MATH135 Lecture Notes - Lecture 36: Polar Coordinate System, Distributive Property, Positive Real Numbers

• School: University of Waterloo
• Department: Mathematics
• Course: MATH135
• Professor: Shane Bauman

Document Summary

= (z + w)( (by pm 3) z w+ z w z z. Let z such that z i, i (by pcj 1) (distributivity property) (by pm 3 and pcj 3) w (if z = a + bi, we know that a (by pm 4) (by pm 2) = z/(1 + z 2 ) (we can multiply by (1 + z 2 )( 1 + 2 )z since z i, i) z z/(1 z z. So z/(1 + z 2 ) = /(1 + 2 ) z(1 + 2 ) = (1 + z 2 ) z + z 2 = + z 2 z + z 2 z 2 = 0 (z ) z (z ) = 0 z (z )(1 z ) = 0. Therefore, z/(1 + z 2 ) if and only if z or |z| = 1 z z z z z z z z z z z (since |z| 0)