Re: how can I return nothing?

Tor Rustad said:

Richard Heathfield wrote:
Michal Nazarewicz said:

richard@xxxxxxxxxxxxxxx (Richard Tobin) writes:

In article <1xgzve9ftnef0.qkfno80gbnsf.dlg@xxxxxxxxxx>,
Coos Haak <chforth@xxxxxxxxx> wrote:

i nor -i is the square root of -1.
What makes you think that?
The same thing I've pointed. In real number domain a square root of
x is defined to be _nonnegative_ number which squared gives x. You
cannot apply analogical definitions to complex numbers since there is
no such think as nonnegative complex number (or nonpositive for that
matter).

Sorry, but I must disagree. I can accept that there is no such thing as
a positive imaginary number, and no such thing as a negative imaginary
number,

Im(z) has a sign, just like Re(z) do.

Yes, but I can *accept* an argument that these signs are at right angles to
the concepts of negative and positive. (I can *also* accept the argument
that the distinction is of no consequence. Just not both at the same
time.)

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. Does that mean that real
numbers cannot be non-negative or non-positive?

A complex number has a (possibly zero but usually not)
real number element which is sufficient to displace it off the zero line
in the complex plane. Most numbers in the complex plane are either to
the right of the zero line (positive) or to its left (negative),
regardless of the value of their imaginary co-ordinate.

Same argument could be used for Im(z) too.

No, it can't, at least not without spinning the complex plane around, thus
invalidating the argument for Re(z) in the instant that one validates it
for Im(z).

An analogous argument involving "up/down" can be made, however.

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.

What matters is whether the property of Im(z) that is analogous to the
property of sign in Re(z) is /also/ to be called 'sign'. Personally, I
think that's a reasonable thing to do, but I don't insist that others
agree with me.

--
Richard Heathfield <http://www.cpax.org.uk>
Email: -http://www. +rjh@
Google users: <http://www.cpax.org.uk/prg/writings/googly.php>
"Usenet is a strange place" - dmr 29 July 1999
.

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