1.3 Conjugation, norm and distance
If \(z = x + iy\) with \(x,y\in \mathbb {R}\), then we set \(\bar z := x - iy\). The complex number \(\bar z\) is called the complex conjugate of \(z\).
If \(z_1\) and \(z_2\) are two complex numbers as in 1.1, then
Moreover, \(z_1^2 = x_1^2 + y_1^2\in \mathbb {R}_{\geq 0} : = [0,\infty )\).
Easy. Just multiply out both sides and check.
We set \(|z| := \sqrt{z\bar{z}}\). This number is called the norm or the magnitude of \(z\).
In general a complex number of the form \(x + 0i\) (with \(x\in \mathbb {R}\)) is said to be real and a complex number of the form \(0 + iy\) (with \(y\in \mathbb {R}\)) is said to be imaginary.
The complex plane is just the set of all complex numbers drawn as the Cartesian plane \(\mathbb {R}^2\). If \(x,y\in \mathbb {R}\), then we put the complex number \(z = x + iy\) at the point \((x,y)\in \mathbb {R}^2\). We write \(\Re z: = x\), which is called the real part of \(z\) and \(\Im z = y\), which is called the imaginary part of \(z\).
The \(x\)-axis is called the real axis, and the \(y\)-axis is called the imaginary axis.
If \(z_1\) and \(z_2\) are two complex numbers as in 1.1, then \(|z_1 - z_2|\) is the distance from \(z_1\) to \(z_2\).