the real and imaginary parts
Every complex number z = x + i y splits cleanly into two real ingredients. The number x is called the real part, written Re z, and the number y is called the imaginary part, written Im z. Reading them off is just a matter of looking: for z = 4 - 7i, Re z = 4 and Im z = -7. Think of it like reading a map coordinate — Re z tells you how far east, Im z how far north.
The single most common stumbling block is this: the imaginary part is the real number y, NOT the quantity i y. For z = 4 - 7i, Im z is -7, a plain real number, not -7i. Both Re z and Im z are always real numbers. With them you can reconstruct z completely (z = Re z + i Im z), and you can also recover them from z using the conjugate: Re z = (z + z-bar)/2 and Im z = (z - z-bar)/(2i).
Splitting into real and imaginary parts is the workhorse move of the whole subject. A complex function f(z) is almost always studied by writing it as f = u + i v, where u and v are ordinary two-variable real functions — and the entire theory of differentiability (the Cauchy-Riemann equations) is a statement about how u and v must be related. So mastering Re and Im now pays off everywhere later.
For z = (2 + i)(1 - 3i) = 2 - 6i + i - 3i^2 = 2 - 5i + 3 = 5 - 5i, we read off Re z = 5 and Im z = -5.
First simplify to the form x + i y, then Re and Im are just x and y.
Re and Im are real-valued, but they are NOT 'linear over the complex numbers': Re(i z) is not i Re z. Treat them as the two real coordinates of z, and watch out for factors of i.