, 7

Multiply them
1*2*3*5*7 = 220.
Add 1 = 221.

Let's call this number M.  Clearly, M is bigger than X.

Now there are two possibilities: Either A) M (221) is prime or B) M is composite.  (We know there are no other possibilities because every number is either prime or composite)

If A is true, then we have found a bigger prime than X.
If B is true, then M can't be divisible by any of the primes we used, because we added one.  So, if we divide by 2, the remainder is 1.  If we divide by 3, the remainder is 1.  If we divide by 5, the remainder is (you guessed it) 1.  So, if it's composite, its prime factors must be bigger than X.  

Since both these conclusions violate the assumption that X is the biggest prime, the assumption must be wrong.  There is no biggest prime.

When mathematicians finish a proof, they sometimes write QED, for quod erat demonstrum (Latin for `which was to be shown').  Euclid couldn't have written this, but one can imagine that he had a pretty elated feeling when he figured this out.  It's beautiful.  What does that mean? Louis Armstrong supposedly said about jazz "if you have to ask, you'll never know" and that may be true here, too.  Why is this proof beautiful? Well, it uses very simple concepts to prove something deep.  It has a certain surprise factor - you don't see the answer coming.  But it also has a certain inevitability once it's done.  I once heard Leonard Bernstein trying to describe why a piece of music was beautiful and he used similar terms.  

We know a lot about primes besides that.  But there are some simple things we don't know.  Here's one: When there are two primes where one is 2 more than the other, they are called twin primes.  The first twin primes are 1 and 3; 3 and 5; 5 and 7; 11 and 13; 17 and 19; 29 and 31.  They keep going for a very long time.  Everyone thinks there are infinitely many. But there's no proof.

Digression: Logarithms

When you raise a number to a power, you multiply it by itself that many times:
2^3 = 2*2*2 = 8.
5^4 = 5*5*5*5 = 625. But mathematicians love to extend the meanings of things, so they invented fractional powers.  The simplest fractional powers are roots. A square root is a number that, when squared, makes the original number
9 ^ 1/2= 3 because 32 = 9.
16 ^ 1/2 = 4 because 42 = 16
But then there are cube roots:
27 ^ 1/3 = 3 because 33 = 27
64 ^ 1/3 = 4 because 43 = 64.

Don't stop there!  If you square a cube root, you get the 2/3 power.  If you cube a square root, you get the 3/2 power.   What fun!
64 ^ 2/3 = 16 because 641/3, squared = 4*4 = 16.

A logarithm is the opposite of a power (or exponent).  A logarithm has a base - by far the most common are 10 and e (more on e in a bit).  In base 10, we usually write it as log(x).
Log(100) = 2, because 102 =100
Log (1000) = 3, because 103 = 1000.

And, since you can have fractional powers you can have the log of any number greater than 0.  

But the most common base for logs isn't 10, it's e, which is approximately 2.718.  You can't figure out what e is exactly (I won't explain why, here) but you can get as close as you want, because

When we take logs to base e, they are called natural logs, and written ln(x).

End digression

Here's another interesting thing about primes: You can't predict, by any known method, what the next one is.  But you can predict about how many there will be below a certain number.  There are a few ways, but the most famous and surprising is one discovered by Gauss (perhaps the greatest mathematician ever).  
 Π(x) ≈x/ln(x)
Π(x) is shorthand for the number of primes less than x.  The wavy equals sign means "approximately equal to".   When I first saw this, I was amazed!  What the ** is going on here? Why are prime numbers, which are integers, related to a fraction that involves logs to the base e, which is an irrational number?  I won't get into that here.  But it's true.  And the approximation gets better as x gets bigger.

Originally posted to plf515 on Fri Jul 14, 2006 at 06:47 AM PDT.


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Comment Preferences

  •  Tip jar (37+ / 0-)

    Tips? Flames? Recommends? Suggestions? Corrections?

    Questions? Answers?

    and aren't you glad this diary is not about the middle east?

    Republicans worry about our souls and their bellies. Democrats worry about their souls and our bellies

    by plf515 on Fri Jul 14, 2006 at 06:43:06 AM PDT

  •  zzzzzzzzzzzzzzzzzzzz (2+ / 3-)
    Recommended by:
    cookiebear, Chamonix
    Hidden by:
    Mikey, weirdscenes, JPete

    'Events are in the saddle and ride mankind.' Emerson

    by deepsouthdoug on Fri Jul 14, 2006 at 06:44:13 AM PDT

  •  woohoo (3+ / 0-)
    Recommended by:
    wozzle, Stand Strong, plf515

    nothing like my daily serving of mathmeatics!

