Complex number generalizations

[314Jason]

Homework Statement

Use de Movire to find solutions for the following:

z^5 = i
z^4 = i
z^3 = i

Find generalization for z^n = x+iy,
where modulus of x+iy is 1

Explore when |x+iy| is not equal to 1

Homework Equations

[rcis(theta)] = (r^n)cis(theta*n)

r = /sqrt(y^2 + x^2)

theta = arctan(y/x)

Finding the roots of complex numbers: (r^1/n)cis(2kpi/n), where k = 0,1,...n-1

The Attempt at a Solution

I tried using moivre's theorem to find the roots of z, but I ended up with theta=arctan(1/0), so it is undefined. Yes, and now I'm stuck. Could the answer possibly be undefined/no solution?

 

[I like Serena]

Welcome to PF, 314Jason!

Homework Statement

Use de Movire to find solutions for the following:

z^5 = i
z^4 = i
z^3 = i

Find generalization for z^n = x+iy,
where modulus of x+iy is 1

Explore when |x+iy| is not equal to 1

Homework Equations

[rcis(theta)] = (r^n)cis(theta*n)

r = /sqrt(y^2 + x^2)

theta = arctan(y/x)

Finding the roots of complex numbers: (r^1/n)cis(2kpi/n), where k = 0,1,...n-1

The Attempt at a Solution

I tried using moivre's theorem to find the roots of z, but I ended up with theta=arctan(1/0), so it is undefined. Yes, and now I'm stuck. Could the answer possibly be undefined/no solution?

arctan(1/0) is indeed undefined, but that only means that theta is either pi/2 or -pi/2.
In your case, for the complex number "i", the corresponding theta is pi/2.

Your problem is arctan. It is not quite the right function to find theta.
What you actually need is atan2(x, y) or atan2(y, x) to find theta.
See http://en.wikipedia.org/wiki/Atan2 for details.
Or otherwise, you could write:
$$\theta = \textrm{Arg}(x+iy)$$
See http://en.wikipedia.org/wiki/Argument_(complex_analysis)" for details.

 

[314Jason]

Oh okay, so atan2(y, x) is the angle between the point (x,y) and the real axis. And when y>0 and x=0, theta = pi/2. YAY!

I've done the all z^3, z^4 and z^5 = i. Turns out, the conjecture for the distance between two adjacent roots (when z^n = i) is the same as when z^n = 1. Which is 2|sin(pi/n)|

Now I have to explore when |x+iy| ≠ 1. On the graph, this the modulus of z only adds to the radius (when mod(z) > 1) or subtracts to the radius (when mod(z) < 1).

In de moivre's theorem, z = r^(1/n) cis(ø + 2kpi)
r = mod(z)
ø = arg(z)
n = number of roots / exponent of z
k = 0, 1, ..., n-1
r^(1/n) is equal to the radius of the circle: x^2 + y^2 = r^(2/n)

THEN I wanted to find the conjecture interms of r, ø, n and k

z = r^(1/n)cis(ø+2kπ) <------------ de moirve
where r = mod(z) = √(x^2 + y^2)
where ø = arg(z) = atan2(x,y) = arctan (y/x)
where n = number of roots / exponents of z
where k = 0,1,...,n-1
distance formula = √(x2+y2)
Point 1(r^(1/n)cos(ø), r^(1/n)sin(ø))
Point 2(r^(1/n)cos(ø+2pi), r^(1/n)sin(ø+2pi))

I plugged the two points into the distance formula and got an answer of 0. I even put it through Wolframalpha (without the r^(1/n)) and got 0.

http://www.wolframalpha.com/input/?i=2cos^2(ø+π)sin^2(π)+-+2sin^2(ø+π)sin^2(π)

^ this is after I applied trig's sum-to-product formulas and extracted r^(1/n) since they're common.

