Complex Numbers and Analysis
Overview
Complex Numbers
{% z = x + iy %}
where we take {% i %} to be a number such that
{% i^2 = -1 %}
Algebraic Properties
{% z_1 + z_2 = (x_1 + x_2) + i(y_1 + y_2) %}
{% z_1 \times z_2 = (x_1 x_2 - y_1 y_2) + i(y_1 x_2 + x_1 y_2) %}
Representing Complex Numbers as Vectors
Complex numbers can be represented as an ordered pair
{% z = (x,y) %}
with the corresponding algebraic Properties
{% z_1 + z_2 = (x_1 + x_2,y_1 + y_2) %}
{% z_1 \times z_2 = (x_1 x_2 - y_1 y_2, y_1 x_2 + x_1 y_2) %}
Implementation
The implementation of compex numbers will typically rely on the vector representation as given above.
The
complex numbers api
represents a complex number as an array of two numbers.
For example, the number {% 1 + 2i %} is represented as
