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 is as follows: \[z+w = (a+bi)+(c+di) = (a+c) + (b+d)i.\]
\( (1+2i) + (-2+i) = -1 + 3i \)
\( i + (3+i) = 3 + 2i \)
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*}
\( (1+2i) (-2+i) = 1(-2+i) + 2i(-2+i) = -2 + i + (-4)i - 2 = (-4) + (-3)i \). For convenience, we simply write \(-4-3i\).
\( 2i (2+i) = 4i + 2i^2 = 4i - 2 = -2 + 4i \).
Let \(z = 2i\) and \(w = -2+i\). Give the answer to each of the following. Express your answer in the form \(a+bi\) with \(a,b\in \mathbb{R}\).
\(z + i w\)
\(z w\)
\(z^3\)