    # Members: 96,326 (as of 10:00pm 7/13). Projected Date of 100,000th member registration: August 5, 2006

    by FleetAdmiralJ on Fri Jul 14, 2006 at 06:47:44 AM PDT

  •  Louis Armstrong: (1+ / 0-)
    Recommended by:

    ...said about jazz "if you have to ask, you'll never know"

    Although I can't find a citation that doesn't include "is said to have replied", so it could be attributed to him wrongly.  I'll see what I can find.

    Great diary, especially for the mathematically un-inclined.

    Saint, n. A dead sinner revised and edited. - Ambrose Bierce

    by pico on Fri Jul 14, 2006 at 06:53:31 AM PDT

  •  Maybe someday (3+ / 0-)
    Recommended by:
    wozzle, Slartibartfast, plf515

    a little something about the Riemann Hypothesis/sequence...

    Lot of hitchhiker's fans here :)

    If you want something other than the obvious to happen - you've got to do something other than the obvious...Douglas Adams

    by trillian on Fri Jul 14, 2006 at 07:02:40 AM PDT

  •  Thank you (6+ / 0-)
    Recommended by:
    wozzle, papercut, Fabian, JPete, pico, plf515

    My 7-year-old son is really into math (his latest obsession is Fibonacci numbers) so I'm going to try later today to explain this proof to him.

    As one who took a lot of advanced math courses in high school & college, but hasn't looked at this in almost 20 years, I really appreciate your effort and look forward to future diaries.

    America will never be destroyed from the outside. If we falter and lose our freedoms, it will be because we destroyed ourselves. --Abraham Lincoln

    by thebes on Fri Jul 14, 2006 at 07:05:02 AM PDT

  •  it's the thinking that's important (0+ / 0-)

    not who thought of it first.  I had the same experience with the realization that the sum of any three successive numbers is divisible by three.

    It turns out that it is a trivial result, and the proof is trivial, too.  But its charm helps it to be an excellent way to show the method of proofs to someone who is a little frightened of them.

  •  Maybe post equations as graphics, for clarity? n/ (3+ / 0-)
    Recommended by:
    wozzle, Stand Strong, plf515
  •  Logs would have been clearer if superscripts (1+ / 0-)
    Recommended by:

    were working.

    Otherwise, It is a great discussion.

    I guess one of the reasons that I will never become a mathematician for real is because of the nature of the 'duh' proof.  I mean, do we really need to prove that there is an infinite number of primes?

    Anyway, I think clock arithmetic (modulo x) is more interesting than primes and should be addressed in this series.

    Trust in God, all others bring data.

    by Mlle L on Fri Jul 14, 2006 at 07:15:09 AM PDT

  •  Math sucks (4+ / 0-)
    Recommended by:
    skrymir, Dr Benway, plf515, 73rd virgin

    Photobucket - Video and Image Hosting

    "If more parents home disciplined [their kids] there would be fewer people I have to smack in public." --Wilzerd Balefire.

    by TheBlaz on Fri Jul 14, 2006 at 07:32:30 AM PDT

    •  Math is just a language (1+ / 0-)
      Recommended by:

      Since we don't learn Math as our first language, it's as hard as any other language we come late to.

      Really.  Math became so much easier when I learned to view it as a language.  I like it because it is usually unambiguous and accurate unlike relativity and Schroedinger's cat and those science things.  Is light a wave or a particle?  Well, it's both - depending on how you look at it.  argh!

      We must never lose it, or sell it, or give it away. We must never let them take it from us.

      by Fabian on Fri Jul 14, 2006 at 07:43:15 AM PDT

      [ Parent ]

    •  The way math is taught (3+ / 0-)
      Recommended by:
      suzq, TheBlaz, arbiter

      in most schools sucks.

      Math is hard, but so are lots of things: Playing the piano, painting a picture.....writing a diary without typos....

      Republicans worry about our souls and their bellies. Democrats worry about their souls and our bellies

      by plf515 on Fri Jul 14, 2006 at 07:49:04 AM PDT

      [ Parent ]

      •  couldn't agree more - (3+ / 0-)
        Recommended by:
        suzq, ignorant bystander, plf515

        My former drummer was a math major and taught me more about algebra in two hours than I managed to learn in seven years.  I always just thought I was stupid, but he proved it was my completely inept teachers who were at fault - and now that I work with programmers I understand WHY they were inept.

