Operations in Cartesian form

In Cartesian form the sum, product and quotient are computed by treating $i$ as a letter and remembering the fundamental rule $i^2=-1$.

Property — Operations in Cartesian form

Given $z_1=a_1+ib_1$ and $z_2=a_2+ib_2$: $\begin{aligned} z_1 + z_2 &= (a_1+a_2) + i(b_1+b_2) \\ z_1 \cdot z_2 &= (a_1 a_2 - b_1 b_2) + i(a_1 b_2 + a_2 b_1) \\ \frac{z_1}{z_2} &= \frac{z_1\cdot\bar{z_2}}{z_2\cdot\bar{z_2}} = \frac{(a_1 a_2 + b_1 b_2) + i(a_2 b_1 - a_1 b_2)}{a_2^2 + b_2^2} \end{aligned}$ where $\bar{z} = a - ib$ is the conjugate of $z$, obtained by changing the sign of the imaginary part.

The product is obtained by applying the distributive property and substituting $i^2=-1$: $(a_1+ib_1)(a_2+ib_2) = a_1 a_2 + i a_1 b_2 + i a_2 b_1 + i^2 b_1 b_2 = (a_1 a_2 - b_1 b_2) + i(a_1 b_2 + a_2 b_1).$

Division is carried out by “rationalising” the denominator, that is by multiplying numerator and denominator by $\bar{z_2}$: the denominator becomes real, $z_2\cdot\bar{z_2} = a_2^2 + b_2^2 \in\mathbb{R}$.

Links

Topics: Complex numbers
Concepts: Conjugate · Cartesian form · Cartesian plane
Methods: Numeri complessi cartesiana
Skills: Calculate · Use formulae