Two complex numbers Z: and z2 are described below: Z1 = 1+ I root 3 z2 = exp (I

Question

Two complex numbers Z: and z2 are described below: Z1 = 1+ I root 3 z2 = exp (I 2 pi / 3) Identify each of the following complex numbers as points (or vectors) on the complex plane, using a well-labeled sketch: z1, z2, z1*, z2*. 1/z1, 1/z2, 1/z*1 and and 1 /z2*. Determine each of the sums .Determine each of the magnitudes Determine each of the following powers of z: and z2: Determine Be mindful of how many fourth roots z2 has and identify each of them graphically on a well-labeled sketch of the complex plane. Express each of your answers in Cartesian form (a + ib). in polar form (re theta, where r > 0). as a real number, as an imaginary number, or graphically in a well-labeled complex-plane diagram, whichever form is less cluttered and more appropriate.

Explanation / Answer

I will give you an idea on how to do these. You should then be able to do the rest yourself.

First, note that z2 = exp(2i/3) = cos(2/3) + i sin(2/3) = -1/2 + i3/2 = 1/2(-1+i3) . Hence |z2| = 1.
Likewise if we wish to write z1 in polar form , note that z1 = 1 + i3 = 2(1/2 + i3/2) = 2 (cos(/3) + isin(/3)) = 2 exp(i/3). Hence |z1| = 2.

Also, the conjugate of z = a+ib is simply a-ib, so you should be able to do all those relevant parts yourself.

Thus in rectilinear form z1 = (1, 3) and z2 = (-1/2, 3/2) in coordinate form.

Also, |zw| = |z||w| for complex numbers z,w. Use this to calculate all the absolute values in the third part.

To calculate powers zn for a complex number z, the simplest way is to use the polar form. If z = r exp(i) then zn = rn exp(in). Thus for instance z16 = 26 exp(i6/3) = 64 exp(2i) = 64.

Likewise, 1/z in polar form is (1/r)exp(-i). This should make the calculations for 9a) a lot simpler.

Finally, to calculate z21/4 note that this corresponds to a complex number whose 4th power is z2. Thus

w = 21/4 exp(i/12) is one such value. Note however, that thre are other value too that satisfy this criterion.

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