Started trying to explain complex numbers, but realized that I might not be right.

complex_numbers.rb

class Float
  # simplify distractingly large numbers that occur from floating point imprecision
  def inspect
    round(8).to_s
  end
end

include Math

# Quantity of a degree in radians, b/c sin and cos expect radians
degree   = PI*1r/180

# Complex numbers have a "real" component and an "imaginary" component, ie (2+3i)
# the real component is 2, the imaginary component is 3
#
# The real component is a number like all the ones you've ever used,
# you can think of it as "whenever I said 2+2=4 I was saying (2+0i) + (2+0i) = (4+0i)"
#
# The imaginary component is multiplied by the unit "i".
# i is usually described as "the square root of -1",
# but makes more sense to me as "rotate 90º".
# Thus 1i = "rotate 90º", and 0.5i = "rotate half of 90º" = "rotate 45º"

# uhhhhh.... I'm not sure this is valid anymore, I stopped writing this so I could
# think about that, but never got back to it, so just going to post this in a gist
# so that I can find it again later if I want

rotation = cos(degree) + sin(degree)*1i
point    = 1 + 0i

# 30º = π/6
30.times { point *= rotation }
sqrt(3)/2  # => 0.8660254
point.real # => 0.8660254
point.imag # => 0.5

# 45º = π/4
15.times { point *= rotation }
sqrt(2)/2  # => 0.70710678
point.real # => 0.70710678
point.imag # => 0.70710678

# 60º = π/3
15.times { point *= rotation }
point.real # => 0.5
point.imag # => 0.8660254
sqrt(3)/2  # => 0.8660254

# 90º = π/2
30.times { point *= rotation }
point.real # => 0.0
point.imag # => 1.0