how can I return nothing?

[Tor Rustad]

Tor Rustad said:

Richard Heathfield wrote:

[...]

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 vector however has direction,
which we typically describe by an angle, which is not a binary concept
(like +- sign).

IMO, your "sign of z", is useless redundant information, and has no
importance in more abstract cases.

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.


An analogy for a program, would be to put a function in the stdlib,
where only R.H. on Earth, wanted/needed to call this function!!


[1] http://www.diva-portal.org/ntnu/abstract.xsql?dbid=1257

 

[Richard Heathfield]

Tor Rustad said:

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.

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.

 

[Peter J. Holzer]

Tor Rustad said:

Ask any mathematician if they'd rather do without the concept of sign.

If you ask any mathematician he will probably tell you that complex
numbers indeed have no order. You can of course define any order you
want but each will be inconsistent with complex arithmetic.

hp

 

[Richard Heathfield]

Peter J. Holzer said:

If you ask any mathematician he will probably tell you that complex
numbers indeed have no order. You can of course define any order you
want but each will be inconsistent with complex arithmetic.

Who said anything about order? I was talking about sign, not order.

The sign of the real part of a complex number tells you whether its
corresponding point on the complex plane is to the right or left of the
imaginary line. The sign of the imaginary part of a complex number tells
you whether its corresponding point on the complex plane is above or below
the real line. This says nothing about the ordering of two complex
numbers.

 

[Richard Tobin]

Who said anything about order? I was talking about sign, not order.

Sign implies order:

a-b < 0 <=> a < b

-- Richard

 

[Keith Thompson]

Sign implies order:

a-b < 0 <=> a < b

I don't see a sign there, just a two-argument "-".

 

[Richard Heathfield]

Richard Tobin said:

Sign implies order:

And two signs?

a-b < 0 <=> a < b

Fine. Given struct foo { int a; int b; } x = { -2, 2 }, y = { 2, -2 };

which comes first, x or y?

 

[Ben Pfaff]

Fine. Given struct foo { int a; int b; } x = { -2, 2 }, y = { 2, -2 };

which comes first, x or y?

x and y are not objects within a single array, so the answer is
undefined

 

[Richard Heathfield]

Ben Pfaff said:

x and y are not objects within a single array, so the answer is
undefined

Indeed. And even if they were, the only ordering imposed on them would be
the trivial one of indexing. There is no ordering inherent in their
values, despite their adequate supply of signs. Signs do not, then, imply
ordering.

 

[Peter J. Holzer]

Peter J. Holzer said:

Who said anything about order? I was talking about sign, not order.

The sign of the real part of a complex number tells you whether its
corresponding point on the complex plane is to the right or left of
the imaginary line. The sign of the imaginary part of a complex number
tells you whether its corresponding point on the complex plane is
above or below the real line.

In both cases it tells you whether the part is less than zero.
"less than" implies order.

And what does your mythical "sign of the complex number" tell you?

It should tell you whether the complex number is less than zero. But it
turns out that it is impossible to define a sign which is consistent
with complex arithmetic.

This says nothing about the ordering of two complex numbers.

Right. Because complex numbers have two signs, not one.

hp

 

[Peter J. Holzer]

I don't see a sign there, just a two-argument "-".

The sign is in the "< 0" part.

hp

 

[Army1987]

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.

Im(z) is usually defined as a real number.
So Im(5 - 4i) is -4, which is a real number and has a sign.
(But that was not your point, I admit.)

 

[Army1987]

(e-mail address removed) (Richard Tobin) writes:

I don't see a sign there, just a two-argument "-".

" < 0" is often the definition of "negative".

 

[Army1987]

I'm not sure how useful that is. It seems to me that a complex number has
*two* signs - one for the real component and one for the imaginary. There
is no particular reason why they should both be the same.

The more common thing is to say that they have an argument, i. e.
an angle which is 0 for positive reals, pi for negative reals
and so on.

With the "two signs" thing, "5 + 12i" and "12 + 5i" have both
signs and magnitude in common, but they're not equal. So such a
notion is not very useful. Anyway, if I really need, I don't
object to talking about the "sign of the imaginary part" of a
complex number.

 

[Richard Heathfield]

Peter J. Holzer said:

"less than" implies order.

Only if there's only one sign. The moment you have two signs, you hit a
precedence issue.

And what does your mythical "sign of the complex number" tell you?

Complex numbers have *two* signs, not one. One is for the real part, and
the other is for the imaginary part. Neither is for the mythical part.

 

[Peter J. Holzer]

Peter J. Holzer said:

Only if there's only one sign. The moment you have two signs, you hit a
precedence issue.

No. To be able to say that one element of a set is "less than" another
element of a set, you must have defined an order between the elements.
That's independent of whether these elements have a sign or not. (For
example, the values representable in an "unsigned int" have an order,
but they have no sign).

You can define an order on the set of all possible char arrays (the
strcmp function does this), even though each char may have a sign.

Similarly, you could define a lexicographic order for complex numbers:
First compare the real part and if they are the same, compare the
imaginary part. But unlike char arrays, complex numbers have a defined
arithmetic, and it turns out that such an order (or any other order) is
not consistent with this arithmetic.

It isn't "two signs" which prevents defining an order on complex
numbers, it's the arithmetic of complex numbers.

Complex numbers have *two* signs, not one. One is for the real part, and
the other is for the imaginary part. Neither is for the mythical part.

So you agree that there is no such thing as a non-negative complex
number?

hp

 

[Army1987]

Peter J. Holzer said:

Only if there's only one sign. The moment you have two signs, you hit a
precedence issue.

Are you suggesting to use the lexicographic order on the complex
numbers?

 

[Army1987]

And what does your mythical "sign of the complex number" tell you?

Somewhere I have seen a definition of "sign of the complex number"
according to which the sign of -3 + 4i is -0.6 + 0.8i.
(Yes, *I* wouldn't call *that* a sign...)

 

[=?iso-2022-kr?q?=1B=24=29CHarald_van_D=0E=29=26=0F]

Somewhere I have seen a definition of "sign of the complex number"
according to which the sign of -3 + 4i is -0.6 + 0.8i. (Yes, *I*
wouldn't call *that* a sign...)

It's consistent with one definition for the real numbers, though: the
sign of zero is zero, the sign of non-zero x is x / abs(x).

 

[Tor Rustad]

Tor Rustad said:

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.