In a nutshell, adding and multiplying complex numbers follow the usual
rules of arithmetic as long as one treats them as polynomials in \(i\) with
the provision that \(i^2\) can always be simplified to \(-1\).

In what follows, \(z = a+bi\) and \(w = c+di\) are complex numbers with
\(a,b,c,d\in \mathbb{R}\).

Addition of complex numbers

Addition is as follows:
\[z+w = (a+bi)+(c+di) = (a+c) + (b+d)i.\]

Examples

\( (1+2i) + (-2+i) = -1 + 3i \)

\( i + (3+i) = 3 + 2i \)

Multiplication of complex numbers

Multiplication is carried out as follows:
\begin{eqnarray*}
z w & = & (a+bi)(c+di) \\
& = & a(c+di)+bi(c+di) \\
& = & ac + adi + bci + bdi^2 \\
& = & ac + adi + bci - bd \\
& = & (ac-bd) + (ad+bc)i.
\end{eqnarray*}

Examples

\( (1+2i) (-2+i) = 1(-2+i) + 2i(-2+i)
= -2 + i + (-4)i - 2 = (-4) + (-3)i \).
For convenience, we simply write \(-4-3i\).