        I hear you asking "seven years?".  Yeah.  I failed Algebra I three times and II twice.  I cherish my math abilities - I had to fight for every bit I got.

        "this new century really sucks ass" - surferal

        by arbiter on Fri Jul 14, 2006 at 08:07:10 AM PDT

        [ Parent ]

        •  I'm convinced (1+ / 0-)
          Recommended by:

          that the only reason I passed algebra in college is because I had a polish last name, my teacher was polish, and I always looked confused and vaguely frightened in class.

          I think she thought I didn't speak english and pitied me enough to give me a C.

          I'll take it.

          "If more parents home disciplined [their kids] there would be fewer people I have to smack in public." --Wilzerd Balefire.

          by TheBlaz on Fri Jul 14, 2006 at 08:32:57 AM PDT

          [ Parent ]

          •  Damn! (0+ / 0-)

            Once again, my ethnic background proves to be of no help.  Lousy French/Scottish ancestors...

            "this new century really sucks ass" - surferal

            by arbiter on Fri Jul 14, 2006 at 08:36:15 AM PDT

            [ Parent ]

            •  I took Calculus 1 (1+ / 0-)
              Recommended by:

              from a professor who was from Poland and had a very strong accent.  I sat next to a guy from Africa (I forget which country) who also had a very strong accent.

              I think there were no sounds in English that they both pronounced the same way.

              Republicans worry about our souls and their bellies. Democrats worry about their souls and our bellies

              by plf515 on Fri Jul 14, 2006 at 08:42:20 AM PDT

              [ Parent ]

        •  Math is not hard (2+ / 0-)
          Recommended by:
          vcmvo2, plf515

          but in the early grades especially it is taught, by and large, by people who believe it is hard and who were not well-taught themselves. Which is kind of like getting all your news from Fox.

      •  I recall reading (2+ / 0-)
        Recommended by:
        papercut, plf515

        about a study of the effects of first impressions about math.  Children who met with success early on went on to love math.  Children who had difficutly went on to have a very negative view of it.  Like all subjects, mathematics requires a certain level of maturity before it can be effectively taught.  This is especially so in a subject that does indeed have "right" answers (at least arithmetic, which is what they were teaching).

        That's not to say that there aren't a lot of bad math teachers!

        What I always have found fascinating is that most people really have no idea what mathematics is, frequently confusing it with arithmetic.  Indeed, most of mathematics isn't about numbers at all (at least not directly)!

        •  Indeed (0+ / 0-)

          most educated adults have some vague idea what a lot of other people do for a living.  Maybe not detailed or exactly right, but something.  e.g., I have some idea
          what doctors, lawyers, architects, physicists, etc. do for a living.

          But most people have not the vaguest idea what mathematicians do.  The idea that more math is being invented every day is foreign to most people, I think.

          Republicans worry about our souls and their bellies. Democrats worry about their souls and our bellies

          by plf515 on Fri Jul 14, 2006 at 09:27:41 AM PDT

          [ Parent ]

    •  as a math impaired person - (1+ / 0-)
      Recommended by:

      who has, by circumstance, been forced to get good at it, I understand the sentiment - and the pic (and caption) describe my feelings perfectly.  I'm not a stupid guy (I would like to think), but sure feel that way every time I've got to laboriously cobble something together for a program.

      "this new century really sucks ass" - surferal

      by arbiter on Fri Jul 14, 2006 at 08:03:41 AM PDT

      [ Parent ]

      •  Me too (1+ / 0-)
        Recommended by:

        I was NEVER good at math, except for Finite Math and Statistics for some reason.

        Those I found incredibly easy, but algebra? Calculus? Geometry?

        Ugh. Not to save my life.

        "If more parents home disciplined [their kids] there would be fewer people I have to smack in public." --Wilzerd Balefire.

        by TheBlaz on Fri Jul 14, 2006 at 08:15:16 AM PDT

        [ Parent ]

        •  Personally (0+ / 0-)

          I found geometry hard, but I am not a visual person.

          I like algebra and number theory, I also like math puzzles and interesting tidbits.

          Republicans worry about our souls and their bellies. Democrats worry about their souls and our bellies

          by plf515 on Fri Jul 14, 2006 at 08:20:19 AM PDT

          [ Parent ]

  •  I find it interesting when (2+ / 0-)
    Recommended by:
    papercut, plf515

    people find beauty in things I find very difficult.  Math is one of those things.  I find beauty in Geometry, but algebra, calculus, I can’t wrap my puny little mind around.

