Complex Numbers

Complex numbers are a funny thing. Complex numbers are numbers that are made up a “real component”, and an “imaginary component”.

Let Z = A complex number

$$Z=X+iY$$

There also exists what is known as the complex conjugate. The complex conjugate is described as:

$$\bar{Z}=X-iY$$

• X and Y are real numbers, with i being an imaginary component. Y here is scaling the imaginary component.

To find the magnitude of Z, you can

$$|Z|=\sqrt{a^2+b^2}$$

$$i=\sqrt{-1}$$ $$i^2=-1$$

Complex numbers can be:

• Added
• Subtracted
• Multiplied
• Divided

Complex numbers can also be found in polar form. Polar form is made up of:

$$z=|z|e^{i\theta }$$

The magnitude of Z, times Euler’s formula. Euler’s formula is:

$$e^{i\theta }=cos\theta +isin\theta$$

The last thing to talk about here, is Demoivre’s Theorem.

$$(e^{i\theta }{)}^n=e^{in\theta }$$

It is a method of proving trigonometric identities.