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We all know that $i$ doesn't have any sign: it is neither positive nor negative. Then how can people use $-i$ for anything?
Also, we define $i$ a number such that $i^2 = -1$. But it can also be seen that $(-i)^2 = -1$. Then in what way is $-i$ different from $i$? If it isn't, then why do we even have something like a $-i$?
The vector $(1,-2)$ is also neither positive nor negative, yet there is value in having the operation $-(1,-2)=(-1,2)$.
Note: there is no natural meaning for "positive" for vectors, this is just an analogy.
You must take care, even though $2^2 = (-2)^2 = 4$ this doesn't mean that $2 = -2$, indeed the problem here (thinking about reals for a while), is that if you consider negative numbers, then the function $f: \mathbb{R}\to\mathbb{R}$ given by $f(x) = x^2$ is not injective.
Since you have created $i$ to be $i=\sqrt{-1}$, then you are obviously allowed to create a new number $-i = -\sqrt{-1}$. Also, if you think on the function $f: \mathbb{C} \to \mathbb{R}^2$ given by $f(a+bi) = (a,b)$, this function identifies $\mathbb{C}$ with $\mathbb{R}^2$, and in this sense, a complex number is a point in the plane. Note that we have $f(i) = (0,1)$ and that $f(-i) = (0,-1)$, so there are two things to note: every number that can be written $ai$ for some $a \in \mathbb{R}$ can be identified with a point in the $y$ axis, and introducing $-i$ we have not just the upper $y$ axis, but the complete one.
EDIT: Because of the comments I thought it would be good to add this edit. Above I've provided a geometrical view of what happens when we introduce $-i$, now I'm just going to say a little about the algebraic point of view. Since every complex number $z \in \mathbb{C}$ is $z=a+ib$ for some $a,b \in \mathbb{R}$ if we do not allow things like $-i$ we would have some issues. For instance, consider quadratic equations with complex conjugate roots: one of them would be a well defined element of $\mathbb{C}$, but the other one wouldn't! Also, it wouldn't be possible to present for each $z$ some $-z$ with the nice property $z + (-z) = 0$ and it turns out that $\mathbb{C}$ wouldn't be a field. Also, if you use the standard operations of sum between complexes and you have $z = a+ib$ and you want the additive inverse $-z$ you can prove rather easily that you must have $-z = (-a) + i(-b)$, so that $-i$ would just really be $i(-1)$, so in practice, the element $-i$ (and any other negative complex in general) appears naturally when we define the complexes as we do and when we introduce the operations as we do.
The imaginary number $i$ was discovered/invented/revealed as a solution the equation $$ x^2+1=0\tag{$\ast$} $$ That equation, and that equation alone, is the link between $i$ and the reals. However, there is another solution to $(\ast)$: $-i=-1\times i$ also satisfies $(\ast)$. Basically, this is because $(-1)^2=1$.
Since the link between $i$ and the reals is the same equation that is satisfied by $-i$, one can easily wonder, "which is which?" When we choose a solution to $(\ast)$, do we get $i$ or $-i$? In a sense, it doesn't matter; both satisfy $(\ast)$ and $i^2=-1$ is the property we use to do math in $\mathbb{C}$.
On the other hand, complex conjugation, swapping $i\leftrightarrow-i$, is an important isomorphism of $\mathbb{C}$. For instance, any polynomial with real coefficients that has $x+iy$ as a root, must also have $x-iy$ as a root. There are many ways to show this, but one of the most basic is by swapping $i\leftrightarrow-i$. This isomorphism does not affect the real coefficients of the polynomial, but it swaps $x+iy\leftrightarrow x-iy$.
In the end, we call one root of $(\ast)$, $i$, and the other, $-i$; it really doesn't matter which. They both satisfy $(\ast)$ and that is what is important.
