Hi,
There is a much simpler way to understand this and that is through the idea that the operator 'j' is the square root of minus 1: j=sqrt(-1).
This produces an imaginary number which we call simply "j" for short. But as we all know, when you multiply the square root of a number by itself you get the original number back. So this leads to the first identity:
j^2=-1
and that is simple enough to understand because we multiplied j*j and that meant we did sqrt(-1)*sqrt(-1) and just got -1 back again. No problem there.
Next comes multiplying that by j again, and this leads to:
j^3=(j*j)*j
and we already know that j*j equals -1 so we get:
j^3=-j
Next we multiply that by j again and we have:
j^4=(-j)*j
or written out slightly different we have:
j^4=(-1)*(j*j)
and we already know what j*j is, it is -1 again, so we get:
j^4=(-1)*(-1)
and of course that equals 1 so we just get j^4=1.
So now we create a little table:
j=sqrt(-1)=j^1=j
j^2=-1
j^3=-j
j^4=1
So what about j^5 then? Well, when we multiply j^4 times j we just get j again, so we see the table expands again:
j^1=j
j^2=-1
j^3=-j
j^4=1
j^5=j
and notice that this is the same as j^1 which is the first entry in the table. What happened here is the values have started to repeat, so we end up with the same repetition of the group:
{j,-1,-j,1}
and so we can immediately expand the table:
j^1=j
j^2=-1
j^3=-j
j^4=1
j^5=j
j^6=-1
j^7=-j
j^8=1
and this goes on forever, so that the value of any power of j is equal to the power of j that has the power minus 4, so we see:
j^(n+4)=j^(n)
and that causes a repeat of j, -1, -j, 1. So those are the only four values we can get.
Now when we look at something like 1/j, that is really j^(-1), and you can create a little table for minus j's too, or just use simple algebra. "j" can also be considered an algebraic constant so we can reduce it based on that too by multiplying the numerator and denominator both by j:
1/j=(j*1)/(j*j)
and again we know what j*1 is and what j*j is:
(j*1)/(j*j)=j/(-1)=-j
So you can use that idea instead of building a table for the negative powers of j.
Viewed as a rotation, multiplying times j means rotating counter clockwise by 90 degrees. Multiplying by j^2 means rotating counter clockwise by 180 degrees, and so on.
Note that as we rotate over and over by 90 degrees that makes the quantity real, then imaginary, then real again, then imaginary, etc.