Open main menu
Home
Random
Log in
Settings
About The Portal Wiki
Disclaimers
The Portal Wiki
Search
Editing
The Road to Reality Study Notes
(section)
Warning:
You are not logged in. Your IP address will be publicly visible if you make any edits. If you
log in
or
create an account
, your edits will be attributed to your username, along with other benefits.
Anti-spam check. Do
not
fill this in!
=== 5.2 The idea of the complex logarithm === Armed with both the cartesian and polar representations of complex numbers, it is now possible to show that the multiplication of two complex numbers leads to adding their arguments and multiplying the moduli. This, for the moduli, converts multiplication into addition. This idea is fundamental in the use of logarithms. We first start with the expression <math>b^{m+n} = b^m \times b^n</math>, which represents the idea of [https://en.wikipedia.org/wiki/Exponentiation exponentiation]. This is easy to grasp for <math>m</math> and <math>n</math> being positive integers, as each side just represents <math>m+n</math> instances of the number <math>b</math>, all multiplied together. If <math>b</math> is positive, this law is then showed to hold for exponents that are positive integers, values of 0, negative, and fractions. If <math>b</math> is negative, we require further expansion into the complex plane. We would need a definition of <math>b^p</math> for all complex numbers <math>p,q,b</math> such that <math>b^{p+q} = b^p \times b^q</math>. If we then define the [https://en.wikipedia.org/wiki/Logarithm logarithm to the base b] as the inverse of the function <math>f(z) = b^z</math> such that <math>z=log_bw</math> for <math>w=b^z</math> then we should expect <math>z=log_b(p \times q) = log_bp + log_bq</math>. This would then convert multiplication into addition and allow for exponentiation in the complex plane.
Summary:
Please note that all contributions to The Portal Wiki may be edited, altered, or removed by other contributors. If you do not want your writing to be edited mercilessly, then do not submit it here.
You are also promising us that you wrote this yourself, or copied it from a public domain or similar free resource (see
The Portal:Copyrights
for details).
Do not submit copyrighted work without permission!
Cancel
Editing help
(opens in new window)