Pi-th Root of -1Date: 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/ |
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