SOLUTION: HOW TO DETERMINE i2=−1 FROM THE EULER`S EQUATION USING LOGARITHM/WITHOUT LOGARITHM? PLEASE TELL ME IN DETAILS.

Algebra ->  Complex Numbers Imaginary Numbers Solvers and Lesson -> SOLUTION: HOW TO DETERMINE i2=−1 FROM THE EULER`S EQUATION USING LOGARITHM/WITHOUT LOGARITHM? PLEASE TELL ME IN DETAILS.       Log On


   

Question 389436: HOW TO DETERMINE i2=−1 FROM THE EULER`S EQUATION USING LOGARITHM/WITHOUT LOGARITHM? PLEASE TELL ME IN DETAILS.
Found 2 solutions by richard1234, Edwin McCravy:
Answer by richard1234(7193) About Me  (Show Source):
You can put this solution on YOUR website!
i%5E2+=+-1 by definition. Or, we use Euler's formula:

e%5Eix+=+cis+x+=+cos+x+%2B+i+sin+x

Letting x+=+pi%2F2,

e%5Ei%28pi%2F2%29+=+cos+%28pi%2F2%29+%2B+i+sin+%28pi%2F2%29+=+i

Squaring both sides,

e%5Ei%28pi%29+=+cos+%28pi%29+%2B+i+sin+%28pi%29+=+-1

This implies i%5E2+=+-1.

Answer by Edwin McCravy(20056) About Me  (Show Source):
You can put this solution on YOUR website!

i%5E2+=+-1 by definition of i

By Euler's equation:

e%5E%28i%2Atheta%29+=+cos%28theta%29+%2B+i%2Asin%28theta%29

Substitute p
 for q

e%5E%28i%2Api%29+=+cos%28pi%29+%2B+i%2Asin%28pi%29

e%5E%28i%2Api%29+=++-1+%2B+i%2A0

e%5E%28i%2Api%29+=+-1

Go back to Euler's equation.

e%5E%28i%2Atheta%29+=+cos%28theta%29+%2B+i%2Asin%28theta%29

Substitute pi%2F2
 for q

e%5E%28i%2Aexpr%28pi%2F2%29%29+=+cos%28pi%2F2%29+%2B+i%2Asin%28pi%2F2%29

e%5E%28i%2Api%2F2%29+=++0+%2B+i%2A1

e%5E%28i%2Api%2F2%29+=+i

Square both sides:

%28e%5E%28i%2Api%2F2%29%29%5E2+=+i%5E2

%28e%5E%28i%2Api%2Fcross%282%29%29%29%5Ecross%282%29+=+i%5E2

e%5E%28i%2Api%29+=+i%5E2

-----------------------
 From the first part we have

e%5E%28i%2Api%29+=+-1

and from the last part we have

e%5E%28i%2Api%29+=+i%5E2

So i%5E2=-1 because things equal to the 

same thing are equal to each other

Edwin