# π 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=aΓx^n we want to be able to solve ALL variants of this equation, no exceptions. To βSolveβ an equation means undoing mathematical steps (βperform the right sequence of INVERSE OPERATIONSβ), to arrive at a number (might need to AUGMENT OUR NUMBER SET to have a place to land).

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 iΒ²=-1, then whenever we happen to get radicals of negatives, we re-write them using βiβ for example β5+β-18 = β5+3β2Γi (from now on itβll just be written 3β2i , but be clear that βiβ isnβt inside the square root, 3β2 is a coefficient multiplying the front of βiβ).

**Complex Number Arithmetic radicals of positives:** We are used to keeping radical parts separate from non-radical parts e.g. (2+β3)(3-β3)+β3 = 6-2β3+3β3-3+β3 = 3+2β3 Notice that if it was (a+b)(a-b)=aΒ²+bΒ² then the βββ would eliminate entirely

**Complex Number Arithmetic radicals of negatives:** Keep the βiβ parts separate e.g. (2+β[-3])(3-β[-3])+β3 = (2+β3i)(3-β3i)+β3 = 6-2β3i+3β3i+3+β3 = 9+β3+β3i Notice that if it was (a+b)(a-b)=aΒ²+bΒ² then the βββ would eliminate entirely