THE DISTANCE CANNOT BE ZERO! What's wrong!? :(

 

[I like Serena]

I think you've got the wrong distance between 2 adjacent roots.
It's not 2|sin(pi/n)|.
It should be |cis(2pi/n)-1|.As for the distance when the mod is not 1.
Note that the distances between successive adjacent roots will not be the same.What you have is the first 2 roots, but you've got their argument (angle) wrong.
Apparently you forgot to divide the angle by n.
It should be:
Point 1(r^(1/n)cos(ø/n), r^(1/n)sin(ø/n))
Point 2(r^(1/n)cos((ø+2pi)/n), r^(1/n)sin((ø+2pi)/n))

 

[314Jason]

I think you've got the wrong distance between 2 adjacent roots.
It's not 2|sin(pi/n)|.
It should be |cis(2pi/n)-1|.

I don't understand how

Here's how I got the generalization:

When z^n = 1, z = cis(2kπ/n), where k = 0,...,n-1

distance formula = √(x^2+y^2)
P1(1, 0)
P2(cos(2pi/n), sin(2pi/n)

d = √[ (cos(2pi/n)-1)^2 + (sin(2pi/n)-0)^2 ]
= √[ cos^2(2pi/n) - 2cos(2pi/n) + 1 + sin^2(2pi/n) ]
= √[-2cos(2pi/n)+2]
= √[-2 ( 1-2sin^2(pi/n) ]+2]v
= √[-2+4sin^2(pi/n)]+2]
= √[4sin^2(pi/n)]
= ± 2[sin(pi/n)] <---- CONJECTURE
Since distance cannot be negative,d = 2|sin(pi/n)|

When z^n = i, z =cis[(π + 4kπ)/2n], where k = 0,...,n-1

distance formula = √(x2+y2)
P1(cos(pi/2n), sin(pi/2n))
P2(cos(5pi/2n), sin(5pi/2n))

d = √[ (sin(5pi/2n)-sin(pi/2n))^2 + (cos(5pi/2n)-cos(pi/2n))^2 ]

Using trig's sum-to-product formula...

= √[ (2cos(3pi/n)sin(2pi/n))^2 + (-2sin(3pi/n)sin(2pi/n))^2 ]
= √[ 4cos^2(3pi/n)sin^2(2pi/n) + 4sin^2(3pi/n)sin^2(2pi/n) ]
= √[ 4(1-sin^2(3pi/n))(sin^2(2pi/n))+ 4sin^2(3pi/n)sin^2(2pi/n) ]
= 2 √ [ (sin^2(2pi/n)-sin^2(2pi/n)sin^2(3pi/n) + sin^2(2pi/n)sin^2(3pi/n) ]
= 2 √ [sin^2(2pi/n)]
= ± 2sin(2pi/n)
Since distance cannot be negative,d = 2|sin(2pi/n)|

As for the distance when the mod is not 1.
Note that the distances between successive adjacent roots will not be the same.

Oh you're right! As n increases, the circle formed from the polygon becomes smaller as well. So the distance is decreasing faster than when |x+iy| = 1. How do I use this information?

What you have is the first 2 roots, but you've got their argument (angle) wrong.
Apparently you forgot to divide the angle by n.
It should be:
Point 1(r^(1/n)cos(ø/n), r^(1/n)sin(ø/n))
Point 2(r^(1/n)cos((ø+2pi)/n), r^(1/n)sin((ø+2pi)/n))

Okay, here's the fix:

d^2 = (r^(1/n)cos((ø+2pi)/n)-r^(1/n)cos(ø/n)^2 + (r^(1/n)sin((ø+2pi)/n-r^(1/n)sin(ø/n))^2