I quickly scanned all the answers and didn't see anyone mention rotations, so I thought I would -- the simplest answer to your question is that $-i$ just represents a rotation of $\frac{\pi}{2}$ clockwise, whereas $i$ represents a rotation of $\frac{\pi}{2}$ counterclockwise. In other words, $i$ and $-i$ are inverses in the group $SO(2)\cong\left\{e^{i\theta}\,|\,\theta\in[0,2\pi)\right\}$ of rotations of the plane. From this perspective, $i$ and $-i$ are about as emphatically "different" as anyone could imagine.
EDIT: I thought I'd add that yet another way of saying what I've already said is that complex conjugation is just a way of swapping the orientation of the plane. (In particular, it is an orientation-reversing isometry.) The seemingly "sign-like" difference between $i$ and $-i$ is really just a difference in the choice of orientation.
After defining a new symbol $i$ such that its square is the negative number $-1$, you can do algebra with it. So, you have an additive operation on the set of symbols like $a+bi$, with $a,b\in \mathbb{R}$. But, if you remember, any element must have an inverse additive and you denote it by $-i$.
Edit:
ps: I suggest you to read about the construction of complex numbers field. How is it defined?
ps1: when you talk about vectors $(a,b)$ (nothing more than ordered pairs) you are identifying $\mathbb{C}$ with $\mathbb{R}^2$.
You are quite right! It is more or less an arbitrary convention that $i$ is represented by the point $(0,1)$ in the plane, and then necessarily $-i$ has to be represented by $(0,-1)$. It could have been the other way around! Regardless of the convention you choose, the PAIR $i, -i$ will always be the same as $(0,\pm 1)$.
In addition to another answer, I know a theorem in which we can extend the order of $\mathbb R$ to $\mathbb C$ such that two following are satisfied:
$P_1$: For all $a\in\mathbb R$ we have only one of $a>0,~a=0,-a>0$.
$P_2$: $a>0,b>0 \to a+b>0$.
In fact, if we let $\alpha=a+ib$ we can have $\alpha>0$ if and only if $a>0$ or $a=0,~b>0$. So for all $\alpha=a+ib$ we have just one of the following facts: $$a>0,a=0,-a>0$$ or $$a>0\to \alpha>0$$ or $$-a>0\to-\alpha>0$$ or if $a=0$ then $$b>0\to \alpha>0\\\\ b=0\to \alpha=0\\\\ -b>0\to-\alpha>0$$. Therefore, for any arbitrary $\alpha=a+ib\in\mathbb C$ we have just one of $$\alpha>0,~ \alpha=0,~-\alpha>0$$ I think this make a bit clear what we need here.
The signs "+" and "-" do not exactly correspond to positive and negative! The negative of a number (or any element) in a group is just the inverse of that element with respect to the group operation. We created $i$ as a solution to $x^{2} + 1 =0$ and extended the class of real numbers. However, this extended class had to be made into a group. So, $-i$ is just the additive inverse of $i$ in $\mathbb{C}$ with $0$ as the identity element.
Complex numbers $\mathbb{C}$ is a set of ordered pairs $(a,b)$ where $a \in \mathbb{R}$ and $b \in \mathbb{R}$ with two arithmetic operations complex addition and complex multiplication defined by \begin{eqnarray} (a_1,b_1) + (a_2,b_2) & = & (a_1 + a_2,b_1 + b_2) \\ (a_1,b_1) \ \cdot \ (a_2,b_2) & = & (a_1 \cdot a_2 - b_1 \cdot b_2, a_1 \cdot b_2 + b_1 \cdot a_2) \ . \end{eqnarray} Here the new operations $+$ and $\cdot$ on the left hand side are defined using known operations $+$ and $\cdot$ on the right hand side defined on $\mathbb{R}$. Note that substraction is defined by the unique opposite element by \begin{equation} a-b = a+(-b) \ . \end{equation} It can be shown that $\mathbb{C}$ is a field, that is a set with two opeations addition and multiplication, write $(\mathbb{C}, +, \cdot)$, that satisfies the field axioms. Consider now the subset $F = \{(a,0)|a \in \mathbb{R}\} \subset \mathbb{C}$. It can be again shown that $F$ is a subfield of $\mathbb{C}$. In addition $F$ is isomorphic to $\mathbb{R}$, written $F \cong \mathbb{R}$ with the isomorphism mapping $\Phi$ defined by $\Phi(z) = \Re{(z)}$, where \begin{equation} \Re{((a,b))} = a \ . \end{equation} A field isomorphism mapping $\Phi$ is always a bijection and preserves all algebraic strucure, that is \begin{eqnarray} \Phi(z_1+z_2) & = & \Phi(z_1) + \Phi(z_2) \\ \Phi(z_1 \cdot z_2) & = & \Phi(z_1) \cdot \Phi(z_2) \end{eqnarray} for every $z_1$ and $z_2 \in F$. For the $\Phi$ defined above it can be shown using the definitions of complex addition and complex multiplication. It can be seen that $\Phi$ maps the unique neutral elements $0 = (0,0)$ and $1 = (1,0)$ of complex addition and multiplication correctly to the corresponding real elements.
