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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 . . . ???!!!
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.
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.
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:
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-)
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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