Finding n roots of unity

image
image
image
image

This leads to Quantization of θ  because there are only specific values that become possible for  θ  when we impose that after n rotations it has to return back to its same starting point. **

And the following are those values:

image

The problem of finding the values of θ for given values of n is more generally known as the N roots of unity.  

We will leave it as an exercise, but the following animation plots all the possible the values of θ for integer values of n=2,3,4,.. from the above equation that we found:

                            Source

And those are the roots of the equation z^n  = z i.e if you start at these points on the unit circle and make n rotations you will get back to the same point that you started with.

Have a good one!

** A more physical way to think about this is matching boundary conditions.
For more insight on how boundary conditions lead to quantization, take a
look at this  post.

*** Finding out the n roots of unity (Video)

If unit vectors always scared you for some reason, this neat little trick  from The story of i by Paul Nahin involving complex numbers is bound to be a solace.

It allows you find the tangential and radial components of acceleration through simple differentiation. How about that! 

Have a good one!

** r = r(t),  θ =  θ(t)

Using Complex numbers in Classical Mechanics

When one is solving problems on the two dimensional plane and you are using polar coordinates, it is always a challenge to remember what the velocity/acceleration in the radial and angular directions (v_r , v_{\theta}, a_r, a_{\theta} ) are. Here’s one failsafe way using complex numbers that made things really easy :

z = re^{i \theta}

\dot{z} = \dot{r}e^{i \theta} + ir\dot{\theta}e^{i \theta} = (\dot{r} + ir\dot{\theta} ) e^{i \theta}

From the above expression, we can obtain v_r = \dot{r} and v_{\theta} = r\dot{\theta}

\ddot{z} =  (\ddot{r} + ir\ddot{\theta} + i\dot{r}\dot{\theta} ) e^{i \theta}   + (\dot{r} + ir\dot{\theta} )i \dot{\theta} e^{i \theta} 

\ddot{z} =  (\ddot{r} + ir\ddot{\theta} + i\dot{r}\dot{\theta}  + i  \dot{r} \dot{\theta} - r\dot{\theta}\dot{\theta} )e^{i \theta} 

\ddot{z} =  (\ddot{r} - r(\dot{\theta})^2+ i(r\ddot{\theta} + 2\dot{r}\dot{\theta} ) )e^{i \theta} 

From this we can obtain a_r = \ddot{r} - r(\dot{\theta})^2 and a_{\theta} = (r\ddot{\theta} + 2\dot{r}\dot{\theta}) with absolute ease.

Something that I realized only after a mechanics course in college was done and dusted but nevertheless a really cool and interesting place where complex numbers come in handy!

 

 

nth roots of unity : A geometric approach

When one is dealing with complex numbers, it is many a times useful to think of them as transformations. The problem at hand is to find the nth roots of unity. i.e

z^n = 1

Multiplication as a Transformation

Multiplication in the complex plane is mere rotation and scaling. i.e

z_{1} = r_{1}e^{i\theta_{1}}, z_{2} = r_{2}e^{i\theta_{2}} 

z_{1}z_{2} = \underbrace{r_{1} r_{2}}_{scaling} \underbrace{e^{i(\theta_{1} + \theta_{2})}}_{rotation}

Now what does finding the n roots of unity mean?

If you start at 1 and perform n equal rotations( because multiplication is nothing but rotation + scaling ), you should again end up at 1.

We just need to find the complex numbers that do this.i.e

z^n = 1

\underbrace{zz \hdots z}_{n} = 1

z = re^{i\theta}

r^{n}e^{i(\theta + \theta + \hdots \theta)} = 1e^{2\pi k i}

r^{n}e^{in\theta} =1e^{2\pi k i}

This implies that :

\theta = \frac{2\pi k}{n}, r = 1

And therefore :

z = e^{\frac{2\pi k i}{n}}

Take a circle, slice it into n equal parts and voila you have your n roots of unity.

main-qimg-6da134e3e5735a9fb92355d53f95e4ed

Okay, but what does this imply ?

Multiplication by 1 is a 360^o/0^o rotation.

drawing

When you say that you are multiplying a positive real number(say 1) with 1 , we get a number(1) that is on the same positive real axis.

