Complex Number

Complex Number : combination of a Real Number and an Imaginary Number.

\[i = \sqrt{-1} \\ i^2 = -1 \\ \text{Comples Number: } a + b \underbrace{i}_{\sqrt{-1}}\]

Complex Plane : 以 Real Number 部份為 x 軸， i 為 y 軸；構成的平面。

據此想像 3 + 5i 為一個向量；則 (3+5i) + (4-3i) = (7+2i)

也可轉變向量 3 + 4i 為 distance = 5, angle = 0.927 radians. 的 polar form.

\[\begin{align} x + iy & = r \ \cos \theta + i \ r \ \sin \theta \\ & = r \big( \cos \theta + i \sin \theta \big) \\ & = r \ cis \ \theta \\ 3 + 4i & = 5 \ cis \ 0.927 \end{align}\]