Complex Numbers

Complex Numbers

Any number that can be written: Z=x+yi x=real party=imaginary parti=imaginary numberNB: Complex roots appear in pairs for equations with real coefficients.If n is even, there is no i in the answer ⇒ i^2=-1If n is odd, there is an i in the answer ⇒ i^3=-i

Addition & SubtractionTo add/subtract 2 complex numbers, add/subtract the real parts and then the imaginary parts.

MultiplicationTreat the same as when expanding normal brackets.

DivisionComplex ConjugateIf Z=x+yi, then the complex conjugate: Z̅=x-yi ZZ̅=x^2+y^2 To divide, write as a fraction and then use the same method as with surds.

Equality of Complex Numbersa+bi and x+yi are only equal if a=x and b=y

Roots of EquationsIf Z=a+bi is a root of a polynomial equation, then its complex conjugate Z̅=a-bi is also a root.Use the factor theorem to find the non-complex root.Use the quadratic formula to find complex roots.

Repeated RootsFor f(x) to have repeated factors/roots, f(a)=0=f'(a)