The Complex Plane?

Question

Plot each complex number in the complex plane and write it in polar form. Express your answer in degrees.

(1) -2

(2) 9(sqrt{3}) + 9i

Answer 1

Complex numbers can be written in the form a+ib where a and b are real numbers. This gives rise to a graphical way to represent complex numbers in what is called the complex plane. Draw a coordinate system as you would for the Cartesian plane but label the horizontal axis the real axis and the vertical axis the imaginary axis. The complex number a+ib is then representated in the complex plane by the point with coordinates (a,b).

PROBLEM 1:
-2
= -2 + 0i
--> (-2, 0)
Plot this point on your graph.

PROBLEM 2:
9√3 + 9i
--> (9√3, 9)
--> (~15.6, 9)
Plot this point on your graph.

Now to write each of these in polar form. Let's let r be the distance from the origin to the point. Then a/r = cos(Θ) and b/r = sin(Θ) so a = r cos(Θ) and b = r sin(Θ). Hence the complex number a+ib can be written r cos(Θ) + i r sin(Θ) or more compactly r (cos(Θ) + i sin(Θ)).

PROBLEM 1:
a = -2 (from above)
b = 0 (from above)
r = 2 (from a² + b² = r²)

a/r = -1
b/r = 0

arccos(-1) = 180 degrees

So putting it all together:
2 ( cos(180) + i sin(180) )

But this simplifies to:
2 cos(180)

Another way to write this is directly as polar coordinates:
(2, 180˚)

PROBLEM 2:
a = 9√3
b = 9
r = 18 (from a² + b² = r²)

a/r = √3/2
b/r = 1/2

arccos(√3/2) = 30 degrees

So putting it all together:
18 ( cos(30) + i sin(30) )

Or,
(18, 30˚)