Re: how can I return nothing?

Richard Heathfield wrote:
Tor Rustad said:

Richard Heathfield wrote:
Tor Rustad said:

Richard Heathfield wrote:
[...]

but I cannot accept that there is no such thing as a non-negative
*complex* number.
Just view a complex number as a vector, the useful properties of a
vector is the direction and magnitude. Those properties fully describe
it in a N-dimensional case too, no matter what coordinate system you
use.
You could use the same argument for real numbers.
No you can't, a scalar is a scalar.

A real number is a point on the real number line. A complex number is a point on the complex plane. If you can call one a vector, you can call the other a vector too. A real number is a special case of a complex number, so if a complex number is a vector, so is a real number.

I think you picture a single real axis.

In theoretical physics (e.g. special relativity), we rather use 3 such axis (x, y and z), as well as an imaginary axis for time.

We can talk about the sign of specific vector component, but defining
sign of a 4-vector, is quite pointless.

Likewise in R^3, vi can multiply a vector by (-1), vi can subtract vectors, in this sense.. sign of a vector have 3 degrees of freedom and "the 3D sign" will not fully specify the direction of a vector anyway.


Why is useful to define a sign of z as the projection along the Re
axis?
Ask any mathematician if they'd rather do without the concept of sign.
OK, I just did, the answer from my brother [1] was why introduce a
definition there is little use of?! He added, in his latest referee
report, he had criticized the author for using too many definitions.

So your brother is prepared to eschew negative numbers? Fine - that makes him a number theorist - but not all mathematicians restrict themselves to number theory.


His field of expertise, is rather in (complex) analysis. :)


Now, please name a mathematician who think that your "sign of z":

sign(z) = Re(z) / |Re(z)|

is a useful definition.

--
Tor <torust [at] online [dot] no>
.