For some practice working with complex numbers:

Calculate
$(1+i ) + (1+4i)$ = ,
$(1+i ) - (1+4i)$ = ,
$(1+i )(1+4i)$ = .
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: $\overline{z}$ and read as "z bar".

Notice that if $z = a +ib$, then

$(z)\left( \overline z\right) = |z|^2 = a^2+b^2$
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 $z = 1+i$ then $\overline z$ = 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.

$\displaystyle \frac{1+i}{1+4i} =$ $+ i$ .

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.