Re: are imaginary exponents "truly" defined that way?

In article
<1167956275.233775.130190@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>
,
jiri.lebl@xxxxxxxxx wrote:

matt271829-news@xxxxxxxxxxx wrote:
I am perpetually confused on this point. Some sources (such as
Mathworld) treat x^(1/2) as single-valued and identically equal to
sqrt(x), and others seem to think that it's multi-valued. There is
similar contradictory evidence about the definition of other fractional
powers - whether they are defined as single-valued (i.e. principal
value is assumed) or multi-valued. You seem fairly certain that
x^(1/2), at least, is properly defined as multi-valued. Is this
definition used by all mathematicians? Is the same true of other
fractional powers?

It is a definition only and depends upon its use. It is nonsensical to
talk about "x^(1/2)" without any context. It is just a formal symbol,
it has no meaning, nor value, nor anything like that on its own. There
is no point in trying to get "one true definition" of this symbol. It
is used in several different contexts and in different contexts it has
different meanings. If you read mathematics and the symbol "x^(1/2)"
is used without it being defined (or implicitly defined), then someone
is trying to pull a fast one on you. Note the following "proof" that
complex numbers are fake:

i = (-1)^(1/2) = (1/-1)^(1/2) = 1 / (-1)^(1/2) = 1/i = i / (i*i) = -i

This is a bit tricky to analyze with the level of
ability attained by someone seeing it for the first
time. The fault lies not where one might expect it.

--
Michael Press
.

Relevant Pages

  • Re: are imaginary exponents "truly" defined that way?
    ... it has no meaning, nor value, nor anything like that on its own. ... is no point in trying to get "one true definition" of this symbol. ... is used in several different contexts and in different contexts it has ... reals subset of the reals (or complex plane) or if you consider it your ...
    (sci.math)
  • Re: from time to time
    ... > following contexts ... ... Its basic meaning in English is "occasionally"; ... > "The regulations from time to time in force..."? ... regulation at the applicable future time is effective. ...
    (alt.usage.english)
  • Re: NIP arrived.
    ... > change in meaning, when it is not. ... and the one excluding it is just as significant as the ... > traffic, some contexts might include the expected or typical traffic, ... > whilst other contexts might include the specific vehicles travelling ...
    (uk.rec.driving)
  • Re: NIP arrived.
    ... specific dynamic nature of the traffic is significant to the change in ... The difference between having one meaning including the street furniture, ... some contexts might include the expected or typical traffic, ...
    (uk.rec.driving)
  • Re: Candy Everybody Wants (was Re: Why do we need let?)
    ... determining which sort of meaning is *intended* for a particular BEGIN ... different meaning in different contexts. ... I define macros using define-syntax and use them by name. ... define-record will have different meaning in different contexts. ...
    (comp.lang.scheme)