    I  know people who find beauty in the Law, probably for similar reasons.  Language.   I know people who find beauty in words, and language where I have a hard time working it out.  

    There is art here, and like many forms of art, this one is audience sensitive.  It takes a certain appreciation to get it.  

    Thank you for helping with that appreciation.

    •  You're very welcome (1+ / 0-)
      Recommended by:

      Probably all art is audience sensitive, but some forms are more so or less so.

      I find nearly all visual art in the last 100 years or so hard to appreciate.  

      Republicans worry about our souls and their bellies. Democrats worry about their souls and our bellies

      by plf515 on Fri Jul 14, 2006 at 08:44:10 AM PDT

      [ Parent ]

  •  Kids and 'what's the biggest number?' (2+ / 0-)
    Recommended by:
    papercut, plf515

    In Grade 3, a classmate absolutely insisted that the largest number was "a dillion and nine," even after being confronted with the inevitable "a dillion and ten." I wonder whatever happened to him.

    "We must not let the terrorists win under any circumstances by changing and being fearful."--Ian Blair, London's police commissioner.

    by Dump Terry McAuliffe on Fri Jul 14, 2006 at 08:45:45 AM PDT

  •  A personal math story (2+ / 0-)
    Recommended by:
    papercut, plf515

    In high school, I wanted to be a theoretical physicist. When it came time time to move to calculus, the math teacher (small school, there was only one) told me I just didn't "have it" for math.

    Stupidly, I took him at his word and stopped taking math-intensive subjects.

    Later in life I have proven to myself that I really just need a better teacher. I have a fancy degree in a non-math subject, because, once I realized that my teacher had been wrong, it was too late to start.

    To this day I follow, and understand much of theoretical physics. I just want to kick that teacher in the ass!

    George W. is NOT an incompetent liar, he's had waaay too much practice for that. (-2.25, -2.56)

    by EclecticFloridian on Fri Jul 14, 2006 at 09:01:39 AM PDT

    •  It's never too late to start. (2+ / 0-)
      Recommended by:
      papercut, plf515

      I know a woman who got her MD license at age 40 and my dad knew a guy who got his law degree and passed the bar at 80.

      •  You can also (0+ / 0-)

        study a lot of things on your own,or by auditing, or extension classes, and so on.

        Most of what I've learned I've learned outside of school

        Republicans worry about our souls and their bellies. Democrats worry about their souls and our bellies

        by plf515 on Fri Jul 14, 2006 at 09:39:37 AM PDT

        [ Parent ]

    •  My story is kind of opposite (1+ / 0-)
      Recommended by:

      Although I was relatively good in math and liked it, it wasn't my top interest, but I had some really, really excellent female math teachers, so I ended up majoring in math and even taught it for a while. For the bottom line was this:  If you got the answer or proof or whatever right, nobody could say it was wrong, even if they hated your guts.

      For instance, one of my social studies teachers was said to have a policy of never giving an A to a girl.  I'm sure he stopped getting away with it a few years after my time, but it was too late to do me any good.

  •  I just got here (0+ / 0-)

    and haven't read through the comments completely.

    But don't you mean to say "suppose the biggest prime is 11" in your example?

  •  Umm (1+ / 0-)
    Recommended by:

    "1" isn't prime.
    I'll admit that this is a convention, but it's the convention.
    2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, ...
    After 43 I have to stop and think.

    •  see my reply to a comment (0+ / 0-)


      For this level of diary, I am not sure this sort of thing matters too much.  But you're right (at least as the convention usually goes).

      Republicans worry about our souls and their bellies. Democrats worry about their souls and our bellies

      by plf515 on Fri Jul 14, 2006 at 09:33:35 AM PDT

      [ Parent ]

  •  Do you plan a segment on computer math? (1+ / 0-)
    Recommended by:

    A short review of 1's and 2's complement arithmetic might be useful to the readers here. Some other topics of interest in computer math might be:

    • binary/octal/hexadecimal number systems
    • Booth's algorithm
    • floating-point operations
    • branch prediction

    I've found that most people have absolutely no idea how their computers perform mathematical calculations using only 1's and 0's. Even many programmers have only a rudimentary knowledge of fundamental digital math functions.

    -6.38/-3.79::'A man is incapable of comprehending any argument that interferes with his revenues.' Descartes

    by skrymir on Fri Jul 14, 2006 at 09:31:26 AM PDT

    •  I can't write this one (0+ / 0-)

      because I don't know anything about it - well, OK, I know what binary and octal and hexadecimal are; I have a vague idea how floating point works.....but I couldn't write this diary.

