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If you ask a MITS (man in the street, not MIT grad) what math is about, chances are he´´ll mention numbers. Not all math is, in fact, about numbers, but some of it is. But what numbers! Mathematicians aren´t content with the numbers everyone knows, nor with what everyone knows about them. They´ve been inventing new kinds of numbers, and finding out new things about them, for thousands of years. Natural numbers, negative numbers, rational and irrational numbers. Imaginary numbers, complex numbers, transfinite numbers and more. If you want to read more, join me below the fold.

The first numbers to be discovered, both by people historically and by individuals in childhood, are called the natural numbers: 1, 2, 3, and so on.People have known about these for a very long time, well back into prehistory; and little kids learn to count before they start school. There´s even some evidence that some animals have some notion of (at least some) numbers. If any part of math is ‘natural´ it´s counting. One of the ways counting can be used is to determine how many of something you have. That leads, naturally enough, to addition. If you have some of something, and you get more, how many have you got? You have to add. Well enough. But what about if you have some of something and lose some? That´s subtraction, and that leads to our first extension of the number system, but first, two short digressions about sets and mathematical operations.

. The notion of a set is basic in math. A set is a collection of objects. It could be any kind of objects: The set of presidents with IQs lower than 100. The set of people in my household, and so on. A set can be empty, that is, have no members (e.g., the set of people named Bush who I have voted for). It can also be infinite (we´ll get into a little more detail on that later).Sets can be specified by list all the members (e.g. {Tom, Jill, Dave}), or by describing the characteristics they have in a way that is clear.Sets are usually written enclosed in curly braces {}, but I´m not going to be very formal about that stuff here. While sets can be anything, in this diary we´ll be concerned about sets of numbers of one sort or another.

Once you have a set of numbers, you can do things to them. These things are called operations, and, since we´re concerned with math, we´ll be talking about mathematical operations. Mostly, if not entirely, we´ll be talking about mathematical operations that take two objects and return a third.Examples of mathematical operations are addition, subtraction, and so on. When a mathematical operation on two elements of a set always returns another object on that set, that operation is said to be closed on that set.This will be clearer with an example or two, given below.

. The operation of addition is closed on the set of natural numbers. If you add two natural numbers, you always get another natural number.3 + 2 = 5. You can be sure of this, even without doing the actual computation: 1,223, 871 + 2,102,876? I don´t know the answer, but it will be another natural number. So, as long as we stick with counting and addition, we´re fine with just the natural numbers. Subtraction, though, is a different story. Sometimes the result is another natural number: 7 –– 5 = 2. But sometimes it isn´t.What is 5 – 7? Hmmmm. What about 5 – 5?Neither answer is part of the natural numbers.For a very long time, mathematicians thought these were nonsense problems. The first person to deal with them sensibly at all appears to have been Brahmagupta, in India, in the 7th century CE; who may also have been the first to use 0 as a solution to an equation. The Greeks, amazing though they were, didn´t get it. So, we have now expanded the number system from the natural numbers to the integers. But we aren´t done.

The next operation to consider is probably multiplication. But that doesn´t pose any threat to the integers; the integers are closed under multiplication. For instance -5*-2 = +10. But once you have the idea of multiplication as repeated addition, it´s natural to think of division, or repeated subtraction, and the integers are not closed under division. For example 5 divided 3 is not an integer. 1 is too little, 2 is too big. For this, we need fractions, also known as rational numbers, because they are ratios. Now we are OK with division. But those mathematicians never leave well enough alone. Multiplication is repeated addition. What, they supposed, would repeated multiplication be like? No problem. They invented exponents. These are written as superscripts, where the superscript indicates how many times to multiply something by itself.

Still, we are okay, as long as we stick to exponents that are positive integers. But, of course, we don't stop there. Since we know that we can take any number and raise it to a power, it´s natural to wonder what number raised to some power, would equal a number. For example, we know that 32 = 9, which means that, if we want to know what number, squared, equals 9, the answer is 3. These are called square roots (root is a synonym, more or less, for solution). They are written as fractional exponents.So

So far so good. But what about, say, the square root of 2? Well, it´s not an integer, because 1 is too small, and 2 is too big. But it´s not a fraction, either, and the proof of that is pretty neat. It´s another reductio ad absurdum proof.That is, we start off by assuming that what we want to show is wrong, then we deduce something absurd from that assumption that :

1. Start with the assumption , we can also assume that p and q have no common factor, because, if they do, we can simply divide it out (e.g. 6/12 = ½ by dividing by 6).

