18.1: Complex numbers
Informally, a complex number is a number that can be put in the form
\[z=x+i\cdot y, \]
where \(x\) and \(y\) are real numbers and \(i^2=-1\) .
The set of complex numbers will be further denoted by \(\mathbb{C}\) . If \(x\) , \(y\) , and \(z\) are as in 18.1.1, then \(x\) is called the real part and \(y\) the imaginary part of the complex number \(z\) . Briefly it is written as
\[x=\text{Re} z \ \ \ \ \text{and} \ \ \ \ y=\text{Im} z.\]
On the more formal level, a complex number is a pair of real numbers \((x,y)\) with the addition and multiplication described below; the expression \(x + i\cdot y\) is only a convenient way to write the pair \((x,y)\) .
\[\begin{aligned} (x_1+i\cdot y_1) + (x_2+i\cdot y_2) &:= (x_1+x_2) + i\cdot(y_1+y_2); \\ (x_1+i\cdot y_1)\cdot(x_2+i\cdot y_2) &:= (x_1\cdot x_2-y_1\cdot y_2) + i\cdot(x_1\cdot y_2+y_1\cdot x_2). \end{aligned}\]