The complex number $i$ is defined by \begin{equation} i = (0,1) \ . \end{equation} This is just a definition and follows Gauss' lead to assimilate $x+iy$ to the ordered pair $(x,y)$. We could of course define $i$ otherwise and still be able to calculate using the new definition. However, the current definition of $i$ is useful. It satisfies $i^2 = -1$, that makes multiplications of the numbers of the form $x+iy$ straightforward. The conventional definition of $i$ is also beautiful. It contains only neutral elements of the real field. There is also another alternative $(0,-1)$ for $i$, but it assimilates $x+iy$ to the ordered pair $(x,-y)$, that contains one additional calculation of opposite element. Hence we keep the conventional definition and calculate \begin{eqnarray} i^2 & = & i \cdot i = (0,1) \cdot (0,1) = (0 \cdot 0 - 1 \cdot 1,0 \cdot 1 + 1 \cdot 0) \\ & = & (-1,0) = (-1,-0) = -(1,0) = -1. \end{eqnarray} The number $1 = (1,0) \in \mathbb{C}$ is the unique neutral element of complex multiplication. Note that in the mapping $\Phi$ its image is $1 \in \mathbb{R}$, that is the neutral element of real multiplication. Note that there is an element $z' = (-a,-b)$ for every $z = (a,b) \in \mathbb{C}$ that satisfies the opposite element axiom: $\exists z'$ s.t. $z+z'=0$, where $0 \in \mathbb{C}$ is a neutral element of complex addition. The complex number $-i$ is now \begin{equation} -i = -(0,1) = (-0,-1) = (0,-1) \ . \end{equation}
Most of the answers already take the whole structure of the complex numbers as given. Therefore let me go to a more elementary level, where we don't yet assume anything about the complex numbers except that they contain the real numbers, and at least one additional number $i$ which solves the equation $x^2+1=0$.
Now consider the equation $x+i=0$. I think you would agree that a set of numbers where we could not solve this equation would not be terribly useful. By definition, we denote the solution of $x+a=0$ with $-a$, so the solution here is $-i$. However that's up to now only a notation; for example, the same way we get $-0$, which we know is the same as $0$. Therefore the question is: What is $-i$?
First, we note that $-i$ cannot be a real number, because otherwise $i$ would be its negative (also a real number), and we already know $i$ is not a real number. So there remain two possibilities: Either $-i=i$, or $-i$ is a different non-real number than $i$.
We also want that the distributive law continues to hold.
Now assume $-i=i$, that is, $i+i=0$. Then we have $-2 = (-1) + (-1) = i^2 + i^2 = i(i+i) = i\cdot 0 = 0$. Now that's obviously wrong, therefore we cannot have $-i=i$. Thus $-i$ is a distinct number from $i$.
Edit: On A.P.'s request, I also add the proof that $-i=(-1)i$.
To prove that $-i=(-1)i$ means to prove that $i+(-1)i=0$, because that's how $-i$ is defined.
