# A few complex number problems

• Nov 1st 2010, 01:54 AM
Paymemoney
A few complex number problems
Hi
The following questions i am having trouble with:

1) Plot the set of points in the complex plane which satisfy $\displaystyle |z+1+j|=2$

2) Express the complex number $\displaystyle 2+2\sqrt{3}j$ in the form re^{j\theta} hence obtain the cartesian form of $\displaystyle (2+2\sqrt{3}j)^6$

This is what i have attempted:
I found out that $\displaystyle 2+2\sqrt{3}j = 4e^{\frac{\pi}{3}j}$

hence i make $\displaystyle (2+2\sqrt{3}j)^6 = (4e^{\frac{\pi}{3}j})^6$

what would i do next?

P.S
• Nov 1st 2010, 02:08 AM
mr fantastic
Quote:

Originally Posted by Paymemoney
Hi
The following questions i am having trouble with:

1) Plot the set of points in the complex plane which satisfy $\displaystyle |z+1+j|=2$

2) Express the complex number $\displaystyle 2+2\sqrt{3}j$ in the form re^{j\theta} hence obtain the cartesian form of $\displaystyle (2+2\sqrt{3}j)^6$

This is what i have attempted:
I found out that $\displaystyle 2+2\sqrt{3}j = 4e^{\frac{\pi}{3}j}$

hence i make $\displaystyle (2+2\sqrt{3}j)^6 = (4e^{\frac{\pi}{3}j})^6$

what would i do next?

P.S

1) The relation can be written $\displaystyle |z - (-1 - j)| = 2$. Geometrically, and you should know this, this defines a circle of radius 2 and center at z = -1 - j.

2) Use an index law to simplify the power. Then convert the resulting polar form into Cartesian form.
• Nov 1st 2010, 02:39 AM
Paymemoney
thank you
• Nov 1st 2010, 08:02 AM
HallsofIvy
Specifically, |a- b| is the distance between points in the plane, interpreted as complex numbers. So |z- a|= r is the set of points whose distance from a is r- that is, the circle with center a and radius r.