# 🚁 Complex Numbers

The emoji for this section is a helicopter, because helicopters rotate, which we will see is the way to understand multiplication by complex numbers.

Complex Number Intuitive Introduction: Take the equation y=a×x^n we want to be able to solve ALL variants of this equation, no exceptions. To ‘Solve’ an equation means undoing mathematical steps (‘perform the right sequence of INVERSE OPERATIONS’), to arrive at a number (might need to AUGMENT OUR NUMBER SET to have a place to land).

$\begin{array} {lrclccccll}\mathbb{N} & y & = & 2\times(8)^3 & & & y & = & \phantom{-}1024 & \\ \mathbb{Z} & y & = & 2\times(-8)^3 & & & y & = & -1024 & \text{Inverse of }+\text{ requires Negative Numbers} \\ \mathbb{Q} & y & = & 2\times(-8)^{-3} & & & y & = & -\frac{1}{512} & \text{Inverse of }×\text{ requires Rational Numbers} \\ \mathbb{R} & y & = & 2\times(8)^{\frac{1}{6}} & & & y & = & 2\sqrt{2} & \text{Inverse of }\hat{} \: \text{ requires Real Numbers} \\ \mathbb{C} & -2 & = & 2\times(x)^2 & & & y & = & \sqrt{-1} & \sqrt{\text{Negatives}} \text{ requires Complex Numbers} \end{array}$

We can take radicals of whole numbers, fractions, and other radicals, so why keep negatives as an exception? If we give ourselves permission to use a mathematical object with the new property i²=-1, then whenever we happen to get radicals of negatives, we re-write them using ‘i’ for example √5+√-18 = √5+3√2×i (from now on it’ll just be written 3√2i , but be clear that ‘i’ isn’t inside the square root, 3√2 is a coefficient multiplying the front of ‘i’).

Complex Number Arithmetic radicals of positives: We are used to keeping radical parts separate from non-radical parts e.g. (2+√3)(3-√3)+√3 = 6-2√3+3√3-3+√3 = 3+2√3 Notice that if it was (a+b)(a-b)=a²+b² then the ‘√’ would eliminate entirely

Complex Number Arithmetic radicals of negatives: Keep the ‘i’ parts separate e.g. (2+√[-3])(3-√[-3])+√3 = (2+√3i)(3-√3i)+√3 = 6-2√3i+3√3i+3+√3 = 9+√3+√3i Notice that if it was (a+b)(a-b)=a²+b² then the ‘√’ would eliminate entirely

Complex Solutions to $$f(z) = az^2+bz+c$$

Inputs and outputs for complex functions: $$\; e^z, z^2, \sqrt{z}, z^4$$