2. Square both sides to get

3. Multiply both sides by q2 to get

4.So, p2 is even.(It is double some other number, so it must be even)

5. So p is even (if you square an odd number, you get an odd number, if you square an even number, you get an even number).

6.q is odd. (p is even, so if q were also even, they would have a common factor of 2, but we assumed no common factor)

7. Since p is even we can set it equal to 2r

8. Substitute p = 2r into the equation in step 3 to get

9. Multiply out to get

10. Divide by 2 to get

11. But that means q is even (see step 5) and we already showed it was odd.

QED

This REALLY bothered the Greeks when they figured it out. They swore everyone who knew it to secrecy, and there´s even supposition that, when one guy blabbed, he was killed. (They took math seriously in those days).

So far, all the numbers we´ve dealt with have been the solution to algebraic equations. For instance, the square root of 2 is the solution to But there are still MORE numbers. There are, in fact, lots of other numbers. There are the transcendental numbers, like π; there are the imaginary and complex numbers; and there are the transfinite numbers. But this is long enough.I´ll talk about those in other diaries.

Sources:I used material from many sources, but the two biggest were Calvin C. Clawson´s Mathematical Mysteries and John Allen Paulos´ Beyond Numeracy. I recommend both.

Originally posted to plf515 on Fri Jul 21, 2006 at 07:48 AM PDT.

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Comment Preferences

  •  Tip jar and thanks (20+ / 0-)

    Thanks to don1one for writing the html code for this.
    Any mistakes in formatting are mine.

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

    by plf515 on Fri Jul 21, 2006 at 07:43:57 AM PDT

  •  Tip (1+ / 0-)
    Recommended by:
    plf515

    I really, really want to read your diaries, but I just can't.  Big chunks of text == very hard to read.  You need to add cool pictures and stuff.  Pythagoras's proof (OK, not really his, but whatev) of the existence of irrational numbers, for instance, is a natural for having a picture of a square with a diagonal.  Look at DarkSyde for an example of how to break up the big chunks with ooh-aah visual stuff.

    Someone who believes in God will believe anything. Base your reality on facts, not myth.

    by RequestedUsername on Fri Jul 21, 2006 at 07:49:11 AM PDT

  •  I really wish this series (2+ / 0-)
    Recommended by:
    sbdenmon, plf515

    Was going when I was taking modern algebra last semester.  Your explinations are as good as anything I have ever seen (although using your proofs for the basis of my work would probably be seen as plagarizing).

    Thanks for this series for us geeks.

    Trust in God, all others bring data.

    by Mlle L on Fri Jul 21, 2006 at 07:56:21 AM PDT

    •  thanks (1+ / 0-)
      Recommended by:
      sbdenmon

      the proofs I've given are in the 'common domain' as it were.  Euclid is not going to complain, nor is Pythagoras, or whichever of his disciples came up with the proof in today's diary.

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

      by plf515 on Fri Jul 21, 2006 at 08:09:43 AM PDT

      [ Parent ]

  •  googled titles (1+ / 0-)
    Recommended by:
    plf515

    If you enjoy reading popular mathematics books, you will probably enjoy Calvin C. Clawson's Mathematical Mysteries. The author has an engaging style of writing, and his enthusiasm for the subject shines through. Topics discussed include prime numbers (of course), number sequences (eg. The Fibonacci sequence), the golden ratio, the proof that square root of 2 is irrational, primes and secret codes, Ramanujan, Goldbach's Conjecture, the Riemann Zeta function, and Godel's incompleteness theorem. There's a good bit of history included throughout, and there is even a chapter that discusses "Numbers and the Occult." The topics do become weightier by the end of the book, and at that point it may be only mathematicians who are still reading. On the whole, most readers of MAA Online will find most of the topics familiar.

    To combat [innumeracy] John Allen Paulos has concocted the perfect vaccine: this book, which is in many ways better than an entire high school math eductation! Our society would be unimaginably different if the average person truly understood the ideas in this marvelous and important book. It is probably hopelessly optimistic to dream this way, but I hope that Innumeracy might help launch a revolution in math education that would do for innumeracy what Sabin and Salk did for polio."

    http://www.cut-the-knot.org/...

