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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 = [i1/2]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 11/4, 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 AB, 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 AB 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)?