There's a slightly less facetious way to make this distinction: (1) is only an automorphism of C as a metric space; (2,3,4) are also automorphisms of C as an additive group; (5) is also an automorphism of C as a ring (or field). APG
Sent: Monday, March 02, 2015 at 11:51 PM From: "Fred Lunnon" <fred.lunnon@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] ln
Below the distinction should be made that 5) constitutes a symmetry of the complex numbers; 1) -- 4) are symmetries of your graph paper!
WFL
On 3/2/15, Mike Stay <metaweta@gmail.com> wrote:
Given an unadorned plane, you get the complex numbers by *choosing* 1) a point for the origin, 2) a line for the real numbers, 3) which side of the line is positive 4) a unit length, and 5) a handedness.
Each of the five choices gives a symmetry: 1) translation, 2) rotation, 3) negation, 4) scaling, and 5) complex conjugation.
On Mon, Mar 2, 2015 at 11:51 AM, Dan Asimov <asimov@msri.org> wrote:
But what if you inadvertently placed -i at (1,0) instead of i.
--Dan
On Mar 2, 2015, at 11:35 AM, Michael Kleber <michael.kleber@gmail.com> wrote:
Sheesh -- of *course* you can tell i from -i. When you draw the complex plane, i is the one at (0,1), and -i is the one at (0,-1). Obviously.
On Mon, Mar 2, 2015 at 2:18 PM, Dan Asimov <asimov@msri.org> wrote:
Yes, good point. Surely it's less trouble if we pretend it makes sense to refer to i and -i separately.
But of course, given that U and V are each a root of
X^2 + 1
, we could equally tell whether they are the same or different according as
UV = -1 or UV = +1
, without referring to either of them separately.
--Dan
On Mar 2, 2015, at 10:19 AM, Eugene Salamin via math-fun < math-fun@mailman.xmission.com> wrote:
When a square root of -1 appears more than once in an exposition, it is necessary to have a way of indicating whether two such occurrences are the same or different square roots. The use of i and -i satisfies that requirement. -- Gene
From: Dan Asimov <dasimov@earthlink.net> To: math-fun <math-fun@mailman.xmission.com> Sent: Sunday, March 1, 2015 7:13 PM Subject: Re: [math-fun] ln
What bothers me is the unqualified use of sqrt(-1) to mean i, since I think that should always be described as a convention.
Sometimes it seems to me we have no right to use a symbol for i, just because there is no way to distinguish between i and -i. Maybe we should be allowed only to refer to both of them at once.
--Dan
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