Pi-th Root of -1

Date: 12/15/2000 at 06:44:14
From: Lukyan Mraz
Subject: Non-integer roots of negative numbers

Hello,

I am trying to find the Pi-th root of a negative number. I know that 
sqrt(-x) = i*sqrt(x) and that (-x)^(1/3) = -(x^(1/3)). I also know 
that the nth root of x = x^(1/n). With pi being transcendental, it 
seems harder than finding something like sqrt(2)-th root. I can't even 
figure out the 1.5th root of a negative number. I think it has 
something to do with the definition of i, and it seems like something 
that isn't dealt with much. Please help me understand these types of 
problems.

Thank you,
Lukyan

Date: 12/15/2000 at 10:29:44
From: Doctor Mitteldorf
Subject: Re: Non-integer roots of negative numbers

Dear Lukyan,

The problem is that there are too many answers. We know that quadratic 
equations have two roots, and cubic equations have three roots, etc. 
In particular, the equation x^2 = -1 has two roots and x^3 = -1 has 
three roots. An equation like x^(5/2) = -1 can be rewritten as 
x^5 = (-1)^2, which has five roots.

But what of the equation x^pi = -1? Usually, functions of an 
irrational number can be defined as the limit of a sequence of 
rationals that get closer and closer to it. For example, the square 
root of pi can be computed as the limit of a series that starts with 
sqrt(3), then sqrt(3.1), then sqrt(3.14), etc. Changes in these 
numbers get smaller and smaller, and they converge on a number that we 
can call the square root of pi.

We can do the same thing with 2^pi. 2^3 is close to 2^3.1, which is 
closer yet to 2^3.14, and the series converges to a real number.

But when we try the same trick with (-1)^pi, the series doesn't 
converge, so we don't know how to define the result. We could start 
with (-1)^3, which is -1. And (-1)^3.1 = (-1)^(31/10) has ten 
different interpretations, only one of them close to -1. Worse, 
(-1)^3.14 = (-1)^(314/100) has 100 different values, and none of them 
is close to -1, because 314 is even while 31 was odd.

We can't let our definition of (-1)^pi depend on whether the fractions 
we use to approximate pi have even or odd numbers in the numerator, so 
we just say that the expression is undefined.

Any expression x^y is undefined if x is negative and y is irrational.

- Doctor Mitteldorf, The Math Forum
  http://mathforum.org/dr.math/