Finding the 4th Roots of -16: Cartesian vs. Polar Form

In summary: But what about the case where k is not one of the integers 0, 1, 2, 3?If k is not one of the integers 0, 1, 2, 3, then the answer will be something other than 2e^i((pi + 2kpi)/4).In that case, you would need to plug in the value of k into the equation and see what happens.In summary, the Homework Statement asks for the 4th roots of -16 in both Cartesian and polar form and plots their positions in the complex plane. The Attempt at a Solution shows how to find the value of r and theta when drawing z=-16 in the complex plane. If k is not one of the integers
  • #1
seboastien
53
0

Homework Statement


Compute the 4th roots of -16 in both Cartesian and polar form and plot their positions in the complex plane.


Homework Equations


z^1/n=(r^1/n)(e^i(theta)/n), (r^1/n)(e^i(theta)/n)(e^i2(pi)/n...


The Attempt at a Solution


How do I find the value of r, and theta??
 
Physics news on Phys.org
  • #2
Draw z=-16 in the complex plane. The distance from the origin to -16 in the complex plane is r and the angle between the positive real axis and the negative real axis rotating counter clock wise is [itex]\theta[/itex].
 
  • #3
How do I draw -16 in the complex plane, when I don't know r or theta?
 
  • #4
Draw the the complex plane and put a dot where -16 is. Then calculate the distance and angle.
 
  • #5
where is -16?
 
  • #6
Do you know where -16 is on the line of real numbers?
 
  • #7
are you saying that the argument is zero and that the modulus is 16?
 
  • #8
The modulus is 16, but the argument is not 0. If the argument was 0 -16 would be placed on the positive real axis, which it clearly isn't.
 
  • #9
okay so you think the argument in pi
 
  • #10
that's not right
 
  • #11
why are you wasting my time?
 
  • #12
Wasting your time? Why would that not be right? You may want to provide some arguments to why this is wrong.

Either way I can tell you that I am not wrong. Perhaps review the the relevant equation you posted before jumping the gun?
 
Last edited:
  • #13
Beacause the answer is apparently 2e^i((pi + 2kpi)/4) where K=0,1,2,3

that's why it wouldn't be right.
 
  • #14
seboastien said:
Beacause the answer is apparently 2e^i((pi + 2kpi)/4) where K=0,1,2,3

that's why it wouldn't be right.

So ...take one of these numbers (say the k=0 one), convert it to Cartesian form, and take its 4th power. You can then check for yourself whether it is right.
 
  • #15
Beacause the answer is apparently 2e^i((pi + 2kpi)/4) where K=0,1,2,3

that's why it wouldn't be right.

It is obvious that every multiple of 2pi added to the original argument will return you to that exact same spot, after all a circle is exactly 2pi radians.
 

What are complex numbers in polar form?

Complex numbers in polar form are a way to represent a complex number using its magnitude (or distance from the origin) and argument (or angle from the positive real axis). They are expressed in the form r(cosθ + isinθ), where r is the magnitude and θ is the argument.

How do you convert a complex number from rectangular form to polar form?

To convert a complex number from rectangular form (a + bi) to polar form, you can use the following formulas:
r = √(a² + b²)
θ = tan⁻¹(b/a)
The magnitude r is the distance from the origin to the complex number, and the argument θ is the angle formed with the positive real axis.

What are the advantages of using complex numbers in polar form?

One advantage of using complex numbers in polar form is that it makes it easier to perform multiplication and division operations. Additionally, it is easier to visualize the complex numbers on a polar coordinate system, which can be helpful in certain applications, such as electrical engineering.

How do you perform addition and subtraction with complex numbers in polar form?

To perform addition and subtraction with complex numbers in polar form, you can use the following formulas:
r = √(a² + b²)
θ = tan⁻¹(b/a)
Then, to add or subtract two complex numbers, you can simply add or subtract their magnitudes and add or subtract their arguments.

What is the conjugate of a complex number in polar form?

The conjugate of a complex number in polar form is a complex number with the same magnitude but an opposite argument. It can be found by changing the sign of the argument (θ) in the polar form. For example, the conjugate of r(cosθ + isinθ) is r(cos(-θ) + isin(-θ)).

Similar threads

  • Calculus and Beyond Homework Help
Replies
6
Views
1K
  • Calculus and Beyond Homework Help
Replies
6
Views
1K
  • Calculus and Beyond Homework Help
Replies
14
Views
2K
  • Calculus and Beyond Homework Help
Replies
1
Views
902
  • Calculus and Beyond Homework Help
Replies
1
Views
787
  • Calculus and Beyond Homework Help
Replies
4
Views
2K
  • Calculus and Beyond Homework Help
Replies
2
Views
1K
  • Calculus and Beyond Homework Help
Replies
7
Views
1K
  • Differential Equations
Replies
4
Views
2K
  • Calculus and Beyond Homework Help
Replies
3
Views
1K
Back
Top