Fundamental Mathematics/Arithmetic/Arithmetic Number/Complex Number

Complex Number

 * Complex_conjugate_picture.svg

Complex Number
 * $$Z = A+iB = |Z| \angle \theta = \sqrt{B^2 + A^2} \angle Tan^{-1} \frac {B}{A} = |Z| e^{i\theta}\,$$

Complex conjugate Number
 * $$Z^* = A-iB = |Z| \angle -\theta = \sqrt{B^2 + A^2} \angle -Tan^{-1} \frac {B}{A}= -|Z| e^{i\theta}\,$$

Operation on 2 different complex numbers

 * {|width=100%


 * Addition || $$(A+iB) + (C+iD) = (A+C) + i (B+D)$$
 * Subtraction || $$(A+iB) - (C+iD) = (A-C) + i (B-D)$$
 * Multilication || $$(A+iB) + (C+iD) = (AC+BD) + i (AD+BC)$$
 * Division || $$\frac{(A+iB)}{(C+iD)} = \frac{(A+iB)}{(C+iD)} \frac{(C+iD)}{(C+iD)} = \frac{(AC+BD) + i (AD+BC)}{(C+iD)^2}$$
 * }
 * Multilication || $$(A+iB) + (C+iD) = (AC+BD) + i (AD+BC)$$
 * Division || $$\frac{(A+iB)}{(C+iD)} = \frac{(A+iB)}{(C+iD)} \frac{(C+iD)}{(C+iD)} = \frac{(AC+BD) + i (AD+BC)}{(C+iD)^2}$$
 * }
 * }
 * }

Operation on complex numbers and its conjugate

 * {|width=100%


 * Addition ||$$(A+iB) + (A-iB) = 2A$$
 * Subtraction || $$(A+iB) - (A+iB) = i 2B$$
 * Multilication || $$(A+iB) + (A-iB) = A^2 - B^2$$
 * Division || $$\frac{(A+iB)}{(A-iB)} = \frac{(A+iB)}{(A-iB)} \frac{(A-iB)}{(A-iB)} = \frac{A^2 - B^2}{(A-iB)^2}$$
 * }
 * Multilication || $$(A+iB) + (A-iB) = A^2 - B^2$$
 * Division || $$\frac{(A+iB)}{(A-iB)} = \frac{(A+iB)}{(A-iB)} \frac{(A-iB)}{(A-iB)} = \frac{A^2 - B^2}{(A-iB)^2}$$
 * }
 * }
 * }

In Polar form
 * $$Z \times Z^* = |Z| \angle \theta \times |Z| \angle -\theta = |Z|^2 \angle(\theta -\theta) = |Z|^2$$
 * $$\frac{Z}{Z^*} = \frac{|Z| \angle \theta}{|Z| \angle -\theta} = 1 \angle 2 \theta$$

Power of Z
Since
 * $$Z \times Z = Z^2 = ( |Z| \angle \theta ) (|Z| \angle \theta) = |Z|^2 \angle (\theta + \theta) = |Z|^2 \angle 2 \theta$$

Hence
 * $$Z^n = Z \times Z \times Z ... = |Z|^n \angle n \theta$$

Since
 * $$Z^* \times Z^* = (Z^*)^2= ( |Z| \angle -\theta ) (|Z| \angle -\theta) = |Z|^2 \angle (-\theta -\theta) = |Z|^2 \angle -2 \theta$$

Hence
 * $$(Z^*)^n = Z^* \times Z^* \times Z^* ... = |Z^*|^n \angle -n \theta$$

Euler formula
$$e^{i\theta} = \cos\theta + i \sin\theta\,$$ of which there is the famous case (for &theta; = &pi;):
 * $$e^{i\pi} = -1\,$$

More generally, $$z = x + yi = r (\cos\theta + i \sin\theta) = r e^{i\theta}\,$$

Eucleur's power can be expressed as complex number
 * $$e^{i\theta} = \cos\theta + i \sin\theta\,$$

Hence, conjugate of the complex number
 * $$e^{-i\theta} = \cos\theta - i \sin\theta\,$$

Adding complex number and its conjugate
 * $$e^{i\theta} + e^{-i\theta} = 2 Cos \theta$$
 * $$Cos \theta = \frac{1}{2} (e^{i\theta} + e^{-i\theta})$$

Minus complex number and its conjugate
 * $$e^{i\theta} - e^{-i\theta} = 2i Sin \theta$$
 * $$Sin \theta = \frac{1}{2i} (e^{i\theta} - e^{-i\theta})$$

de Moivre's formula
$$z^n = (\cos(x) + i \sin(x))^n = \cos(nx) + i \sin(nx)= r e^{in\theta}\,$$

for any real $$x$$ and integer $$n$$. This result is known as

Reference

 * Complex Number