    •  Clawson has written a bunch (0+ / 0-)

      of books with simiar themes.  

      anyone who likes my series will probably like his books

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

      by plf515 on Fri Jul 21, 2006 at 08:19:05 AM PDT

      [ Parent ]

    •  Innumeracy.... (1+ / 0-)
      Recommended by:
      plf515

      while an excellent book, won't create a revolution in math education or analytical thought.

      Unfortunately....

      The best start for a decent revolution in math education would be for a complete overhaul on how primary and secondary teachers view mathematics.

      By the time I see students at the university level, their preconceptions of mathematics are either that it is either the foundation for all rational thought (excellent attitude) or that it is something to struggle through until your requirements for your major are met (by far the most prevalent view).

      I do get the chance to turn people around on math in my position, but not nearly enough....

      Really, WWFSMD?

      by sp0t on Fri Jul 21, 2006 at 08:23:41 AM PDT

      [ Parent ]

      •  primary math (0+ / 0-)

        I remember when my kids started math in kindergarten.  They started with estimation.  My response was "Egad!  They've turned MATH into another touchy-feely, all-answers-are-right subject!"

        Me, I'd start 'em with number theory.

        "True individual freedom cannot exist without economic security and independence."
        -FDR

        by Leggy Starlitz on Fri Jul 21, 2006 at 08:49:17 AM PDT

        [ Parent ]

        •  Actually.... (1+ / 0-)
          Recommended by:
          plf515

          I often play number games with my 7 and 9 year olds, and have ben doing this for years now.  Nothing special, just simple stuff to ge them to reason stuff out.

          They are both quite good t it now, and like the puzzle-like feeling of math.

          To stay on your side, number theory is quite fun for kids.

          I taught my seven year old that 1 plus 1 is 10 (when you have only two symbols to work with, namely 0, and 1), and 2 plus 2 is 11 (when you have only three numbers, or base three arithmetic).

          They eat this stuff up!

          Really, WWFSMD?

          by sp0t on Fri Jul 21, 2006 at 09:24:46 AM PDT

          [ Parent ]

          •  You know about gallon problems? (0+ / 0-)

            My dad used to do these with us....

            If you have a 6 gallon can and a 2 gallon can, how do you get 4 gallons?

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

            by plf515 on Fri Jul 21, 2006 at 02:49:26 PM PDT

            [ Parent ]

      •  Students actually fall into (0+ / 0-)

        one of those two camps (foundation for all rational thought or a struggle) by the time I get them in 6th grade...

        I really do get to change a few minds about this every year... I love showing them the REAL answers to "Where am I ever going to use this?"  :-)  

        Our country can survive war, disease, and poverty... what it cannot do without is justice.

        by mommyof3 on Fri Jul 21, 2006 at 07:53:00 PM PDT

        [ Parent ]

        •  I think the bigget problem (0+ / 0-)

          with primary school math is that many (most?) elementary school teachers don't like math, don't get math, don't think math is interesting.  They communicate this not only by not being able to MAKE it interesting, but by body language and so on.

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

          by plf515 on Sat Jul 22, 2006 at 03:41:49 AM PDT

          [ Parent ]

  •  Very nice exposition.... (2+ / 0-)
    Recommended by:
    plf515, 73rd virgin

    I guess you plan on continuing this series?  A very good idea.  Many people have a genuine interest in easily explained, yet possibly deep mathematics, yet have built-in fears or repulsions that are difficult to get around.

    The history is indeed fascinating.  You have started well also.

    •  thanks (1+ / 0-)
      Recommended by:
      sp0t

      yes, I plan to do one a week.

      I forgot to add links to my earlier ones - I am experimenting with the best way to format them.  Don1one was very nice to format these in html.

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

      by plf515 on Fri Jul 21, 2006 at 08:40:14 AM PDT

      [ Parent ]

  •  Actually.... (2+ / 0-)
    Recommended by:
    Leggy Starlitz, plf515

    I will put forward that Math has almost nothing to do with numbers.

    It is a pure formal way of thinking that allows for a logically consistent train of ideas to develop.  The formal structure of numbers is simply a basic set of building blocks to build on.

    I would recommend to all another great book on mathematics and its relation to art and music:

    Douglas Hofstadter's  "Godel, Escher, and Bach:   An eternal Golden Braid

    •  GEB is fantastic! n/t (0+ / 0-)

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

      by plf515 on Fri Jul 21, 2006 at 08:41:00 AM PDT

      [ Parent ]

      •  His follow up book ia also good.... (1+ / 0-)
        Recommended by:
        plf515

        I believe it is called the same thing as his former column:  "Metamagical Themas", and anagram of the famous "Mathematical Games" column of Martin Gardner that ran in Scientific American years ago.

