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Mathematical Trivia

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  • #46
    Re: Mathematical Trivia

    Yes, an excellent formula.

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    • #47
      Re: Mathematical Trivia

      That formula has always blown my mind. What seem (to me at least) to be three random math entities: pi, natlogbase and an (the) imaginary number . . . ???!!!

      Comment


      • #48
        Re: Mathematical Trivia

        Originally posted by Marlow
        That formula has always blown my mind. What seem (to me at least) to be three random math entities: pi, natlogbase and an (the) imaginary number . . . ???!!!
        I think that the key conclusion is that these are not three random math entities; they are fundamental concepts at their core.

        I have heard that a true mathematics whiz thinks of this equation as obvious -- not obvious to me and not a true mathematics whiz.

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        • #49
          Re: Mathematical Trivia

          Originally posted by 26mi235
          I have heard that a true mathematics whiz thinks of this equation as obvious
          Suffice to say that this is NOT obvious to me, not remotely, not even a little...

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          • #50
            Re: Mathematical Trivia

            Originally posted by kuha
            Originally posted by 26mi235
            I have heard that a true mathematics whiz thinks of this equation as obvious
            Suffice to say that this is NOT obvious to me, not remotely, not even a little...
            It is derived from the following formula:

            e^ix = cos x + i?sin x

            And that formula is derived from comparing the power series expansions of the exponential function (e^x) and the trigonometric functions

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            • #51
              Re: Mathematical Trivia

              Thanks for all of the above, but I will stick with:

              1 + 1 = 2

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              • #52
                Re: Mathematical Trivia

                It's i^i = e^(-pi/2) [actually, e^(-pi/2 + 2*pi*n), n integer, just like e^[(pi+2*pi*n)*i] = -1]

                But yes, it's amazing that this value is a real number.

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                • #53
                  Re: Mathematical Trivia

                  Originally posted by 26mi235

                  I have heard that a true mathematics whiz thinks of this equation as obvious -- not obvious to me and not a true mathematics whiz.
                  This also is obvious to a true mathematician.

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                  • #54
                    Re: Mathematical Trivia

                    Originally posted by bambam
                    It is derived from the following formula:

                    e^ix = cos x + i?sin x

                    And that formula is derived from comparing the power series expansions of the exponential function (e^x) and the trigonometric functions
                    In physics, the factor e^ix is called a complex phase. One can think of it as the hand of a clock (of length 1) that is free to swing in a circle. The value of x fixes the angle at which the hand is positioned. When x=0, it's horizontal, pointing in the positive direction (3 o'clock), and its value is +1. When x= pi, it points to 9 o'clock -- the opposite direction -- and hence has a value of -1.

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                    • #55
                      Re: Mathematical Trivia

                      If possible, in layman's terms, what the F*** does pi (a math constant that relates a circle's circumference to its radius) have to do with logarithms (a function that relates a number's base to an exponent) to an imaginfreakinary number (the square root of a negative number, which, very simply, can NOT exist, because any number times itself MUST be positive!!).
                      :evil: :evil: :evil: :evil: :evil: :evil: :evil: :evil:

                      edit, oh I see JRM tried to do that. It ALMOST makes sense (nah, just kidding!). :shock: :roll: :shock: :roll:

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                      • #56
                        Re: Mathematical Trivia

                        Originally posted by Marlow
                        If possible, in layman's terms, what the F*** does pi (a math constant that relates a circle's circumference to its radius) have to do with logarithms (a function that relates a number's base to an exponent) to an imaginfreakinary number (the square root of a negative number, which, very simply, can NOT exist, because any number times itself MUST be positive!!).
                        There is no universal rule that says "any number times itself must be positive". That's a specific case of a more general multiplication rule for imaginfreakinary -- or complex -- numbers, which states that "a complex number times its conjugate is positive".

                        In other words, any complex number (call it "z") can be written as

                        z = a + i b

                        where "a" and "b" are real numbers. The complex conjugate of "z" is called "z*", and by definition it is

                        z* = a - i b

                        When you multiply them together, you get

                        z z* = a^2 + b^2 ,

                        which is real and positive. But if you try zz, or z* z*, you get something else that is still a complex number.

                        A real number is a complex number with b = 0 (and so z = z*, which gives the rule you stated).

                        The answers to your other questions are a bit more intricate, and I'm on vacation. Check out a complex analysis book! 8-)

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                        • #57
                          Re: Mathematical Trivia

                          Originally posted by JRM
                          The answers to your other questions are a bit more intricate, and I'm on vacation. Check out a complex analysis book! 8-)
                          Cop out!! Thanks anyway. I kinda/sorta followed your explanation.
                          I may have related this story before, but I went to college to be a math major, having successfully navigated two years of calculus in high school and was therefore exempt from taking it in college, so I skipped straight on to 'Number Theory'', which I was very sad to learn had no numbers in it, just proofs with thetas and rhos and sigmas. That was the end of that!

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