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Thread: Numbers

  1. #1
    Join Date
    Aug 2003
    I am often confused when confronted with trancsendensal numbers, like e, pi etc. It's confusing.

    Like, why does pi have the value is does? Why does 'e' ? Why is it called natural base? I know, that growth of bacteria, radioactive decay, all are exponential with base 'e' . But why not 2, 4 or some other number? Why 'e'?

    Well, I hope this is not another silly question.

  2. #2
    Join Date
    Sep 2003

  3. #3
    StarLab Guest
    Wait a minute...according to the last site, e^pi is isn't it ironic that e^i*pi is -1?

  4. #4
    Join Date
    Apr 2004
    Hi StarLab,

    It says, if you further read the link, that this is a special case of Eular's identity which states that:
    e^ix= cosx + i sinx
    given that x=pi

    the equation will become e^i(pi) = cos(pi) + i sin(pi)

    Since cos(pi) = -1 and sin(pi) = 0

    you'll get e^i(pi) = -1

  5. #5
    Join Date
    Apr 2004
    Originally posted by rahuldandekar@Dec 8 2004, 02:55 PM
    Like, why does pi have the value is does?
    Pi is the ratio between the circle's circumference and diameter. It's constant (value=3.14) for all circles.

    You can prove this to yourself with a circle, a piece of tape and a ruler. Look around your house and find something circluar: a jar lid, a CD, a plate - whatever you can find that is circular, the bigger the better. Measure its diameter (the width across the center of circle) with the ruler. Now wrap a piece of tape around around the circle and cut or mark the tape so that it is exactly as long as the outer edge (the circumference) of the circle you are measuring. Measure the piece of tape. With a calculator divide the length of the tape by the diameter you measured for the circle. The answer you will get, if you have measured accurately, is always 3.14 ( estimated to 3 significant figures )

  6. #6
    Guest Guest
    Yes, but how will you prove that theoritically. Like, using equations or graphs. I got a good estimate using sine and cosine functions, but even these are measured ones. Is it possible to prove it theoritically?

  7. #7
    Join Date
    Aug 2003
    That was me. By the way, why is 'e' called natural base?

  8. #8
    Join Date
    Apr 2004
    Here's a link that hopefully would answer some of your questions:
    The number "e"

    It talks about the history of "e" and how did Eular work its value.

    Here's another link that shows the value of "e" to 10,000 decimal places if you're interested
    value of "e"

  9. #9
    Join Date
    Oct 2004
    e is defined as the number given by the series:

    1+1+1/2! + 1/3!+1/4!..... on to infinity (4! means 4x3x2x1 etc)

    e^x is given by the series:

    1+x+x^2/2!+x^3/3! +x^4/4!....

    this is special because it is the only function that exists that when you differentiate/integrate it it comes to exactly the same thing (I don't know if you know about calculus, but try differentiating the above series, it comes to the same thing)
    e works out at about 2.7, and because it is so easy to do calculus with we put all exponential relationships in terms of e, even if they are really base 2 or base 4...

    for instance
    if y=2^x, y=e^xln2 (ln 2 is about 0.69)
    we express it in the second form because it's easy to differentiate

    But still, theres no obvious reason to me why e^i*pi should equal zero (I follow the logic from eulers identity, but still it is pretty amazing&#33

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