Finding the Smallest n for Real z^n in Complex Numbers

In summary, z^{n} is a real number for n = 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.
  • #1
nayfie
50
0
I have solved part a, I just have no idea how to go about doing part b. If anybody could point me in the right direction, that would be greatly appreciated!

Homework Statement



a. Express [itex]z = \frac{1 + \sqrt{3}i}{-2 -2i}[/itex] in the form rcis[itex]\theta[/itex]

b. What is the smallest positive integer [itex]n[/itex] such that [itex]z^{n}[/itex] is a real number? Find [itex]z^{n}[/itex] for this particular [itex]n[/itex].


Homework Equations



[itex]z^{n} = r^{n}cis(n\theta)[/itex]

[itex]\frac{z_{1}}{z_{2}} = \frac{r_{1}}{r_{2}}cis(\theta_{1} - \theta_{2})[/itex]

The Attempt at a Solution



Part A

[itex]z_{1} = 1 + \sqrt{3}i[/itex]

[itex]r = |z_{1}| = \sqrt{4} = 2[/itex]

[itex]\theta = arccos(\frac{1}{2}) = \frac{\pi}{3}[/itex]

[itex]z_{1} = 2cis(\frac{\pi}{3})[/itex]

[itex]z_{2} = -2 -2i[/itex]

[itex]r = |z_{2}| = \sqrt{8} = 2\sqrt{2}[/itex]

[itex]\theta = arccos(\frac{-2}{2\sqrt{2}}) = \frac{3\pi}{4}[/itex]

[itex]z_{2} = 2\sqrt{2}cis(\frac{3\pi}{4})[/itex]

[itex]\frac{z_{1}}{z_{2}} = \frac{2}{2\sqrt{2}}cis(\frac{\pi}{3} - \frac{3\pi}{4}) = \frac{1}{\sqrt{2}}cis(\frac{-5\pi}{12})[/itex]

Part B

No idea :(
 
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  • #2
nayfie said:
I have solved part a, I just have no idea how to go about doing part b. If anybody could point me in the right direction, that would be greatly appreciated!

Homework Statement



a. Express [itex]z = \frac{1 + \sqrt{3}i}{-2 -2i}[/itex] in the form rcis[itex]\theta[/itex]

b. What is the smallest positive integer [itex]n[/itex] such that [itex]z^{n}[/itex] is a real number? Find [itex]z^{n}[/itex] for this particular [itex]n[/itex].


Homework Equations



[itex]z^{n} = r^{n}cis(n\theta)[/itex]

[itex]\frac{z_{1}}{z_{2}} = \frac{r_{1}}{r_{2}}cis(\theta_{1} - \theta_{2})[/itex]

The Attempt at a Solution



Part A

[itex]z_{1} = 1 + \sqrt{3}i[/itex]

[itex]r = |z_{1}| = \sqrt{4} = 2[/itex]

[itex]\theta = arccos(\frac{1}{2}) = \frac{\pi}{3}[/itex]

[itex]z_{1} = 2cis(\frac{\pi}{3})[/itex]

[itex]z_{2} = -2 -2i[/itex]

[itex]r = |z_{2}| = \sqrt{8} = 2\sqrt{2}[/itex]

[itex]\theta = arccos(\frac{-2}{2\sqrt{2}}) = \frac{3\pi}{4}[/itex]

[itex]z_{2} = 2\sqrt{2}cis(\frac{3\pi}{4})[/itex]

[itex]\frac{z_{1}}{z_{2}} = \frac{2}{2\sqrt{2}}cis(\frac{\pi}{3} - \frac{3\pi}{4}) = \frac{1}{\sqrt{2}}cis(\frac{-5\pi}{12})[/itex]

Part B

No idea :(

Hint: What property does a real number have (what is its imaginary component equal to)?

Based on an argand diagram (a two dimensional graph with real and imaginary part), what is the argument (angle) equal to?
 
  • #3
Oh right. So they want me to find the smallest value of [itex]n[/itex] such that [itex]arg(z)[/itex] = [itex]k\pi[/itex] (so that the imaginary part = 0)?
 
  • #4
Okay it turns out that's the way to solve the question.

Thank you for the help!
 

What is a complex number in polar form?

A complex number in polar form is a way of representing a complex number using its magnitude (or distance from the origin) and its angle. It is written in the form r(cosθ + i sinθ), where r is the magnitude and θ is the angle.

How is a complex number converted into polar form?

To convert a complex number to polar form, you can use the formula r(cosθ + i sinθ) = a + bi, where a and b are the real and imaginary components of the complex number. The magnitude r can be calculated using the Pythagorean theorem (r = √(a^2 + b^2)), and the angle θ can be found using inverse trigonometric functions (θ = tan^-1(b/a)).

What is the significance of the angle in polar form?

The angle in polar form represents the direction of the complex number in the complex plane. It is also known as the argument or phase of the complex number. The angle θ is measured counterclockwise from the positive real axis to the line connecting the origin and the complex number.

Can a complex number have multiple polar forms?

Yes, a complex number can have infinitely many polar forms. This is because the angle θ can be represented as θ + 2πn, where n is an integer. This results in the same complex number with a different angle. However, the magnitude r will remain the same.

How are complex numbers in polar form used in mathematics and science?

Complex numbers in polar form are used in various fields of mathematics and science, including engineering, physics, and signal processing. They are particularly useful in representing and analyzing periodic phenomena, such as waveforms and oscillations. They are also used in calculating electrical currents and voltages in AC circuits.

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