Multiplication by (-1) is a 180^o rotation.

drawing-1

When you multiply a positive real number (say 1) with -1, then we get a number (-1) that is on the negative real axis

The act of multiplying 1 by (-1) has resulted in a 180o transformation. And doing it again gets us back to 1.

Multiplication by i is a 90^o rotation.

drawing-2

Similarly multiplying by i takes 1 from real axis to the imaginary axis, which is a 90o rotation.

This applies to -i as well.

That’s about it! – That’s what the nth roots of unity mean geometrically. Have a good one!

 

Tricks that I wish I knew in High School : Trigonometry (#1)

I really wish that in High School the math curriculum would dig a little deeper into Complex Numbers because frankly Algebra in the Real Domain is not that elegant as it is in the Complex Domain.

To illustrate this let’s consider this dreaded formula that is often asked to be proved/ used in some other problems:

cos(nx)cos(mx) =  ?

Now in the complex domain:

cos(x) = \frac{e^{ix} + e^{-ix}}{2}

And therefore:

cos(mx) = \frac{e^{imx} + e^{-imx}}{2}

cos(nx) = \frac{e^{inx} + e^{-inx}}{2}

cos(mx)cos(nx)  = \left( \frac{e^{imx} + e^{-imx}}{2} \right) \left(  \frac{e^{inx} + e^{-inx}}{2} \right)

cos(mx)cos(nx)  = \frac{1}{4} \left( e^{i(m+n)x} + e^{-i(m+n)x} + e^{i(m-n)x} + e^{-i(m-n)x}   \right)

cos(mx)cos(nx)  = \frac{1}{2} \left( \left( \frac{e^{i(m+n)x} + e^{-i(m+n)x}}{2} \right) + \left( \frac{e^{i(m-n)x} + e^{-i(m-n)x}}{2} \right)   \right)

cos(mx)cos(nx)  = \frac{1}{2} \left( cos(m+n)x + cos(m-n)x   \right)
And similarly for its variants like cos(mx)sin(nx) and sin(mx)sin(nx) as well.

****

Now if you are in High School, that’s probably all that you will see. But if you have college friends and you took a peak what they rambled about in their notebooks, then you might this expression (for m \neq n):

I =  \int\limits_{-\pi}^{\pi} cos(mx)cos(nx) dx \\

But you as a high schooler already know a formula for this expression:

I =  \int\limits_{-\pi}^{\pi} \left( cos(m+n)x + cos(m-n)x   \right)dx \\

I =  \int\limits_{-\pi}^{\pi} cos(\lambda_1 x) dx + \int\limits_{-\pi}^{\pi} cos(\lambda_2 x) dx \\ 

where \lambda_1, \lambda_2 are merely some numbers. Now you plot some of these values for lambda i.e (\lambda = 1,2, \hdots) and notice that since integration is the area under the curve, the areas cancel out for any real number. drawing.png

and so on….. Therefore:

I =  \int\limits_{-\pi}^{\pi} cos(mx)cos(nx)dx = 0

This is an important result from the view point of Fourier Series!

n roots of unity

When one is dealing with complex numbers, it is many a times useful to
think of them as transformations. The problem at hand is to find the n
roots of unity. i.e

image

As is common knowledge z = 1 is always a solution.

Multiplication as a transformation

Multiplication in the complex plane is mere rotation and scaling. i.e

image
image

Now what does finding the n roots of unity mean?

If
you start at 1 and perform n equal rotations( because multiplication is nothing but rotation + scaling ), you should again end up
at 1.

We just need to find the complex numbers that do this.i.e

image
image
image
image
image

This implies that :

image

And therefore :

image

Take a circle, slice it into n equal parts and voila you have your n roots of unity.

image


Okay, but what does this imply ?

Multiplication by 1 is a 360o / 0o rotation.

When
you say that you are multiplying a positive real number(say 1) with 1 ,
we get  a number(1) that is on the same positive real axis.

image


Multiplication by (-1) is a 180o rotation.

