Re: 0th root

On Oct 16, 10:12 pm, Phil Oberforcher <ph...@xxxxxxxxxxxxx> wrote:
Help settle a bet.

What's the 0th root of a number?

The discussions in this thread brings to mind that every operation has
its mechanics (how to crunch the operands) and its applicable range
(the set of operands for which the crunching is defined). For the "n-
th root of x" operation there are two operands: n and x. The
mechanics is (can be) defined by x^(1/n). The usual range on n is any
non-zero complex number. The usual range on x is any complex number.
There is also the special case that the n-th root is not defined by
x^(1/n) for x = 0 and for n being any purely real and non-positive
number. To extend the range of the n-th root operation beyond the
ususal range just stated, one must also carefully define the mechanics
(number crunching) to be applied. In the present example, the n-th
root of x cannot be defined by x^(1/n) for n = 0, but must be defined
by some ancilary mechanics, several of which have been given in this
thread.

- MO
.

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