What symmetry property in complex space is related to |a+ib| = |b+ia|?

Question

What symmetry property in complex space is related to the fact that the absolute value of numbers $|a+ib| = |b+ia|$ are equals?

Answer 1

reading the comment of fleablood and Widawensen about the numbers laid of a circle, I think that the correct answer is:

The wanted property is that the absolute value of a number must be invariant under any axis rotation. In this case all numbers $z_1 = a + ib$, $z_2 = -a + ib$, $z_3 = a - ib$, $z_4 = -a - ib$, $z_5 = b + ia$, $z_6 = -b + ia$, $z_7 = b - ia$ and $z_8 = -b - ia$ and other infinity have the same absolute value.

Answer 2

If you identify the complex space $\mathbb{C}$ with $\mathbb{R}^2$ by sending $z = a + ib$ to $(a,b)$, the norm of a complex number $z$ is the same as (regular, Euclidean) the norm of the vector $(a,b) \in \mathbb{R}^2$. Consider the following two maps:

1. The conjugation map $T \colon \mathbb{C} \rightarrow \mathbb{C}$ given by $T(a + ib) = T(z) = \overline{z} = a - ib$.
2. The multiplication by $-i$ map $S \colon \mathbb{C} \rightarrow \mathbb{C}$ given by $S(z) = S(a + ib) = (-i)z = (-i)(a + ib) = b - ia$.

The map $T$ is not a $\mathbb{C}$-linear map but under the identification of $\mathbb{C}$ with $\mathbb{R}^2$, it corresponds to an $\mathbb{R}$-linear reflection map across the $x$-axis which preserves the norms of vectors. The map $S$ is a $\mathbb{C}$-linear map that corresponds under the identification of $\mathbb{C}$ with $\mathbb{R}^2$ to a rotation map by $\frac{\pi}{2}$ degrees clockwise which also preserves norms of vectors.

Hence, the composition $T \circ S$ also preserves the norms of vectors and

$$ (T \circ S)(a + ib) = T(b - ia) = b + ia $$

which implies that $|a + ib| = |b + ia|$ (which of course can be verified directly using the definition of the norm).

Answer 3

In $\mathbb R^2$, the map $(a,b) \mapsto (b,a)$ is reflection in the 45-degree line $y=x$. This map is (of course) an isometry of the plane, so it is an isometry of $\mathbb C$.