When you multiply a positive real number (say 1) with -1, then we get a number (-1) that is on the negative real axis 

The act of multiplying 1 by (-1) has resulted in a 180o transformation. And doing it again gets us back to 1.

image



Multiplication by i is a 90o rotation.

Similarly multiplying by i takes 1 from real axis to the imaginary axis, which is a 90o rotation. 

This applies to -i as well.

image

so on and so forth,

Have a great day!

fuckyeahphysica:

In mathematics there is a concept known as ‘Conformal Mapping’ which allows you convert a given shape to a completely different one by making a transformation.

In the joukowski transform you take all the points on a circle and apply the following transform:

image

And the resulting transformed points resemble an aerofoil shape. Pretty cool huh ?

** Conformal mappings are a really cool topic in complex analysis but also equally extensive. If you want to know more about them click here

EDIT:

I forgot to mention that the transformation of a circle into an aerofoil is a very special case and the transformation of other shapes would yield distinct shapes. ( very different from an aerofoil )

Here’‘s what i mean:

Different circle

image

Triangle

image

Ellipse

image

Source : AT&T Archives

–. — — -.. / -.. .- -.–

fuckyeahphysica:

Only Time will tell – A Complex Number Tribute.

I was in High School when the notion of complex numbers was fed into my vocabulary. None of it made sense. “ Why on earth did they have to invent a new Number System? Uh.. Mathematicians !!  “, One of my friends remarked. And as distressing as it was, we weren’t able to comprehend why!

image

There are certain elegant aspects to complex numbers that are often overlooked but are pivotal to understanding them. Of the top of the chart – the events that led to the invention / discovery of complex numbers. To shed some light on these events is the crux of this post.

A date with history.

There were quotidian equations such as x² + 1 =0 which people wanted to solve, but it was well known that the equation had no solutions in the realms of real numbers. Why, you ask ?. Well, quite intuitively the addition of a square real number ( always positive ) and one was never going to yield 0.

image

And also, as is evident from the graph, the curve does not intersect the x- axis for a solution to persist.

For the ancient Greeks, Mathematics was synonymous with Geometry. And there were a legion geometrical problems which had no solutions, peculiar quadratics like  x² + 1 =0 were branded the same way.

“ Why make up new numbers for the sole purposes for being solutions to Quadratic equations? “. This was the rationale that people stuck with.

The Real Challenge.

image

Quadratics, per se were easy to solve.  A 16th-century mathematician’s redemption was confronting a cubic equation. Unlike Quadratics, cubic equations pass through the x axis at least once, so the existence of solution is guaranteed. To seek out for them was the challenge.

image

The general form of a cubic equation is as follows:

f (x) = au³  + bu² + cu + d

If we divide throughout by “a”, it simplifies the equation and substituting x  = u – ( b / 3a )  gets rid of the squared term. Thus, we obtain:

x³ – 3px – 2q = 0

A mathematician named Cardano is attributed for coming up with the solution for the above equation as :

x = ³√( q + √ ( q²  – p³ ) )  +  ³√( q – √ ( q²  – p³ ) )

This equation is perfectly legit. But when p³ > q² it yields incomprehensible solutions.

Bombelli’s “Wild Thought”.

image

The strangeness of the formula enticed Bombelli. He considered the equation x³  = 15x + 4. By virtue of inspection, he found out that x = 4 satisfied the equation. But, plugging values into the cardano’s equation, he obtained:

x = ³√ ( 2 + 11 i ) + ³√ ( 2  – 11 i )    where i = √-1

Wait a minute! The equation is hinting that there exists no solution in the real numbers domain, but in contraire x = 4 is a solution!!

In a desperate attempt to resolve the paradox, he had what he called it as a ‘wild thought’. What if the equation could be broken down as

x = ( 2 + n i ) + ( 2 – n i )

This would yield x = 4 and resolve the conflict. It might sound magical, but when he tested out his abstraction, he was indeed right. From calculations, he obtained the value for n as 1. i.e

2 ± i =  ³√( 2  ± 11 i )

This was the birth of Complex Numbers. 

image

By treating a quantity such as 2 + 11√-1, without regard for its meaning in just the same way as a natural number, Bombelli unlike no one before him had come up with a modus operandi for dealing with such intricate equations, which were previously thought to have no solutions.

