3.2. Complex numbers

A complex number can be represented as a sum \(a+b\mathrm{i}\) where \(a\) and \(b\) are real-valued numbers, and \(\mathrm{i}\) is the imaginary unit, defined via

(3.43)\[\mathrm{i}^2 = -1\]

The complex conjugate of a complex number is indicated with a superscript asterisk and is defined as

(3.44)\[(a + b\mathrm{i})^* = a - b\mathrm{i} \qquad (A\mathrm{e}^{\mathrm{i}\phi})^* = A\mathrm{e}^{-\mathrm{i}\phi}\]

(Here, \(a\), \(b\), \(A\) and \(\phi\) are real-valued.)

The norm of a complex number is

(3.45)\[|c| = \sqrt{c^*c} = \sqrt{(a-b\mathrm{i})(a+b\mathrm{i})} = \sqrt{a^2+b^2}\]