Conceptual question about the imaginary number $i$

Question

One of the first things we see in our first complex analysis class is the standard way of introducing us to the imaginary unit $i$ which is to think of a solution to the equation $$x^2=-1$$ Obviously, since a real number has the same sign, multiplying it by itself will result in a positive quantity. And so since the imaginary unit breaks that rule, can we think of $i$ as having two different signs? Is this idea linked to a more general one in say abstract algebra? Thanks

Answer 1

Rather than thinking of $i$ as having two different signs, think of it as having no sign at all. The complex plane is different than the real line in many respects, one of which is the loss of an ordering. You can long longer speak of positive and negative numbers. It can be proven algebraically (this is not hard) that $\mathbb C$ is a non-orderable field. By introducing a solution to the equation $X^2=-1$ we are forced to abandon the nice algebraic property of the order of the reals. We gain something, we loose something else.

Answer 2

Complex numbers don't have a sign.

As you'll probably see, it's often convenient to visualize complex numbers as points on the $xy$-plane, where the $x$-axis is the real numbers and $i$ is the point $(0,1)$. Positive numbers are the $x$-axis with $x>0$, or directly right of the origin. So asking whether $i$ is positive or negative is like asking whether the direction from $(0,0)$ to $(0,1)$ is the same as the direction to $(1,0)$ or the same as the direction to $(-1,0)$. The answer is neither; it's in an entirely different direction.

Answer 3

The concept behind $i$ is that it is not a real number, but a quantity which can operate on real numbers. Since $i$ doesn't exist in the real number line, it doesn't have to follow the same rules. There is no real number equal to $i$, but if we look orthogonal to the real line, we can find $i$ in what we call the Complex Plane.

Answer 4

I think it's worth pointing out that $i$ and $-i$ are interchangeable. They are different things, and they are both solutions to the equation $x^2 = -1$, and on the complex plane they are drawn in different directions away from the horizontal axis, and that's about all there is to it. We indicate one with a minus sign not because it's lesser and the other is greater, but because they represent opposite quantities and they cancel each other when added, like just like positive and negative real numbers.

When you're working with complex numbers, "norm" or "magnitude" is a more useful concept than "sign". The field of complex numbers is not ordered, meaning you can't really say if one number is greater than another, although you can compare their distance and angle from the origin. (You'll learn this soon enough when you get to Euler's formula.)