_______________________________________
u + v / 2
(ø+2π + ø)/2n = 2ø+2π)/2n = (ø+π)/n

u - v / 2
(ø+2π - ø)/2n = π/n
_______________________________________


d^2 = (r^(1/n) * 2cos((ø+π)/n)sin(π/n))^2 + (-r^(1/n) * 2sin(ø+π)sin(π))^2
= 4r^(2/n) [ cos^2((ø+π)/n)sin^2(π/n) - sin^2((ø+π)/n)sin^2(π/n) ]
= 4r^(2/n) [ (1-sin^2((ø+π)/n))sin^2(π/n) - sin^2((ø+π)/n)sin^2(π/n) ]
= 4r^(2/n) [ sin^2(π/n) - sin^2((ø+π)/n)sin^2(π/n) - sin^2((ø+π)/n)sin^2(π/n) ]
= 4r^(2/n) [ sin^2(π/n) - 2sin^2((ø+π)/n)sin^2(π/n) ]
= 4r^(2/n) [ sin^2(π/n) (1 - 2sin^2((ø+π)/n) ]
= 4r^(2/n) [ sin^2(π/n) (cos^2((2ø+2π)/n) ]
= 4r^(2/n) [ sin^2(π/n) (cos^2((2ø+2π)/n) ]
d = 2r(1/n)|sin(π/n)cos((2ø+2π)/n)|

Is this right? And how do I prove if its true??

Btw, THANK SO MUCH for all this, I would have gotten this one wrong if you hadn't pointed it out.

 

[I like Serena]

My bad.
Your distance of 2|sin(pi/n)| is right.

I don't understand how

Here's how I got the generalization:

When z^n = 1, z = cis(2kπ/n), where k = 0,...,n-1

distance formula = √(x^2+y^2)
P1(1, 0)
P2(cos(2pi/n), sin(2pi/n)

d = √[ (cos(2pi/n)-1)^2 + (sin(2pi/n)-0)^2 ]

There it is: d = |cis(2pi/n)-1|
But nevermind. Your distance is right.

= √[ cos^2(2pi/n) - 2cos(2pi/n) + 1 + sin^2(2pi/n) ]
= √[-2cos(2pi/n)+2]
= √[-2 ( 1-2sin^2(pi/n) ]+2]v
= √[-2+4sin^2(pi/n)]+2]
= √[4sin^2(pi/n)]
= ± 2[sin(pi/n)] <---- CONJECTURE
Since distance cannot be negative,d = 2|sin(pi/n)|

When z^n = i, z =cis[(π + 4kπ)/2n], where k = 0,...,n-1

distance formula = √(x2+y2)
P1(cos(pi/2n), sin(pi/2n))
P2(cos(5pi/2n), sin(5pi/2n))

d = √[ (sin(5pi/2n)-sin(pi/2n))^2 + (cos(5pi/2n)-cos(pi/2n))^2 ]

Using trig's sum-to-product formula...

= √[ (2cos(3pi/n)sin(2pi/n))^2 + (-2sin(3pi/n)sin(2pi/n))^2 ]
= √[ 4cos^2(3pi/n)sin^2(2pi/n) + 4sin^2(3pi/n)sin^2(2pi/n) ]
= √[ 4(1-sin^2(3pi/n))(sin^2(2pi/n))+ 4sin^2(3pi/n)sin^2(2pi/n) ]
= 2 √ [ (sin^2(2pi/n)-sin^2(2pi/n)sin^2(3pi/n) + sin^2(2pi/n)sin^2(3pi/n) ]
= 2 √ [sin^2(2pi/n)]
= ± 2sin(2pi/n)
Since distance cannot be negative,d = 2|sin(2pi/n)|

Good!
And nice work!

Btw, there's also a geometrical interpretation and proof.
If you draw the roots in the complex plane, it should become obvious that the distance between the roots is the same for z^n=1 and for z^n=i.

Oh you're right! As n increases, the circle formed from the polygon becomes smaller as well. So the distance is decreasing faster than when |x+iy| = 1. How do I use this information?

Never mind.
The distance between the roots is equal after all.

FYI, the distance between root k and (k+1) is:
|r^(1/n)\cis((ø+2(k+1) pi)/n) - r^(1/n)cis((ø+2k pi)/n)|
But if you work it out, you'll see it's the same distance.

