How Do You Convert Complex Numbers Between Cartesian and Polar Forms?

[vorcil]

Take,

$$P = 4e^{-j\frac{\pi}{3}}$$

$$Q = 4-3j$$

$$R = 2e^{j\frac{\pi}{2}}$$

$$S = 5$$

note: I'm using j to be a complex number, it's equivalent to i in mathematics

-

A: express p q r s in both cartesian (a+ib) and polar (re^itheta) forms

B: sketch p q r and s on the complex plane

--------------------------------------------------------------
I'm not too certain as to how P and R work,
I know from the taylor series that r(cos(theta)+isin(theta)) = re^itheta

P is already in polar form, $$P = 4e^{-j\frac{\pi}{3}}$$
so the Cartesian form,
4(cos(pi/3)+j(sin(pi/3)))

(a+jb) = $$4 cos\frac{\pi}{3} = a$$ and $$4 sin \frac{\pi}{3} = b$$

so it is 2+j(3.464) or $$2+j\sqrt{12}$$
NOTE I AM NOT SURE, BECAUSE THE ORIGINAL EQUATION SHOWS -J
SO SHOULD I PUT, $$2-j\sqrt{12}$$ ??
----------------------------------------------------------------
$$Q = 4-3j$$
Q is already in cartesian form,

|z| q = 5, = root 4^2 + -(3^2)
$$\theta = cos^{-1} \frac{4}{5} = 0.643 rad$$

so the polar form is , 5cis(0.643) = 5(cos(0.643) + jsin(0.643)) = 5e^(j0.643)

----------------------------------------------------------------
$$R = 2e^{j\frac{pi}{2}}$$
R is already in polar form,

2 = |z|

2 cos(pi/2) = 0,
2 sin(pi/2) = 2,
this means if i wear to imagine the vector R, it would be going straight up,

so in cartesian form, the equivalent equation of $$R = 2e^{j\frac{pi}{2}}$$ is j2

-------------------------------------------------------------------------------------------------
S=5
I'm assuming S is already in cartesian form, since it is 5+j0

|z| =5
the angle it makes with the x axis, is 0,
so i think the equation for S in polar form is, $$5e^{j * 0 }$$

 

[vorcil]

Determine numberical answers for each of the following:
either in cartesian form or in polar form with the angle in degrees

------------------------------------------------------------------------

1) (PQ)^1/2
2) (R/P)^1/3

-------------------------------------------------------------------------------------

 

[vorcil]

i hate macs,

I calculate pq to be,

$$20e^{-j(\frac{\pi}{3} - 0.643)}$$

how do i square root it?

$$\sqrt{20e^{-j(\frac{\pi}{3} - 0.643)}}$$

 

[ammontgo]

=>P is already in Exponential form. To convert to Cartesian:

Euler's Identity:
[PLAIN]https://dl.dropbox.com/u/4645835/MATH/EulersID.gif

so:
[PLAIN]https://dl.dropbox.com/u/4645835/MATH/Pcart.gif

=>Q is already in Cartesian; so convert to Polar:
[PLAIN]https://dl.dropbox.com/u/4645835/MATH/Qpol.gif

=>Convert Q to Exponential:
[PLAIN]https://dl.dropbox.com/u/4645835/MATH/Qexp.gif


I think these are right and I hope they help.

 

[HallsofIvy]

Take,

$$P = 4e^{-j\frac{\pi}{3}}$$

$$Q = 4-3j$$

$$R = 2e^{j\frac{\pi}{2}}$$

$$S = 5$$

note: I'm using j to be a complex number, it's equivalent to i in mathematics

-

A: express p q r s in both cartesian (a+ib) and polar (re^itheta) forms

B: sketch p q r and s on the complex plane

--------------------------------------------------------------
I'm not too certain as to how P and R work,
I know from the taylor series that r(cos(theta)+isin(theta)) = re^itheta

P is already in polar form, $$P = 4e^{-j\frac{\pi}{3}}$$
so the Cartesian form,
4(cos(pi/3)+j(sin(pi/3)))

No, [itex]\theta= -\frac{\pi/3}[itex], not $\frac{\pi}{3}$.

(a+jb) = $$4 cos\frac{\pi}{3} = a$$ and $$4 sin \frac{\pi}{3} = b$$

so it is 2+j(3.464) or $$2+j\sqrt{12}$$
NOTE I AM NOT SURE, BECAUSE THE ORIGINAL EQUATION SHOWS -J
SO SHOULD I PUT, $$2-j\sqrt{12}$$ ??
----------------------------------------------------------------
$$Q = 4-3j$$
Q is already in cartesian form,

|z| q = 5, = root 4^2 + -(3^2)
$$\theta = cos^{-1} \frac{4}{5} = 0.643 rad$$

so the polar form is , 5cis(0.643) = 5(cos(0.643) + jsin(0.643)) = 5e^(j0.643)

----------------------------------------------------------------
$$R = 2e^{j\frac{pi}{2}}$$
R is already in polar form,

2 = |z|

2 cos(pi/2) = 0,
2 sin(pi/2) = 2,
this means if i wear to imagine the vector R, it would be going straight up,

so in cartesian form, the equivalent equation of $$R = 2e^{j\frac{pi}{2}}$$ is j2

-------------------------------------------------------------------------------------------------
S=5
I'm assuming S is already in cartesian form, since it is 5+j0

|z| =5
the angle it makes with the x axis, is 0,
so i think the equation for S in polar form is, $$5e^{j * 0 }$$

 

[ammontgo]

It is possible for an expression to fit into more than one category or into none of the categories. I'm not sure how this works anymore..as I did this a long time ago. But I think S satisfies exponential and cartesian.