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18.1: Complex numbers

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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}