Okay, here's the fix:

d^2 = (r^(1/n)cos((ø+2pi)/n)-r^(1/n)cos(ø/n)^2 + (r^(1/n)sin((ø+2pi)/n-r^(1/n)sin(ø/n))^2

_______________________________________
u + v / 2
(ø+2π + ø)/2n = 2ø+2π)/2n = (ø+π)/n

u - v / 2
(ø+2π - ø)/2n = π/n
_______________________________________


d^2 = (r^(1/n) * 2cos((ø+π)/n)sin(π/n))^2 + (-r^(1/n) * 2sin(ø+π)sin(π))^2

I'm afraid you have the wrong trig identities here.
You should use:

cos u - cos v = -2sin[(u+v)/2] sin[(u-v)/2]
sin u - sin v = 2cos[(u+v)/2] sin[(u-v)/2]

= 4r^(2/n) [ cos^2((ø+π)/n)sin^2(π/n) - sin^2((ø+π)/n)sin^2(π/n) ]
= 4r^(2/n) [ (1-sin^2((ø+π)/n))sin^2(π/n) - sin^2((ø+π)/n)sin^2(π/n) ]
= 4r^(2/n) [ sin^2(π/n) - sin^2((ø+π)/n)sin^2(π/n) - sin^2((ø+π)/n)sin^2(π/n) ]
= 4r^(2/n) [ sin^2(π/n) - 2sin^2((ø+π)/n)sin^2(π/n) ]
= 4r^(2/n) [ sin^2(π/n) (1 - 2sin^2((ø+π)/n) ]
= 4r^(2/n) [ sin^2(π/n) (cos^2((2ø+2π)/n) ]
= 4r^(2/n) [ sin^2(π/n) (cos^2((2ø+2π)/n) ]
d = 2r(1/n)|sin(π/n)cos((2ø+2π)/n)|

Is this right? And how do I prove if its true??

What you have is a proof.
What you need is a way to verify independently there are no calculation mistakes.

Geometrically (that is, if you draw it), you can tell that the distance must be:
2r^(1/n)|sin(π/n)|

Btw, THANK SO MUCH for all this, I would have gotten this one wrong if you hadn't pointed it out.

No problem!

 

[314Jason]

I don't understand how

When z^n = i, z =cis[(π + 4kπ)/2n], where k = 0,...,n-1

distance formula = √(x2+y2)
P1(cos(pi/2n), sin(pi/2n))
P2(cos(5pi/2n), sin(5pi/2n))

d = √[ (sin(5pi/2n)-sin(pi/2n))^2 + (cos(5pi/2n)-cos(pi/2n))^2 ]

Using trig's sum-to-product formula...

= √[ (2cos(3pi/n)sin(2pi/n))^2 + (-2sin(3pi/n)sin(2pi/n))^2 ]
= √[ 4cos^2(3pi/n)sin^2(2pi/n) + 4sin^2(3pi/n)sin^2(2pi/n) ]
= √[ 4(1-sin^2(3pi/n))(sin^2(2pi/n))+ 4sin^2(3pi/n)sin^2(2pi/n) ]
= 2 √ [ (sin^2(2pi/n)-sin^2(2pi/n)sin^2(3pi/n) + sin^2(2pi/n)sin^2(3pi/n) ]
= 2 √ [sin^2(2pi/n)]
= ± 2sin(2pi/n)
Since distance cannot be negative,d = 2|sin([STRIKE]2[/STRIKE]pi/n)|

I noticed a mistake. For z^n = i, I got 2|sin(2pi/n)| but for z^n = 1, I got 2|sin(pi/n)|. I'm sure the later is correct. I've tried and tried again, but I just can't find the mistake. And I KNOW it must be the same length in either casses.


Also, for z^n = x+iy, when |x+iy| ≠ 1, I got d = 2r^(1/n)|sin(pi/n)|. But, I don't know how to show my generalization on a graph. I don't know what to do with the "r" value :( ...Is there any other way I can prove this is true? I need to explore a variety of different methods to make sure.

 

[I like Serena]

I noticed a mistake. For z^n = i, I got 2|sin(2pi/n)| but for z^n = 1, I got 2|sin(pi/n)|. I'm sure the later is correct. I've tried and tried again, but I just can't find the mistake. And I KNOW it must be the same length in either casses.

d = √[ (sin(5pi/2n)-sin(pi/2n))^2 + (cos(5pi/2n)-cos(pi/2n))^2 ]

Using trig's sum-to-product formula...

