Complex Numbers

Complex Numbers

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

Slide 2

• Addition & Subtraction
• To add/subtract 2 complex numbers, add/subtract the real parts and then the imaginary parts.
• Multiplication
• Treat the same as when expanding normal brackets.

Slide 3

• Division
• Complex Conjugate
• If 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 Numbers
• a+bi and x+yi are only equal if a=x and b=y

Slide 4

• Roots of Equations
• If 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 Roots
• For f(x) to have repeated factors/roots, f(a)=0=f'(a)