Cube Root of Unity 

Why is this math community so much better than others that i have been involved in? Here, on this channel, nobody judges, nobody brags,.... its a utopia of curiosity and good vibes.

I am a math teacher in Greece. You explain all these very . Congratulations.

but the way you were so excited to teach, omg. i smiled everytime you smiled and looked into the camera😭. first time watching your video, gonna continue for sure. love from India!

Sir I really understood the basics of this topic. At first I was really scared of this topic. But now after I got to know this topic explained with simplicity, I became a fan of your teaching. Thank you very much!!!!!!!!!!

We can generalise, that the sum of consecutive 3 powers of omega is always equal to zero. Btw, your views should be in millions. The way you teach is just awesome, even a 5th std kid would understand a 10th std concept by seeing your videos!! Hats off sir 🙏🙏

You're such a great teacher! I had so much difficulty in understanding this topic but you explained it so well and so patiently. Truly helpful! Please keep posting videos and maybe start some courses on RU-vid as well because you're really a great teacher!

What's also interesting is that it doesn't matter which complex root is called omega. If you square it, the result is the other root which can be called omega-squared. The results are fascinating indeed.

Again... unbelievable. Especially when you put these solutions in a vector form. When you add these 3 vectors you end up at the origin.

The only thing missing here is that (ω²)² = ω⁴ = ω³·ω = 1·ω = ω, or in other words, each of ω and ω² is the square of the other. Repeatedly squaring flips back and forth from one to the other. There's also that -ω is a sixth root of unity, but that's out of the scope of this video.

Sir🙏 , i am from India thank you for this amazing video which solves my all problems those are roaming in my mind.❤❤

I'm a class 12 student from indian I don't know about the cube root of unity after I seen this video I understand about it and this video is underrated

Does this guy just know everything?

This was fabulous, sir! I had trouble understanding from my own textbooks what unity meant in a case full of examples pertaining to it. The video helped marvelously!

so simple yet so intriguing. Man mind blown

Man.... What it ended...oh god 12 min gone so quickly ....man your teaching style is just mesmerizing ...i just loved it and also it cleared my doubt ❤

Why did i understand this so easilyyy You're the best!!

Thank you SO MUCH for verifying a complex solution involving an odd power!

You should have finished the video with a graphic representation of the 3 roots on a complex plane

we can get 'n' th root of any number. Just take a circle with radius of the number in argand plane and divide 2pi by n and every {2zpi/n} z belongs to integer 0

Fascinating stuff. Entertaining mathematics.

vieta's formula says it all. great video anyway! (i've been thinking about this since i started watching your videos: if there's an ASMR maths channel, you're definitely the one!)

This is elegant and beautiful

What's also amazing is when you look at these on the argand diagram and see how they relate when they are written in polar form :D

Thank you for the video, hope you discuss the curvature of a curve in a simplistic way.

These last properties of the cube roots (they multiply to 1 and sum to 0) is a property of all nth roots of 1, where n is any natural number.

No this is really insane wow!!

Great video as usual, I have a question: is Omega the only number (besides 1) that equals the fourth power of itself (I noticed that omega^2 * omega^2 = omega)? This also implies that (omega^(2^k))^3 = 1 for any positive integer k, right? And the cool thing about this formula is that it also encompasses the basic case 1^3=1 by picking k=0!

The magnitude of omega is 1 and argument of 120 degrees (2pi/3 radians), so repeated mulitplication by omega 'spins' us around the origin, reaching (1, 0i) every third step.

Thank u soo much sir, it really helped a lot ❤

z^3 +0*x^2+0*x-1=0 The coefficient of x^2 is equal to 0, therefore according to Vieta's formulas, the sum of the roots is 0

this was a beautiful video

Bravo from Greece.

Thank you for the nice video. 💥 The grapical representation of 1, ω, ω² on the complex plane, as it seems, will be good thing to provide visual proof of sum 1 + ω + ω² = 0 and product 1 ∙ ω ∙ ω² = 1.

We can generalise those to all nth root of unity. I am not a math teacher so I cant explain every thing but let me give you some tools. Imagine the unity cercle in complex plan. center or the cercle is 0.0 and it goes through 1,0 if you want the all the nth root of 1 you just need to divide the cercle into n equal parts. to add all of those you just need to add the vectors from 0.0 to those roots, and since they are by construction balanced around the circle, sum is 0. For the multiplication I dont have a visual proof to mind.

Cool video

Bravo Prime Newtons! That's serious stuff. Power 3

Amazing...

Excellent.....

Which class omega is studied in

With (u+v)^3=u^3+v^3+3uv(u+v) and Vieta formulas for quadratic x_{1}+x_{2} = -b/a x_{1}x_{2} = c/a we can solve cubic equation Cube roots of unity help us to find all roots of cubic equation not just one of them

Power System Protection Engineers use this daily. We just don't call it "omega" and the square root of negative 1 is j.

in my opinion, representing the numbers on complex plan, as vectors would more show why the sum = 0 but formally all the video is great. polynom of degree n will always have n solutions either in R or C (and at times even more if k*2pi involved) Ps : your accent is so crystal clear to foreigners... it's a no accent or from anywhere ?)

(a+bi)³ = 1 (a+bi)(a²-b²+2abi) = 1 a³-3ab²+3a²bi-b³i = 1 a³-3ab²+i(3a²b-b³) = 1 a³-3ab² = 1 and 3a²b-b³ = 0 3a² = b² and a³-9a³ = 1 a³ = -1/8 and a = -1/2 3(1/4) = b² and b = ±sqrt(3)/2 The complex answers are (-1±isqrt(3))/2

Each of the complex cube roots of unity is both the square and the square root of the other .

You forgot to mention, that (ω²)² = ω, which leads to (ω²)³ = ω * ω² = ω³, which you already showed, is 1.

Thanx

very well, I wanted to write.

Which grade omega is studied in

Do you know Probability and Statistics

Now divide them 😈

What a great teacher!!

But, I thougt that n sqrt could use at real number.

I have a unity Shirt lol

Hi brother

😯😮😶🙂😀😃😍

I desagree. If I'm wrong, I would like someone with more knowledge to correct me. Thank you. In the video, in my understanding, the calculation of a root is being confused with the solutions or roots of an equation. The root symbol is reserved only for the principal root of a number or, if desired, the root of the principal argument complex. Thus, if x is equal to the cube root of 1 then x=1. There are no more values. Cube root is a function. And a function only returns one value. A example about I'm telling. Nobody disputes that √4 = 2. No one who knows anything about mathematics can claim that √4=-2. This is false. Since the symbol √ is reserved for the main root or positive branch (if real). According to the chain of reasoning in the video, the following happens. √4 = x we square (correct) (√4)² = x² It is correct to cancel the root and the square since 4 is positive 4 = x² x² - 4 = 0 (x + 2) (x - 2) = 0 Therefore, x + 2 = 0 → x = -2 x - 2 = 0 → x = 2 Therefore x = -2 is a solution of the equation x² - 4 = 0, but it is not a solution of √4 = x, because √4 ≠ -2, as everybody knows. In the original equation in the video, cube root of 1 only has one value that satisfies it, that is, x=1. The problem appears in the second step, when it is rised to 3 power snd transform in another different equation x³ = 1. In this step two extra (imaginary numbers) solutions have been added. They must be wasted.

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