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…

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⎯⎯

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.