We obviously have, using the fact that $x=1x$ for all $x$: $$i +(-1)i = 1i + (-1)i =(1+(-1))i = 0i = 0$$
Note that you can prove essentially the same way that $i(-1) = -i$ (in case you take into account that multiplication might not turn out to be commutative, which actually would not be that unusual, although for complex numbers multiplication actually is commutative)- Therefore you also get $$(-i)^2+1 = i(-1)(-1)i+1 = i^2+1 = 0$$
Now for the proof of $(-1)i=-i$ as well as above when proving $-i\ne i$, I've used $0i=0$, which seems obvious, but to be sure, let's prove this as well. We have $$0i = (0+0)i = 0i+0i$$ Now we add $-(0i)$ to both sides of the equation (remember, the demand that for each $x$ we have a $-x$ which fulfils $x+(-x)=0$ was how we arrived at $-i$ at the first place), to get for the left hand side $0i + (-(0i)) = 0$, and for the right hand side $0i + 0i + (-(0i)) = 0i + 0 = 0i$ (note that in this, I've made another assumption, namely that addition is still associative even if expressions with $i$ are involved; but that's also a natural assumption which you'd not give up unless necessary).
Note that ultimately, all I needed was
The first two are given by the problem, while the rest are very fundamental assumptions which you wouldn't give up lightly.
The imaginary unit $i$ is just a unit, like $m$ (meters), $s$ (seconds), or $\hat{u}$ (a unit vector). Any unit can be multiplied by a scalar, so $-i$ is jus the result of multiplying the imaginary unit by $-1$, no different than $4i$, $6i$, or $2.5i$.
Comparing $\mathbb{C}$ with vectors in $\mathbb{R}^2$ the $i$ can be viewed as describing a direction in 2D space (vertical, if you want), and it gets its magnitude and orientation (i.e. sign) from the coordinate that is a real number. It turns out that defining $i$ as the $\sqrt{-1}$ (which doesn't exist in $\mathbb{R}$), we get a field extension of $\mathbb{R}$ that is algebraically very useful and has the "geometrical properties" of the usual $\mathbb{R}^2$ vectorspace.
You need to impose an orientation on the plane. Since we mathematicans like counterclockwise to be the positive direction, we think of $i$ as being "one-up" from zero. You could equally well work with the opposite. It's a "tom(ay)to vs. tom(ah)to" situation. I refer all readers to this (humorously)....
The complex number $i$ is the representation of an angle of $\pi/2$ between the real axis and imaginary axis and $-i$ mean that the angle is $-\pi/2$. So there is a huge difference between $i$ and $-i$ as they have $\pi$ radians phase difference.
If we say that $i^2$ = -1, then $(-i)^2 = (-1)^2(i)^2 = 1(-1) = -1$ as well. i can't be zero, since $0^2 = 0$, not -1. So, $i$ and $-i$ are distinct. You can't have one without the other, and they are defined identically. However, they are in effect defined as a pair: There are two numbers which squared give you -1; we call one of them $i$, and that makes the other $-i$.
Make your choice about which is which and stick to it for your equation or expression, and all will be well. If you like, you can change your mind about which is which every alternate Thursday (except on leap day), and as long as you're consistent within an equation that's just fine. Just don't tell anyone you did it; they'll only be confused.
I have only very recently learned about imaginary numbers, so I don't have any fancy advanced concepts to impart to you like the other people who have answered so far, but I can totally understand your confusion, as I went through it myself.
The way it was explained to me is that you need $-i$ so that then $-1$ has two square roots just like $1$ does. Then $i$ is the principal square root of $-1$, and $-i$ is the other square root, just as $1$ has two square roots ($-1$ and itself).
One of the things you can use numbers like $-1$ for is for symmetry in the complex plane. Look for example at the diagram near the top of this page: http://mathworld.wolfram.com/GaussianPrime.html You should see a little cross at the center. The numbers that make up that cross are (clockwise) $i$, $1$, $-i$, $-1$, plus $0$ in the center (or maybe it's $-i$, $1$, $i$, $-1$, I don't know for sure).
Before you spend too much effort mulling over the "meaning" of $-i$, you should consider the fact that if $x = -2$, then the "meaning" of $-x$ is something that has the value $2$.