While complex numbers per se still remained mysterious, Bombelli’s work on Cubic equations thus established that perfectly real problems required complex arithmetic for their solutions.This empowered people to venture into frontiers which were formerly unexplored.

And for this triumph Bombelli is regarded as the Inventor of Complex Numbers.

Fun fact

A moon crater was named after Bombelli, honoring his accomplishments.

image

PC: NASA, gpr, realitycrowdtv

Thanks for reading! Hope you enjoyed reading it as much as I enjoyed writing it.

Only Time will tell – A Complex Number Tribute.

I was in High School when the notion of complex numbers was fed into my vocabulary. None of it made sense. “ Why on earth did they have to invent a new Number System? Uh.. Mathematicians !!  “, One of my friends remarked. And as distressing as it was, we weren’t able to comprehend why!

image

There are certain elegant aspects to complex numbers that are often overlooked but are pivotal to understanding them. Of the top of the chart – the events that led to the invention / discovery of complex numbers. To shed some light on these events is the crux of this post.

A date with history.

There were quotidian equations such as x² + 1 =0 which people wanted to solve, but it was well known that the equation had no solutions in the realms of real numbers. Why, you ask ?. Well, quite intuitively the addition of a square real number ( always positive ) and one was never going to yield 0.

image

And also, as is evident from the graph, the curve does not intersect the x- axis for a solution to persist.

For the ancient Greeks, Mathematics was synonymous with Geometry. And there were a legion geometrical problems which had no solutions, peculiar quadratics like  x² + 1 =0 were branded the same way.

“ Why make up new numbers for the sole purposes for being solutions to Quadratic equations? “. This was the rationale that people stuck with.

The Real Challenge.

image

Quadratics, per se were easy to solve.  A 16th-century mathematician’s redemption was confronting a cubic equation. Unlike Quadratics, cubic equations pass through the x axis at least once, so the existence of solution is guaranteed. To seek out for them was the challenge.

image

The general form of a cubic equation is as follows:

f (x) = au³  + bu² + cu + d

If we divide throughout by “a”, it simplifies the equation and substituting x  = u – ( b / 3a )  gets rid of the squared term. Thus, we obtain:

x³ – 3px – 2q = 0

A mathematician named Cardano is attributed for coming up with the solution for the above equation as :

x = ³√( q + √ ( q²  – p³ ) )  +  ³√( q – √ ( q²  – p³ ) )

This equation is perfectly legit. But when p³ > q² it yields incomprehensible solutions.

Bombelli’s “Wild Thought”.

image

The strangeness of the formula enticed Bombelli. He considered the equation x³  = 15x + 4. By virtue of inspection, he found out that x = 4 satisfied the equation. But, plugging values into the cardano’s equation, he obtained:

x = ³√ ( 2 + 11 i ) + ³√ ( 2  – 11 i )    where i = √-1

Wait a minute! The equation is hinting that there exists no solution in the real numbers domain, but in contraire x = 4 is a solution!!

In a desperate attempt to resolve the paradox, he had what he called it as a ‘wild thought’. What if the equation could be broken down as

x = ( 2 + n i ) + ( 2 – n i )

This would yield x = 4 and resolve the conflict. It might sound magical, but when he tested out his abstraction, he was indeed right. From calculations, he obtained the value for n as 1. i.e

2 ± i =  ³√( 2  ± 11 i )

This was the birth of Complex Numbers. 

image

By treating a quantity such as 2 + 11√-1, without regard for its meaning in just the same way as a natural number, Bombelli unlike no one before him had come up with a modus operandi for dealing with such intricate equations, which were previously thought to have no solutions.

While complex numbers per se still remained mysterious, Bombelli’s work on Cubic equations thus established that perfectly real problems required complex arithmetic for their solutions.This empowered people to venture into frontiers which were formerly unexplored.

And for this triumph Bombelli is regarded as the Inventor of Complex Numbers.

Fun fact

A moon crater was named after Bombelli, honoring his accomplishments.

image

PC: NASA, gpr, realitycrowdtv

Thanks for reading! Hope you enjoyed reading it as much as I enjoyed writing it.