# Useful Formulae

50 COMPLEXNUMBERS into the imaginary realm The familiar real numbersare contained within the larger realm of complex numbers. These are constructed by starting with the imaginary unit which is denotedi,and which satisfies unlike any real number i2 1 or i 1 Given any two real numbers aand b,the quantity a bi is called a complex number. Some complex equations are shown below a bi c di if and only if a c andb d a bi c di a c b di a bic di ac bd ad bc i i Polar representationuses the angle and radius r z rcos irsin rcos isin The exponential function exmay be extended into the complex plane using Eulers equation ei cos i sin From this we have both the mathematical gemei 1, and DeMoivres Theorem,for powers of a complex number z zn rein rnein rncosn isinn a bi c di ac bd c2 d2 bc ad c2 d2 51