# Rotations in Complex Plane

by chaotixmonjuish
Tags: complex, plane, rotations
 P: 4,512 You're notation is still hard to follow. For instance, the letter z is usually used to express a complex number. z = x+iy. There are some basic tools you need to perform operations on complex numbers. 1 Euler's Equation. $$\ e^{i \theta} = cos(\theta) + i sin(\theta)$$ Where $$x=cos(\theta)$$ and $$y= sin(\theta)$$, a number in the form $$X+iY$$ can be expressed in the form $$\ Z e^{i \Theta}$$. (In this case 'Z' is a magnitude, a real positive value--so much for conventions.) X,Y,Z, and theta are all real valued numbers, and Z is positive. 2 Complex Conjugation. The complex conjugate of $$\ X+iY$$ is $$\ X-iY$$. You just negate the imaginary part to get the complex conjugate. 3 Division. $$c = a+ib$$ $$z = x+iy$$ What is the value of c/z expressed in the form X+iY ? $$\frac{c}{z} = \frac{a+ib}{x+iy}$$ Multiply the numerator and denominator by the complex conjugate of the denominator. $$\frac{c}{z} = \frac{(a+ib)(x-iy)}{(x+iy)(x-iy)}$$ $$\ \ \ \ \ \ = \frac{(a+ib)(x-iy)}{x^2 + y^2}$$ $$\ \ \ \ \ \ = \frac{(ax+by) + i(bx - ay)}{x^2 + y^2}$$ $$\ \ \ \ \ \ = \frac{ax+by}{x^2 + y^2} + i \frac{(bx - ay)}{x^2 + y^2}$$