What are complex numbers and why have them?

arithmetic starts with 0,1,2,3,4....  (natural number)
    3 - 8  (undefined)
Miss Flanigan (grade 4)   
    3 - 8 = -5  (integers)
Miss Eisenberg (grade 5)
    fractions: allow you do divide by anything not zero 
    ratainal number

real numbers extend rationals.

x^2 + 1 

Extend the real nubers with i
i^2 = -1
Complex numbers!

a + bi where a, b are real 


add

z = 5 + i
w = 4 - 3i

z + w = 9 - 2i

combine like terms; non=i and i terms are unlike

subract

z - w = 5 + i - (4 - 3i)
      = 1 + 4i   

subtract and combine like terms      

multiply

(a + bi)*(c + di) 

= ac + bci + adi + bdi^2

= ac - bd + i(bc + ad)

FOIL then use i^2 = -1



conjugate

if z = a + bi
conjugate(z) = a - bi

z*conjugate(z) = (a + bi)*(a - bi) = a*a + b*b
absolute value  = sqrt(a*a + b*b) = math.hypot(a,b)


Do complex numbers have multiplicative inverses?


divide
z/w  = z*multiplicativeInverse(w)



powers  (integer powers)

(a + ib)*(a + ib) = a*a + b*b - 2abi

If power is negative, take recipricoal of the base
and change sign of the power

z^(-3) = (1/z)^3.