On the definition of the number $i$

Question

$i$ is defined as a root of the polynomial $x^2+1$. However, both $i$ and $-i$ obey this definition.

So, basically, there should be no way to distinguish the two (in the sense that replacing $i$ and $-i$ everywhere would yield the exact same results), right? But, operators such as $\Im$ (imaginary part) do let us distinguish between the two! How do we define the action of $\Im$ as well-defined?

Also, does this mean that the theory of complex numbers relies on the fact that we can distinguish between the roots? Trying to make sense of this makes me a bit confused, but I believe there is a simple explanation.

Secondly, does the definition assume that there are only $2$ complex roots to $x^2+1$? I'm having a hard time understanding "what comes before what", the fundamental theorem of algebra, the definition of $i$, the theory of complex numbers...?

This is the first time I've thought of the fact that $i$ is not uniquely defined.

I would love for a clear explanation. Thank you.

Answer 1

Good question! This is a subtle point that's easy to miss and points to a lot of rich mathematics.

So, basically, there should be no way to distinguish the two (in the sense that replacing $i$ and $-i$ everywhere would yield the exact same results), right?

Yes, that's correct. This is a good example of Galois theory: precisely because there is, in some sense, no way to distinguish between $i$ and $-i$, complex conjugation, which swaps them, is a symmetry of the theory, and preserves true statements about complex numbers (although it would be a bit annoying to spell out exactly which statements, e.g. "the imaginary part is positive" is not preserved, but "this complex number is a root of this polynomial" is).

But, operators such as $\Im$ (imaginary part) do let us distinguish between the two! How do we define the action of $\Im$ as well-defined?

We have to make a choice, and different ways of defining the complex numbers make the choice in different ways. If you define the complex numbers as the quotient ring $\mathbb{R}[x]/(x^2 + 1)$ then we simply decide that $x$, in this quotient ring, is the object we're going to call $i$ (even if we could also equally well have chosen $-x$). My preferred definition is to define the complex numbers as the subalgebra of $M_2(\mathbb{R})$ (the $2 \times 2$ real matrices) of matrices of the form

$$\left[ \begin{array}{cc} a & -b \\ b & a \end{array} \right].$$

Here the two square roots of $-1$ are the rotation matrices $\left[ \begin{array}{cc} 0 & 1 \\ -1 & 0 \end{array} \right]$ and $\left[ \begin{array}{cc} 0 & -1 \\ 1 & 0 \end{array} \right]$, which perform a $90^{\circ}$ clockwise resp. counterclockwise rotation of $\mathbb{R}^2$. You can pick either one of these to be $i$ but the standard preferred choice is the second one, the counterclockwise rotation. So we see from this that choosing which square root of $-1$ to call $i$ amounts to choosing an orientation of the plane.

Note that in the second definition, even without choosing which square root of $-1$ to call $i$, we can still define complex conjugation (it negates $b$) which means we can still write down the real part $\frac{z + \bar{z}}{2}$. We can write down the imaginary part in the sense that we can compute $\frac{z - \bar{z}}{2}$, but the result is not a real number but an element of a $1$-dimensional real vector space which we haven't chosen a basis of. Making a choice of $i$ fixes a basis of this vector space and then we can identify it with $\mathbb{R}$ by dividing by our choice of $i$.

Secondly, does the definition assume that there are only $2$ complex roots to $x^2+1$?

No, you don't need to assume that, although it is true and follows from general facts about fields (if you believe that the complex numbers form a field).