Background
Imaginary numbers are of the form $ a+bi $, where $ a $ is the real part, $ b $ is the imaginary part, and $ i $ is a number with the property that $ i^2=-1 $. As Electrical Engineers, we substitute $ j $ for $ i $ because $ i $ is usually reserved for current; this ideally makes things less confusing.
Complex numbers are used to simplify analysis of circuits that would otherwise require us to write high-order differential equations.
Properties
Since we know that $ j=\sqrt{-1} $, we can deduce the following cyclic pattern:
$ j^{4n}=j $
$ j^{4n+1}=-1 $
$ j^{4n+2}=-j $
$ j^{4n+3}=1 $
From this we see that the pattern $ \{j, -1, -j, 1\} $ repeats as $ j $ is raised to higher and higher powers.
