Rationale that ( i ) = (1/2) + i (1/2)

Assuming that
  ( i ) = (1/2) + i (1/2)
we can square both sides to get
  i = [ (1/2) + i (1/2) ]²
  i = [ (1/2) + 2(1/2) i + (1/2) i² ]
  i = [ (1/2) + i + (1/2) (-1) ]
  i = i (which is a true statement)

This is not a proof, but simply evidence that the formula is correct.