The Elusive Imaginary
We know: (1)^{1/2} = i. But then we also have:
(1)^{2/4} = [(1)^{2}]^{1/4} = [1]^{1/4} = 1
(1)^{2/4} = [(1)^{1/4}]^{2} = [i^{1/2}]^{2} = i
And....
(1)^{3/6} = [(1)^{3}]^{1/6} = [1]^{1/6} = i
(1)^{3/6} = [(1)^{1/6}]^{3} = [i]^{3} = i
What's going on here? Are exponents never allowed to be ratios of numbers with common factors? But then what about...
(1)^{3/2} = [(1)^{3}]^{1/2} = [1]^{1/2} = i
(1)^{3/2} = [(1)^{1/2}]^{3} = [i]^{3} = i
Huh? Can you explain....?
Source: "ttwngcbt" on Community.livejournal.com/mathematics, 9/17/2006
Hint: What happens on your graphing calculator? Can you draw a picture? Any connection between the three forms...that will help resolve this confusion? What does the included diagram (from MathWorld) have to do with anything?
Solution Commentary: Vartiations of replies to "ttwngcbt"...do you agree with any of them?
First, "fuwang" says: When you get to 1^{1/4}, there are four fourth roots of 1: 1, i, i, and 1. The ambiguity lies in the choice of the particular root.
Second, "jestingrabbit" says: In general, if you write something like A^{B}, its ambiguous, especially if A isn't a positive real or B isn't a natural number. If you restrict A to being a positive real then A^{B} is well defined for all real B, if you accept that the range of your function is the positive reals. Similarily, what is meant when B is a nonnegative integer is also relatively unambiguous. But in general, its not as well defined as you might think...Slight correction: If B is negative that's also unambiguous.
And who can forget "e_to_the_ipi"'s response:
(1)^{1/2} = i or i
(1)^{1/2} = 1 or 1
(1)^{1/4} = [(1)^{1/2}]^{1/2} = (1)^{1/2} or (1)^{1/2} = 1 or 1 or i or i.
Again, are these correct explanations (and complete...satisfying)?
