Question

The complex conjugate of (1+i) is (1−i). In general to obtain the complex conjugate reverse the sign of the imaginary part. (Geometrically this corresponds to finding the "mirror image" point in the complex plane by reflecting through the x-axis. The complex conjugate of a complex number z is written with a bar over it: z⎯⎯ and read as "z bar".

Notice that if z=a+ib, then

(z)(z⎯⎯)=|z|2=a2+b2
which is also the square of the distance of the point z from the origin. (Plot z as a point in the "complex" plane in order to see this.)

If z=1+3i then z⎯⎯ =  and |z| =  .

You can use this to simplify complex fractions. Multiply the numerator and denominator by the complex conjugate of the denominator to make the denominator real.

1+3i1−i=  +i  .

Two convenient functions to know about pick out the real and imaginary parts of a complex number.

Re(a+ib)=a (the real part (coordinate) of the complex number), and
Im(a+ib)=b (the imaginary part (coordinate) of the complex number.  Re and Im are linear functions -- now that you know about linear behavior you may start noticing it often.

Answer 1