To: "usernamehere"@primenet.com (real name here)
Subject: Re: math philosophy
Newsgroups: alt.algebra.help
Organization: Florida State University
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In article <32A475B0.7B95@primenet.com> you wrote:
: i came across this formula in some of my reading. i would like to find
: out if anyone else has put thought into this, and if so, let me know what
: this means to you:
: e^(i(pi)) + 1 = 0
: please email me your comments: araicard@primenet.com
: my name is Aron.
It is very neat identity of 5 constants. Basically, it reflects the fact
that multiplication of complex numbers corresponds to rotations.
Each complex number has an angle and a magnitude. Suppose the complex
number z1 has magnitude m1 and angle a1. Suppose the complex number z2
has magnitude m2 and angle a2. Then to construct the product z1 z2, start
at z1. That means, draw the line from the origin that makes the angle a1
with the x axis, and has length m1. Rotate this line through an
additional angle of a2. Multiply its length by the magnitude m2. You
have now constructed the magnitude and angle of the product z1 z2.
Now consider the numbers on the unit circle. They all have magnitude = to
1 since the line from the origin has length equal to 1. For these
numbers, multiplication is rotation.
Now since multiplication of the numbers on the circle corresponds to
addition of angles, we see that the angle of a complex number on the unit
circle is something like a logarithm. Logarithms also have the property
that multiplication of numbers corresponds to addition of their
logarithms.
Let z be the complex number that has angle of 1 radian and magnitude = 1.
Then z^2 will have an angle of 2 radians.
sqrt(z) will have an angle of 1/2 radians.
In general, for any real number r, z^r will have an angle of r radians.
So what is the logarithm of z? There must be some number w such that e^w
= z.
w = pi i.
This is a remarkable formula that is worth deep discussion.
rose@garnet.acns.fsu.edu To be sure I see your response, use e-mail.
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