The arbitrary of assigning the labels $\mathbf{i}$ and $\mathbf{-i}$

Question

I have heard "Every equation involving complex numbers and expressions retains its truth value if every complex number/variable is replaced by its complex conjugate."

But, I don't understand how this can be the case. An intuitive reason I recently checked up on is that "the labels $\mathbf{i}$ and $\mathbf{-i}$ are really arbitrary, there is no way to know which is which."

I am not sure I follow. Just because

1. $-(-i) = i$
2. $i, -i$ are roots of the same equation $x^{2} + 1 = 0.$

doesn't prove that $\mathbf{i}$ and $\mathbf{-i}$ are truly arbitrary, right?

So, why are the labels $\mathbf{i}$ and $\mathbf{-i}$ arbitrary? Is there any intuitive way of understanding this?

Answer 1

The difference between the labels $i$ and $-i$ is whether or not you like to draw your $y$-Axis from bottom to top or from top to bottom. It is the difference whether or not you say clockwise or counter-clockwise the mathematical positive direction of rotation. A matter of convention.

Answer 2

I think the statement carries less meaning that a first reading suggests.

I would interpret this is the following way: You are given some variables $z_1,...,z_n$ and an equation. Let $A$ be the set of values of $(z_1,...,z_n)$ such that the equation is true.

Then $(z_1,...,z_n) \in A$ iff $(\overline{z_1},...,\overline{z_n}) \in \overline{A}$.

Answer 3

Depending on the algebraic structure under consideration, we have units which are elements $u_i$ such that in the same structure is an element $u_i^*$ such that $u_i\cdot u_i^* = 1$. As our algebraic structure gets more involved we may make other demands like the norm of $u_i$ must be $1$. In a sense not unlike how we can factorize integers uniquely into primes up to ordering we can say things like the GCD of 24 and 14 is 2 up to multiplication by a unit since the set of all multiples of $-2$ is the same as the set of all multiples of $2$ (in, say, $\mathbb{Z}$). This doesn't mean all units are indistinguishable. $i$ is different from $-i$ without question.