# Aplety

## Complex numbers

You can find there three aplets which illustates basic operations with complex numbers: addition, multiplication and finding roots.

This is the formula for addition of complex numbers given in algebraic form:$$(a_1 + i \cdot b_1) + (a_2 + i \cdot b_2) = (a_1+a_2) + i \cdot (b_1+b_2)~,$$ where $a_1,a_2, b_1, b_2 \in \mathbb{R}$ and $i$ denotes the imaginary unit.
This is the formula for multiplication of complex numbers written in algebraic form: $$(a_1 + i \cdot b_1) \cdot (a_2 + i \cdot b_2) = (a_1\cdot a_2 - b_1\cdot b_1) + i \cdot (a_1\cdot b_2 +a_2 \cdot b_1)~.$$
The multiplication of complex numbers given in the trigonometric form is given by the formula: $$\left(\rho_1(\cos(\phi) + i \cdot \sin(\phi)) \right) \cdot \left(\rho_1(\cos(\psi) + i \cdot \sin(\psi)) \right) =$$ $$(\rho_1\cdot\rho_2)(\cos(\phi+\psi) + i \cdot \sin(\phi+\psi))~.$$
The n-th degree root of a complex number $z$ is any complex number $u$, such that $u^n = z$. If $n>0$ is a natural number, and $z$ is a complex number other than zero, then there are exactly $n$ different n-th degree roots of the number $z$ of degree $n$. You can test this in the following diagram: