Playing around with imag. numbers.

[flatmaster]

We define i=Squrt(-1). However, from what math I know, we just use i as a tool in complex exponentials, complex analysis, etc. I curious what happens if you actually, take powers of i. Does this actually mean anything? Is it mathematically interesting. What is i^(1/2) other than just i^(1/2)?

 

[mathman]

i (like all complex numbers) has a polar representation. i=e^i(pi/2). Roots of i can be obtained by dividing the exponent. For square root divide by 2. To get all the roots use the fact that e^i(2npi)=1, for all n. So to get the other square root, use n=1.

 

[flatmaster]

Ok. I see how to calculate powers of i now, but is this at all mathematically or physically ineteresting?

 

[mathman]

Ok. I see how to calculate powers of i now, but is this at all mathematically or physically ineteresting?

My answer is yes, but to explain it would require descriptions of lots of physics theory as well as mathematics.

 

[CRGreathouse]

Ok. I see how to calculate powers of i now, but is this at all mathematically or physically ineteresting?

Extremely interesting. kth roots of unity (of which powers i^(1/n) are a special case) have a wide variety of applications.

 

[HallsofIvy]

i^(1/2)= sqrt(2)/2+ i sqrt(2)/2 or sqrt(2)/2- i sqrt(2)/2.

 

[Tobias Funke]

"What is i^(1/2) other than just i^(1/2)"

Do you mean what are physical interpretations of it, or what are other ways to write it? If it's the latter, then knowing the geometric interpretation of complex number multiplication helps a lot. Multiplying two complex numbers corresponds to adding their angles (with the positive x-axis) and multiplying their absolute values.

Now it's easy to see what two numbers squared give i. They must have magnitude 1, and since i makes an angle of 90 degrees, one root must make an angle of 45 degrees. Another solution would be the number on the unit circle that makes an angle of 225 degrees. According to the fundamental theorem of algebra, those are the only two solutions.

 

[flatmaster]

"What is i^(1/2) other than just i^(1/2)"

Do you mean what are physical interpretations of it, or what are other ways to write it? If it's the latter, then knowing the geometric interpretation of complex number multiplication helps a lot. Multiplying two complex numbers corresponds to adding their angles (with the positive x-axis) and multiplying their absolute values.

Now it's easy to see what two numbers squared give i. They must have magnitude 1, and since i makes an angle of 90 degrees, one root must make an angle of 45 degrees. Another solution would be the number on the unit circle that makes an angle of 225 degrees. According to the fundamental theorem of algebra, those are the only two solutions.

This is what I was looking for. A geometrical way to express the analytical expression. I guess I didn't explain that very well.