What is the power of a complex number?

modulus of q and the principle argument of q. So we have got two complex numbers we raise
them to various powers and divide, get a new complex number what is the modulus or the
length if you like of that new complex number and what is the principal argument of that new
complex number. So the ideas we are going to use will involve the polar exponential form. Okay
so this is the polar exponential form r is a distance or a length called the modulus and θ is an
angle. So let us draw a picture of these two complex numbers on the same complex plane. So
here we have got the imaginary axis; here we have got the real axis. Okay let us plot z first so 1
+ i so we go one unit here and one unit up there so that is going to be z and w we go one unit
this way and √3 units this way. Okay so this should be your w. Okay so if we draw some line
segments joining those points to the origin then to put these two complex numbers in polar form
we need an angle to the positive real axis and we need a length. Okay now the way you can
come up with those things you can do visually or by applying various formulae, but I like to do it
visually. So if you look here we can have a right angle triangle associated with z. Okay this
length is 1 this length is 1; by Pythagoras this length has to be √2 and we know from basic
trigonometry basically we have got a triangle where these two angles must be equal and they
are going to be π/4. So the length or the modulus of z is √2 and the angle to the positive real
axis is π/4. Now similarly for w you have got another right angle triangle which is something like
this: We know that the base is 1 the height is √3, and again, by Pythagoras, the hypotenuse
which is the length that we want is going to be 2. Again, this is another special triangle from
trigonometry again one of the two special triangles. This angle here is going to be π/6 and this
angle here is going to be π/3 now this angle here is going to be this smaller angle down here;
what we want is this angle here so we know the whole thing has to be π/2 so this angle here
would be actually π/3 (sixty degrees if you like) because we started the positive real axis and we
rotated clockwise the opposite way to go to z it would be negative. So let me just write some of
these things down so from our diagram – now if you want to use the formulae to compute this
you know you can do that but for me I always like to draw diagrams when possible because it
gives you a simple visual way to interpret things. Okay so z=r e^(i x θ) where r is the modulus or
the length and θ is an angle to the positive real axis. So here we are going to have √2 and the
angle is π/4. Okay similarly what about w well w is two units from the origin and the angle,
rotating from the positive real axis clockwise you have to rotate π/3 or 60 degrees in the
clockwise direction, so that is going to mean a negative sign here. Okay so we can now use
these things use these forms these polar trig forms to calculate q and then we can look at its
modulus and it is principal arguments. So let us work out q first and then we will worry about its
length okay so (z^6)/(w^5) so that is just this to the 6 all over this to the power of 5 and then we
can just use our index or power laws you would distribute the 6 distribute the 5 accordingly.
Okay so √2 to the power of 6 that is just the same as 2^3. e^(i x (π/4))^6 well that just e^i x (6 x
π)/4) so you just multiply and on the bottom were going to get 2 to the power of 5 times e (-i x (5
x π)/3). So what we want to do is simplify this. So obviously the 2's will cancel and you will get 2
squared on the bottom which is going to be 4. You can use the power laws or the index laws to
bring that up to the top and then simplify we are going to get e^(π x i x (3 x (1/6))) so you just
might want to put in a step there where you simplify this okay but this is what you should get.
Now without doing anything more to this I can immediately solve part a) because this number
here in the polar exponential form denotes a length or a modulus so the modulus of q is going to
be 1/4 okay so that is a) done. 0:36-8:00.
Principal argument. Okay so principal argument is the angle associated with q that lies between
negative π and π. If you look at this angle here you have got 3 and 1/6 π. That is not between
negative π and π okay. So what we can do is basically look at this and try to break it down a bit.
So 3 π and 1/6. Some books have the a principal argument between 0 and 2π but I
am not going to do it that way. 8:00-10:10.
Timestamp:
Part a - 0:36
Part b - 8:00