Is -i negative?

Question

It's pronounced "negative i," yet I'm told complex numbers aren't negative or positive. Is the - just arbitrary and unreflective of -i's sign?

I pronounce it "minus $i$". Of course if $x$ is a negative number, say $x=-4$, then $-x$ (pronounced "minus $x$") is a positive number. — Angina Seng, Apr 12 '19 at 05:00
$i$ doesn't have a "sign", neither does $-i$. But there are two complex numbers that square to minus one, and they are additive inverses, so if we call one of them $i$, we have little choice but to call the other one $-i$. — Gerry Myerson, Apr 12 '19 at 05:08
@LordSharktheUnknown but i is a constant, so negative i implies that one version of i is positive and the other negative (even if i is the negative one) — Alec, Apr 12 '19 at 05:14
@alec935 Is it your insistence on pronouncing $-$ as "negative" the source of your problem? The unary minus sign maps an object to its additive inverse. If you had a vector $v$ in three-dimensional space, would you insist one of $v$ and $-v$ was positive, and one negative? All this notation means is that they have the same length and point in opposite directions. — Angina Seng, Apr 12 '19 at 05:20
@alec935 It implies nothing of the sort. The complex numbers do not come with an order relation. — Robert Israel, Apr 12 '19 at 05:20

score 4 · Answer 1 · answered Apr 12 '19 at 15:21

To extend the real numbers so that the equation $x^2=-1$ has a root is the same as requiring that $x^2+1$ factorizes—say as $(x-a)(x-b)$. It is easy to see that $a$ and $b$ cannot be the same: they are two things. However, they are related: they must satisfy $a+b=0$. Another way of saying this is $a=-b$, or we could equally well say $b=-a$. Each is the negative of the other; but there is no way of deciding that one is primary and the other is derived from it—for example that $a$ is “positive” and $b$ is “negative”. These terms have no meaning off the real line. Rather than having to deal with two names (such as $a$ and $b$), along with the condition that each is the negative of the other, it is convenient just to arbitrarily appoint one to have a simple name, and attach a minus sign to it to denote the other: thus $\mathrm i$ and $-\mathrm i$.

While convenient, this choice of symbolism has the unfortunate effect of suggesting that $\mathrm i$ is in some sense the primary “square root of minus one” while $-\mathrm i$ is derived from it. But we can no more put them in order than we can put an order on Tweedledum and Tweedledee (or perhaps I should say Tweedledee and Tweedledum).

rschwieb · Accepted Answer · 2019-04-12T14:26:52.337

To paraphrase what has transpired in the comments, and bring some closure to the question: it looks like an ambiguity of understanding of "-/minus/negative."

Minus

Foremostly, $-$ is used as a binary operation: $a-b$, which one could comfortably call "minus." Since your question only involves the unary use of $-$, we'll skip this case in favor of the other two.

Negative

On one hand, we have concrete representation of real numbers and their notion of order which we have traditionally affixed the "-" prefix to the ones less than zero and called them negative numbers. In this case we talk about $-2$ and $3$ in $\mathbb R$ and there isn't any ambiguity about where they lie, because we follow the convention that the negative numbers are the ones below $0$.

This is wholly due to the total order on $\mathbb R$, which of course not all rings have. Rings that can't be totally ordered include things like $\mathbb C$ and finite rings.

Additive inverse

On the other hand, "-" the notational prefix for "additive inverse," connects pairs of elements (not necessarily distinct elements) by the relationship that the two elements add up to $0$. So, we can talk about $x\in\mathbb R$ and $-x\in \mathbb R$ without ever deciding if $x$ lies above $0$, below zero, or exactly at zero. The symbol in front doesn't presuppose anything about the ordering of $\mathbb R$, just its addition operation.

The problem here, perhaps, is that we don't have a snappy name we use consistently that connotes this usage. People say "minus x" and "negative x" interchangeably, without considering that the first suggests a binary operation and the second suggests an ordering.

A more accurate name for $-x$ would be "(additive) inverse of x," but it's a bit of a mouthful when reading expressions for a basic algebra class.

Thought experiment

Let's pretend we're in a universe where take the whole situation and transfer it over to multiplication in $\mathbb R$. We have the notation $x^{-1}$ which we're going to confuse in the same way.

In an ordered ring (like $\mathbb R$) call an element $x$ "tiny" if $0<|x|<1$, and "magnitudinous" if $|x|>1$ ($1$ is neither small nor large, and $0$ is conspicuously missing, and we'll let it stay that way.) In our new system, we'll also never write a tiny positive number starting with "$0.$" or "$-0.$", our concrete representation of it will just be as an inverse of a magnitudinous number: e.g. there is no $0.5$, only $2^{-1}$. So, you can immediately and conveniently recognize "tiny" numbers as those with $-1$ floating up and to the right, and one might pronounce $2^{-1}$ as "tiny 2$."

The analogous situation in this thought experiment is that people refer to $x^{-1}$ ambiguously as "divided by $x$" and "tiny $x$," but they only call it "the multiplicative inverse of $x$" on formal occasions.

Your question

I'm told complex numbers aren't negative or positive. Is the - just arbitrary and unreflective of -i's sign?

The first statement is correct in general (although one might still call the positive real numbers inside $\mathbb C$ positive.)

It is arbitrary in the sense that it is a notational choice. For example if you have any complex number $\alpha$, you could locate its additive inverse $\beta$ and write $\alpha=-\beta$ and it doesn't change anything about $\alpha$ besides the way it looks on paper.

Finally $i$ doesn't have a "sign" in the sense that real numbers "have a sign," because $\mathbb C$ can't be split up into a positive and a negative piece like the real numbers can, for the reasons mentioned before.