complex numbers

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introduction

It was first created to support the fundamental theorem of calculus, since some quadratics, such as x² + 1 = 0 do not have any roots, however they needed 2 (or 1 repeated root). Hence i was created.

i² = −1

0 is an imaginary and real number

if Z = a + bi where i is the imaginary number then z̄

diary

intuition

Thing of the complex_numbers as a vector with a special multiplication defined to it, but to build the complex_numbers rigorously actually needs a lot of sophisticated ideas and such to it.

complex_numbers:~ ℂ = ℝ² + field multiplication ## rigour polar form is z = r(cos(θ) + isin(θ))

exponential form is z = e^iθ

regular form = a +bi

let u₁, u₂, v₁, v₂ ∈ ℝ, then the field multiplication for a complex_number in vector form is defined by $\begin{pmatrix} u_1\\ u_2 \end{pmatrix} \cdot \begin{pmatrix} v_1\\ v_2 \end{pmatrix} = \begin{pmatrix} u_1v_1 - u_2v_2 \\ u_1v_2 + v_1u_2 \end{pmatrix}$

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