2.2.1 Imaginary Number Definition
Square roots of negative numbers fail to yield real results. The imaginary unit satisfies the equation $i^2=–1$.
$$i^2=-1 \therefore i=\sqrt{-1}$$
2.2.2 Integer Powers of Imaginary Units
Powers of $i$ always reduce using a modulus of $4$
$$i^x=i^{(x \bmod 4)}$$
Even powers simplify into real numbers
$$i^{-2}=i^2=i^6=-1\qquad i^{-4}=i^0=i^4=1$$
Odd powers simplify into imaginary numbers
$$i^{-3}=i^1=i^5=i\qquad i^{-1}=i^3=i^7=-i$$
It follows that imaginary numbers with real integer exponents are a
periodic series of $i, –1, –i, 1$
Examples
$$i^3=i^2\cdot i=-i\qquad \frac{1}{i}=i^{-1}=-i$$
2.2.3 Complex Addition
Complex numbers add the same way as expressions with variables
$$(a+b·i)\pm(c+d·i)=(a\pm c)+(b\pm d·i)$$
Example
$$5-3·i+(-7+9·i)=-2+6·i$$
2.2.4 Complex Conjugation
Complex conjugation functions the same as real conjugation with the sign reversal on the imaginary part
$$z=x+y·i\qquad z^*=x-y·i$$
2.2.5 Magnitude of Complex Numbers
$$|z|^2=z·z^*=x^2+y^2$$
2.2.6 Complex Division
$$\frac{a+b·i}{c+d·i}=\frac{a+b·i}{c+d·i}·\frac{c-d·i}{c-d·i}$$
$$=\frac{a·c+b·d+(b·c-a·d)·i}{c^2+d^2}$$
