for any pair of complex numbers and similarly .
Write z= a+bi and w=c + di. Using the definition of complex conjugate we can easily check the two statements by using the formulas given above for multiplication and division. For example,
But we may also notice that the conjugate of a number in standard form is obtained by changing i to -i. Since (-i)(-i) = (i)(i) = -1, the real part of an arbitrary algebraic expression is not affected by changing i to -i and the imaginary part changes sign. So it doesn't matter if we do algebraic manipulations first and then change i to -i or if we first change i to -i and then do the algebra. We will get the same answer.
Do you need more help? Please post your question on our S.O.S. Mathematics CyberBoard.
Author: Michael O'Neill