Solving Complex Number Question z^3+i=0 using z^n=|z|^(n) x e^((i)(n)(theta))

In summary, to find z, you can use the equation z^3+i=0 and solve for z by first writing -i in the form of r.e^(i theta), then dividing theta by 3. There are multiple solutions for z.
  • #1
Ry122
565
2
Find z
Question:
z^3+i=0
My attempt:
z^3=-i
use z^n=|z|^(n) x e^((i)(n)(theta))
n = 3
|z|=1
theta = -pie/2
Is this correct?
 
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  • #2
if z=i

z^2=-1 z^3=-1*i >>-i
 
  • #3
… one step at a time …

Ry122 said:
z^3+i=0
My attempt:
z^3=-i
use z^n=|z|^(n) x e^((i)(n)(theta))

Hi Ry122! :smile:

You must be much more logical, or you'll make mistakes. :frown:

Do it one step at a time.

You know z^3=-i.

So - first step - write -i in the form r.e^(i theta).

What is it?

Then divide theta by 3. :smile:

Oh … and how many different solutions are there? :rolleyes:
 

What is a complex number?

A complex number is a number that can be written in the form a + bi, where a and b are real numbers and i is the imaginary unit (√-1).

What does z^3+i=0 mean?

This is an equation in which a complex number z is raised to the third power and added to the imaginary unit i, resulting in a value of 0. The goal is to solve for the value of z that satisfies this equation.

What is the z^n=|z|^(n) x e^((i)(n)(theta)) formula used for?

This formula is used to convert a complex number from its rectangular form (a + bi) to its polar form (|z|e^(iθ)), where |z| is the magnitude or absolute value of the complex number and θ is the angle in radians.

How do you solve a complex number equation using the z^n=|z|^(n) x e^((i)(n)(theta)) formula?

To solve the equation z^3+i=0, we can first rewrite it as z^3=-i. Then, using the formula z^n=|z|^(n) x e^((i)(n)(theta)), we can determine the magnitude and angle of z. In this case, the magnitude is 1 and the angle is -π/2. Therefore, z=1 x e^(-iπ/2) or z=e^(-iπ/2).

What are some practical applications of solving complex number equations?

Complex numbers are used in many fields of science and engineering, such as electrical engineering, physics, and signal processing. They are particularly useful in solving problems involving alternating currents, oscillations, and vibrations.

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