The Math Forum

Ask Dr. Math - Questions and Answers from our Archives
Associated Topics || Dr. Math Home || Search Dr. Math

Modern Algebra

Date: 07/10/97 at 17:55:34
From: Cathy Fyle
Subject: Modern algebra

Our professor has asked us to show that the natural log of i 
to the 1/2 is equal to i times pi over 4. I have not had complex 
numbers nor do I know how to go about showing this is true.  
Can you help?

Date: 07/11/97 at 17:27:51
From: Doctor Anthony
Subject: Re: Modern algebra

If you are new to complex numbers, I think your professor is expecting 
rather a lot of you to work with natural logs and complex numbers.  
However, let us see what can be done.

   ln(i^(1/2)) = (1/2)ln(i)

Now let  ln(i) = x

             i = e^(x)

But we also have the identity:  

 i = cos(pi/2) + i.sin(pi/2)  =  e^(i.pi/2)   

and equating the two expressions for i

        e^(x) = e^(i.pi/2)   so the powers of e must be equal

            x = i.pi/2

       (1/2)x = i.pi/4

   (1/2)ln(i) = i.pi/4

  ln(i^(1/2)) = i.pi/4

-Doctor Anthony,  The Math Forum
 Check out our web site!   
Associated Topics:
College Modern Algebra

Search the Dr. Math Library:

Find items containing (put spaces between keywords):
Click only once for faster results:

[ Choose "whole words" when searching for a word like age.]

all keywords, in any order at least one, that exact phrase
parts of words whole words

Submit your own question to Dr. Math

[Privacy Policy] [Terms of Use]

Math Forum Home || Math Library || Quick Reference || Math Forum Search

Ask Dr. MathTM
© 1994- The Math Forum at NCTM. All rights reserved.