Complex Numbers 1

Complex Numbers 1

1. i is defined as the root of -1. It is imaginary

2. i^2 = -1

3. A complex number has a real part and an imaginary part i.e. z = 5 + 7i

   • Real part = 5 & imaginary part = 7(times i)

4. If a quadratic equation has a complex root, the other root is the complex conjugate pair of that root

   • zz* = always real, as is z+ z*
   • z^2 + bz + c = 0 has roots (alpha, beta)
     • (z-alpha)(z-beta) = 0
     • z^2 - (betaz)-(alphaz)+(alphabeta) = 0
     • b = - (alpha + beta)
     • c = (alpha * beta)

5. Factor Theorem

   • If f(x) is a polynomial then,
     • if f(p) =0, x = p is a root f(x) and (x-p) is a factor
     • (x--p) is a factor of f(x), x = p is a root of f(x) and f(p) = 0
   • Box method

6. Complex conjugates are a reflection of eachother in the real axis of the argand diagram

   • Modulus is the distance of a complex number to the origin
   • The argument is an angle measured from the positive real axis - Will always be between 180 and -180