18.03 Lecture Notes - Lecture 5: Unit Circle, Wwzz, List Of Trigonometric Identities
Document Summary
Complex numbers provide a tool for expressing aspects of the real world. Anything with an amplitude and a phase is secretly a complex number, and it is worthwhile bringing that secret identity out in the open. We think of the real numbers as filling out a line. The point up one unit from 0 is written i (but written j by many engineers). Addition and multiplication by real numbers is as vectors. The new thing is i^2 = -1 . For example foil: (1 + i)(1 + 2i) = 1 + 2i + i - 2 = -1 + 3i. Every complex number can be written as a + bi with a and b real. a = re(a+bi) the real part. = im(a+bi) the imaginary part: nb this is a real number. Maybe complex numbers seem obscure because you are used to imagining numbers by giving them units: 5 cars, or -3 degrees.
and its complex conjugate 
on the same complex plane.