        All good stuff, these....

        Really, WWFSMD?

        by sp0t on Fri Jul 21, 2006 at 09:13:33 AM PDT

        [ Parent ]

    •  G.E.B. (1+ / 0-)
      Recommended by:
      plf515

      One of the greatest books I have ever read.  

      Takes months to wrap your head around it, but it's SO entertaining!  And it teaches a lot about THINKING.

      "True individual freedom cannot exist without economic security and independence."
      -FDR

      by Leggy Starlitz on Fri Jul 21, 2006 at 08:50:06 AM PDT

      [ Parent ]

    •  As an engineer (1+ / 0-)
      Recommended by:
      plf515

      I think of math as a way to predict the future.

      We model the real world with math and then use those models to create predictions on how things will work.

      I'm a new high school math teacher, having spent 20 years as a software engineer. It's interesting to see what has and has not changed over the years.

      One of the most alarming things is watching a student make a mistake in inputing 20 / 3 into the calculator and think that .666 is a reasonable answer.

  •  Good job on the proof! (2+ / 0-)
    Recommended by:
    plf515, 73rd virgin

    Now can you do the same proof for an arbitrary nth-root of k (assuming that k is not an nth-power of an integer. :-)

    On a further note:

    This REALLY bothered the Greeks when they figured it out. They swore everyone who knew it to secrecy, and there´s even supposition that, when one guy blabbed, he was killed. (They took math seriously in those days).

     By the time of the renaissance, mathematics had become a bar game.  Prove/solve this or buy the drinks.  They almost ran the first guy to write a math book there out of town.

    Robyn

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

    by rserven on Fri Jul 21, 2006 at 08:31:18 AM PDT

  •  I gotta go for a while (0+ / 0-)

    I just got a call from my siter that my mom broke her hip and is in the hospital

    I gotta go visit her, will be back in a few hours.

    talk amongst yourselves.  

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

    by plf515 on Fri Jul 21, 2006 at 08:52:40 AM PDT

  •  Thanks; this was neat. (2+ / 0-)
    Recommended by:
    plf515, mommyof3
    I remember reading Bridges to Infinity by Michael Guillen for a college seminar on imagination. (I have no idea how that work is viewed by mathematicians.)

    I felt cheated that math had essentially been taught wrong to me.

    My favorite mathematical concept: factorials!

    "'Cause growing up is awfuller than all the awful things there ever were."

    by Waterbug on Fri Jul 21, 2006 at 08:53:26 AM PDT

  •  Surreal and hyperreal numbers (2+ / 0-)
    Recommended by:
    plf515, poserp

    Knuth defined and studied "surreal" numbers. These are "more numerous" than real numbers. It will be interesting if you write about these after you get to the reals.

    Great diary!
    •  Used for counting moves left to win a game (1+ / 0-)
      Recommended by:
      plf515

      Certain kinds of games, anyway. See this Wikipedia entry:

      http://en.wikipedia.org/...

      They're very interesting for games, as they can be used to work out exactly how far ahead of or behind your opponent you may be. Surreal numbers also have birthdays, and several other fun properties.

      The link above is a bit of a heavy read, but since when was math anything but heavy? Consider, for example, the havoc wreaked by adding 1 to a number -- you can totally modify its properties (for instance, adding 1 to 7 (a prime) produces 8 (a square)) and all you did was add 1. If we consider 1 as an operator rather than a number, it's somewhat chaotic -- we can't predict everything that will happen when we use the "1" operator on a number. In my view that's pretty heavy...

    •  THey sound cool! (0+ / 0-)

      I hadn't heard of them.

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

      by plf515 on Fri Jul 21, 2006 at 02:41:04 PM PDT

      [ Parent ]

  •  Some notes on the series (0+ / 0-)

    Hi all

    I plan to do 1 of these a week.

    All are welcome to participate, both math geeks (there are several here who know more math than I), or neophytes.

    Suggestions for future diaries are welcome.

    Guest diaries are also welcome, just clear it with me.

    Have fun!

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

    by plf515 on Fri Jul 21, 2006 at 02:43:47 PM PDT

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