      Would you like to?  Or do we have another volunteer?

      We could work out a date to post it.

      Republicans worry about our souls and their bellies. Democrats worry about their souls and our bellies

      by plf515 on Fri Jul 14, 2006 at 09:35:41 AM PDT

      [ Parent ]

      •  I'm just an engineer, not a mathemetician. (1+ / 0-)
        Recommended by:

        I know how binary math works because I used it extensively, and even implemented it, in my work, but I can't prove or derive any of it from a theory standpoint. Let me think about it a little and see what I think I can explain without getting all twisted up.

        -6.38/-3.79::'A man is incapable of comprehending any argument that interferes with his revenues.' Descartes

        by skrymir on Fri Jul 14, 2006 at 03:35:09 PM PDT

        [ Parent ]

  •  Crucial stuff for computer security systems. (1+ / 0-)
    Recommended by:

    factoring, prime numbers

    Factoring and prime numbers are used in one of the most commonly used public key systems, the RSA system.

    In asymmetric cryptography it’s essential that an attacker cannot use one key, for example the public key, to find the value of the private key. In order to do this, a one-way, or trapdoor mathematical function is required. This is a mathematical function that’s easy to do in one direction but is very difficult, or impossible to reverse. Factoring prime numbers is like this. While it is easy to find the product of two large prime numbers, finding the unique value of those two prime numbers, called factoring, is very difficult if all that you know is the product.

    Very large prime numbers are used, because the larger the prime number, the more difficult factoring becomes.

    "Patriotism is supporting your country all the time, and your government when it deserves it." - Mark Twain

    by cyberKosFan on Fri Jul 14, 2006 at 09:38:34 AM PDT

  •  The number 1 is not prime (0+ / 0-)

    The number 1 has no factors, but it also isn't prime. There are several reasons explained at the link above, which is an excellent resource generally for information about prime numbers. Essentially number theorists and others have deterimined that the number 1 has too many special properties to group it with the primes. It used to be considered prime, but over time the realization of its special status has changed how it's classified. Consider the definition of prime numbers, as quoted on the page I linked above:

    'An integer greater than one is called a prime number if its only positive divisors (factors) are one and itself.'[emphasis original, style changed]

    Also consider that the number 1 has no unique multiplicative representation; it can be expressed as 1 * 1, 1 * 1 * 1, etc. This property of "identity" is specific to 1 and isn't possessed by any of the other primes, or any other integer for that matter.

  •  Prime Links -- links concerning primes and math (1+ / 0-)
    Recommended by:

    I've collected a number of links over the years to useful and interesting resources on math, specifically on primes. Here are a few that I think may be of general interest:

    1.) MIT's OpenCourseWare. MIT has a number of their classes' notes and other resources online. Challenging stuff, but the professors are good and generally not as obtuse as you might think:

    2.) Prime Pages. Hosted by the University of Tennessee at Martin and compiled by professor Chris Caldwell. A great FAQ about primes, plus lists of primes, curios, definitions and explanations of various prime-related theorms and subjects, etc.:

    3.) The Online Encyclopedia of Integer Sequences. Created by Neil J. A. Sloane at AT&T, an extension of his book "Handbook of Integer Sequences". Over 100,000 integer sequences are included in the database, with more added by professional and amateur mathmeticians all the time. If you run into an interesting sequence of integers, you're likely to find out more about it here:

    4.) Hypermath. Hosted by Georgia State University department of Physics and Astronomy. This is a handy site for understanding how various branches of math and physics are linked. There are two main sites, hyperphysics and hypermath, along with subsections on Algebra and applied statistics, all geared towards applications in physics. It's a great reference, but also has a number of good charts and images demonstrating various concepts in math and physics:



    5.) Mathworld. Hosted by Wolfram Publishing. While I don't think cellular autonomae really offer a "New Kind of Science", I do think this is a great site for learning more formally about math. If you need to know the equation for something, this is where you look it up. Has lots of other stuff too, including bios of famous mathemeticians. Descriptions of topics are concise, with all of the important equations included:

    And, to keep this comment somewhat topical, here's a link to their page on prime numbers:

  •  look at the area bounded by... (0+ / 0-)

    ...the graph of y = 1/x and the x axis vertically and x = 1 and x = c (a constant > 1) hprizontally.  e is the value for c that makes that area be 1.

    Teacher's Lounge opens each Saturday, sometime between 10am and 12 noon EST

    by rserven on Fri Jul 14, 2006 at 10:00:45 PM PDT

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