In any polynomial involving i, i.e. $ c1*i^n+c2*i^{n-1}+...+c $ we can express the even powers of i as either 1 or -1. Thus, any polynomial in i can be expressed as $ c1+c2*i $ where c1 and c2 are any real number constants. This also establishes the set {1, i} as a basis for C as a vector space over all real numbers.
More importantly for our class, euler's formula: $ e^{i\pi}=\cos(\theta)+i*\sin(\theta) $ also $ e^{-i\pi}=\cos(\theta)-i*\sin(\theta) $
