I am wondering if there is any difference between non-positive and negative numbers?
I think that negative numbers mean "negative real numbers"
and "Non-positive numbers" are negative real numbers and complex numbers.
Is it right?
Since there is no way to tell complex numbers are negative or positive.
So when It says non-positive, I think complex numbers are included.
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$\begingroup$"Positive numbers", "Negative numbers", "Nonpositive" and "Nonnegative" only make sense when talking about the real numbers (or some other field with a positive and negative decomposition). It does not work in the context of complex numbers.
Positive (real) numbers: $\mathbb{R}^+ = (0,\infty)$
Negative (real) numbers: $\mathbb{R}^- = (-\infty, 0)$
Non-positive (real) numbers: $\mathbb{R}^-\cup\{0\} = (-\infty,0]$
Non-negative (real) numbers: $\mathbb{R}^+\cup\{0\} = [0,\infty)$
That is to say, the difference between positive and non-negative numbers is that for positive numbers it does not include zero, whereas for non-negative it does include zero.
When no distinction is made, it is assumed that the person means real numbers.
$\endgroup$ 1 $\begingroup$I have never seen "non positive" to include complex numbers. Although I see how you could see it that way. In my experience non positive means negative numbers and 0. So yes there is a difference because negative numbers don't include 0.
$\endgroup$ $\begingroup$I don't think I had ever heard the term "non-positive number" used in any way whatsover. But I suppose that theoretically you can use any term to mean whatever you want (including counter-intuitive meanings) as long as you define it. But for the sake of clarity and not causing needless confusion, I strongly urge you to avoid using "non-positive numbers" to mean negative real numbers and complex numbers having nonzero imaginary part.
Let's run a few examples:
- 4 is a positive number.
- $-7$ is a negative number, it is not positive.
- $i$ is a purely imaginary number with a real part of 0. But one could argue that $\Im(i) = 1$ and that that is a positive number. Similarly, $-i$ is a purely imaginary number but we could say it has a negative imaginary part, that $\Im(-i) = -1$.
- $1 - 3 \sqrt{-2}$ has a positive real part (I say that without any hesitation) and we could say that it has a negative imaginary part.
There is nothing wrong whatsoever with saying, for a given complex number $z$, that $\Re(z)$ is positive, negative or 0. But it is problematic to say that $\Im(z) \neq 0$ is positive or negative. Given $x \in \mathbb{R}$, do you consider that $\Im(xi) = x$ and that therefore $\Im(z) \in \mathbb{R}$? Or do you consider $\Im(xi) = xi$?
Maybe you're thinking of the similar term "non-negative number," which almost always means numbers such that $\Re(x) \geq 0$ and $\Im(x) = 0$. That term is often useful. For example,
All non-negative numbers have real square roots.
Negative numbers have imaginary square roots. Positive numbers don't. And 0 doesn't either, so in this application you want to lump it with the positive numbers. Under what circumstance would you want to lump together 0, the negative numbers and all numbers $z$ satisfying $\Im(z) \neq 0$?
$\endgroup$ 5 $\begingroup$You could declare it to mean that, but you shouldn't. When there is more than one way to extend a concept, professional mathematicians tend to balk, whether the extension feels natural to an amateur or not. And if the extension feels unnatural (as this one does to me), it can cause confusion even if you take care to define it right at the beginning.
Visualize the real number line with $0$ in the middle, the negative numbers to the left and the positive numbers to the right (you can orient it differently in your mind if you want, but then you'll have to adjust some of the following words). Then, if $x < 0$, that means $x$ is to the left of $0$, it is negative; and if $x > 0$, that means $x$ is to the right of $0$, it is positive.
Now visualize the complex plane coming into focus with the real number line as its central horizontal axis. The numbers $x$ on the real number line have $\Im(x) = 0$, and $\Re(x) \neq 0$ except for $x = 0$. Perpendicular to the real number line, you have the imaginary number line. The numbers $y$ on the imaginary number line have $\Re(y) = 0$, and $\Im(y) \neq 0$ except for $y = 0$.
Maybe we could call those complex numbers such that $\Re(z) > 0$ "positive" and those with $\Re(z) < 0$ "negative." But to do that feels wrong because we're ignoring the imaginary part of the number. The real and imaginary number lines divide the complex plane into four quadrants. Thus, a complex number $z$ such that $\Re(z) \neq 0$ and $\Im(z) \neq 0$ is in one of four quadrants, which we could call "positive-positive," "positive-negative," "negative-positive" and "negative-negative."
But this quadrant concept does not seem to be too useful. For example, in algebraic number theory, the concept of norm is much more useful. The norm of an algebraic integer in an imaginary quadratic integer ring (such as $1 + i$ in $\mathbb{Z}[i]$, for example) is either $0$ or a positive real integer. Numbers like $-1 + i$ and $1 - i$ are called "associates," but since they have the same norm, it's not of much concern which one is in which quadrant.
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