= √[ (2cos(3pi/n)sin(2pi/n))^2 + (-2sin(3pi/n)sin(2pi/n))^2 ]

The mistake is right here.
You forgot to divide (u+v) and (u-v) by 2.

Also, for z^n = x+iy, when |x+iy| ≠ 1, I got d = 2r^(1/n)|sin(pi/n)|. But, I don't know how to show my generalization on a graph. I don't know what to do with the "r" value :( ...Is there any other way I can prove this is true? I need to explore a variety of different methods to make sure.

If you draw a graph for z^n=1, you find a regular polygon.
For z^n=i, you find the same polygon, only rotated by 90 degrees.

For z^n=x+iy, again you find the same polygon, but rotated over Arg(x+iy) and scaled by r^(1/n).
And if you're interested, here's another way to show your result.
It depends on Euler's formula, which is the generalization of De Moivre's theorem.

Euler's formula is:
$$e^{i \phi} = \cos \phi + i \sin \phi$$
Which can be rewritten as:
$$\cos \phi = {1 \over 2}(e^{i \phi} + e^{i \phi})$$$$\sin \phi = {1 \over 2i}(e^{i \phi} - e^{i \phi})$$Starting with your problem, you have:
z^n = i
The solution as you basically already know is:
$$z = (e^{i ({\pi \over 2} + 2k\pi)})^{1 \over n} = e^{i ({\pi \over 2} + 2k\pi) {1 \over n}}$$

The distance between the first and second root is:

$$d = |e^{i ({\pi \over 2} + 2\pi) {1 \over n}} - e^{i ({\pi \over 2}) {1 \over n}}|$$
$$d = |e^{i ({\pi \over 2} + \pi) {1 \over n}}(e^{i {\pi \over n}} - e^{-i {\pi \over n}})|$$
$$d = 1 \cdot |(e^{i {\pi \over n}} - e^{-i {\pi \over n}})|$$
$$d = |2i \sin{\pi \over n}|$$
$$d = 2 |\sin{\pi \over n}|$$

 

[314Jason]

If you draw a graph for z^n=1, you find a regular polygon.
For z^n=i, you find the same polygon, only rotated by 90 degrees.

For z^n=x+iy, again you find the same polygon, but rotated over Arg(x+iy) and scaled by r^(1/n).

Oooo. Is there a way to prove this? I drew the graphs and they same to be rotating all over the place.

If I were to make this the "original polygon" (z^n = 1 with ø = 0)
Then this (z^n = i with ø = π/2) would rotate it 90/n degrees anti-clockwise.

E.g. When z^3 = x+iy, the "original polygon" would rotate 90/3 = 30 degrees anti-clockwise.

I feel like I need to adress this pattern ("rotated over Arg(x+iy) and scaled by r^(1/n)") but I don't know how I can show it or prove it mathematically!

And if you're interested, here's another way to show your result.
It depends on Euler's formula, which is the generalization of De Moivre's theorem.

Euler's formula is:
$$e^{i \phi} = \cos \phi + i \sin \phi$$
Which can be rewritten as:
$$\cos \phi = {1 \over 2}(e^{i \phi} + e^{i \phi})$$$$\sin \phi = {1 \over 2i}(e^{i \phi} - e^{i \phi})$$


Starting with your problem, you have:
z^n = i
The solution as you basically already know is:
$$z = (e^{i ({\pi \over 2} + 2k\pi)})^{1 \over n} = e^{i ({\pi \over 2} + 2k\pi) {1 \over n}}$$

The distance between the first and second root is:

$$d = |e^{i ({\pi \over 2} + 2\pi) {1 \over n}} - e^{i ({\pi \over 2}) {1 \over n}}|$$
$$d = |e^{i ({\pi \over 2} + \pi) {1 \over n}}(e^{i {\pi \over n}} - e^{-i {\pi \over n}})|$$
$$d = 1 \cdot |(e^{i {\pi \over n}} - e^{-i {\pi \over n}})|$$
$$d = |2i \sin{\pi \over n}|$$
$$d = 2 |\sin{\pi \over n}|$$

Wow this is great! Thanks!

But I don't quite understand what you did here:

The distance between the first and second root is:
$$d = |e^{i ({\pi \over 2} + 2\pi) {1 \over n}} - e^{i ({\pi \over 2}) {1 \over n}}|$$

Is there a rule to find the distance between the two points with Euler? I haven't learned Euler.

Also, I happen to study Mathematical Induction in class, and that's pretty much the only proving I know how to do. But it doesn't seem to work here! Since you must have a lot more experience with this, from the methods listed in the link below, what do you think I can use?

http://en.wikipedia.org/wiki/Mathematical_proof#Direct_proof

 

[I like Serena]

Oooo. Is there a way to prove this? I drew the graphs and they same to be rotating all over the place.

If I were to make this the "original polygon" (z^n = 1 with ø = 0)
Then this (z^n = i with ø = π/2) would rotate it 90/n degrees anti-clockwise.

E.g. When z^3 = x+iy, the "original polygon" would rotate 90/3 = 30 degrees anti-clockwise.

I feel like I need to adress this pattern ("rotated over Arg(x+iy) and scaled by r^(1/n)") but I don't know how I can show it or prove it mathematically!

The "original" polygon has its points on the unit circle.
The angle at the origin between successive points is 2pi/n.
Therefore the polygon is regular, meaning all its edges have the same length.

When x+iy has an angle ø, the first root is at ø/n, and on the circle with radius r^(1/n).
Each successive root is on the same circle and has an angle 2pi/n further along.
Therefore the polygon is regular, meaning all its edges have the same length.

I believe this would suffice as a proof.

Wow this is great! Thanks!

But I don't quite understand what you did here:

The distance between the first and second root is:
$$d = |e^{i ({\pi \over 2} + 2\pi) {1 \over n}} - e^{i ({\pi \over 2}) {1 \over n}}|$$

Is there a rule to find the distance between the two points with Euler? I haven't learned Euler.

A complex number represents a point in the complex plane, meaning it has an x-coordinate (the real part) and an y-coordinate (the imaginary part).
The distance between 2 of these points is simply the absolute value of the difference of the 2 complex numbers.

The exponential expression is simply another way to write a complex number, using its distance to the origin and the angle with the real axis.
The exponential expression is also another way to express De Moivre's theorem.

Also, I happen to study Mathematical Induction in class, and that's pretty much the only proving I know how to do. But it doesn't seem to work here! Since you must have a lot more experience with this, from the methods listed in the link below, what do you think I can use?

http://en.wikipedia.org/wiki/Mathematical_proof#Direct_proof

In practice I basically use only 3 proofs: forward proof, proof by contradiction, and full induction.
Actually I rarely see full induction except in a math context that is dedicated to it.
I guess I do use the others too, but I'm not really aware that I'm doing that.
I haven't studied how you would call the specific ways of proving things.

 

[314Jason]

Hi! Sorry for the late reply, I've been busy completing a ton of assignments.

I talked to my teacher about proving the conjecture and he said he expects me to use mathematical induction, and that its the "best, valid proof".

I just have no idea how I could apply that here. According to this: http://people.richland.edu/james/lecture/m116/sequences/induction.html

I need to make the left hand side equal the right hand side. But my conjecture is (the distance between 2 adjacent roots of z^n=1) = 2|sin(pi/2)|

From the examples I found online, its generally listing out the values (in this case the distances) and seeing the pattern, then forming a conjecture from the values observed. And proving it by making the expressions equal.

For example, I have to find the distances of n=3, n=4, n=5, etc.

n distance
3 √3
4 √2
5 1.755
6 1
...

And then see from the values above that its 2|sin(pi/2)|. THEN I'll prove it by simplifying the distance formula. But the thing is, I can't form a conjecture just by looking at those values!

 

[I like Serena]

Uum... didn't you already proof this for the general case with Direct proof?

I'm not sure what else there is to do.
I do not see how to proof this by full induction.

Listing specific cases for a sequence of n-values makes the conjecture likely.
I guess this is also a form of induction, but it is no proof.