Exponential form to find complex roots | Imaginary and complex numbers | Precalculus | Khan Academy

[@cpanati]

Connect the 3 roots for X^3 = 1 and you get a triangle. Connect the 4 roots for X^4 = 1 and you get a square. Connect the 5 roots of X^5 = 1 and you get a pentagon. The 6 roots for X^6 = 1 and you get a hexagon. Heptagon, Octagon, et. You are now seeing the symmetry that leads into the subject of Group Theory.

[@guidad542]

Man, I know this is coming 2 years late but thank you! This helped me gain tremendous insight and really gain much-needed intuition!

[@isaactyrone4844]

Why do all the sum of all the solutions to the equation = 0?

[@charleshudson5330]

@@isaactyrone4844 When you set an equation in X equal to zero, you are asking that equation "Where do you cross of X axis?" You are asking the equation to tell you specific information. Only the points when the equation goes across the X axis.

[@isaactyrone4844]

@@charleshudson5330 but let’s say I have a complex equation to the 7th power. Therefore I should get 7 solutions. Some imaginary. Some real. Why is it that the sum of all the solutions will equal 0?

[@tapanbaral5862]

Mind boggling

[@c4talin94]

Man, it's 2 a.m. before a test and here you are! Saving my ass again, as you always did. I can't find the words to thank you enough!

[@mojorn8837]

c4talin94 4 years later and I’m going through the same thing!

[@user-rd5nc1nb9f]

@@mojorn8837 same here

[@promisechuks6445]

5years later, at 2am on a Monday morning, I'm here doing the same thing before a test.

[@Blue_Yuan]

@@promisechuks6445 2months later at 1.38am morning，im here before a test dude

[@kevinqu640]

I was gonna like your comment cause I'm experiencing the same but since u got 69 likes I'm just gonna comment instead ; )

[@Crossfire9211]

I find the repetition helps instil the information, gives myself time to get his point before he moves on. Thumbs up for every vid man, no one does this as good as you.

[@shizaep]

this vid is a complete bless still

[@philipbernstein5069]

man, i was struggling with this in math all last week, and you made it crystal clear to me in ten minutes....great job

[@zanepotts]

I learned in 10 minutes what I had been trying to learn through an entire hour and half lecture. Thanks man!!!

[@ImAllInNow]

Think in Tau, even more easy. 2 pi / 3 becomes tau/3 which is so obviously 1/3 of a full rotation.

[@PercentageSign]

this video is golden for EE if you never get taught this until it's too late, thank you so much

[@Jemiiiimmaaa]

So true

[@trbone76]

That’s how 2pi/3 is expressed in complex numbers. cos(2pi/3) corresponds to the real part of the number because the real axis is horizontal and the cosine of an angle gives you the horizontal component. sin(2pi/3) corresponds to the imaginary part of the number because the imaginary axis is vertical and the sine of an angle gives you the vertical component. cos(2pi/3) (real) is -1/2 and sin(2pi/3) (imaginary) is sqrt(3)/2 (basic trig), so the complex number can be written as -1/2 + i*sqrt(3)/2.

[@katherinemays9856]

This is probably the coolest thing I've ever seen.

[@danieljumbo5875]

He makes Maths so cool

[@chickmagnent231]

You are GREAT!!! My math teacher is good but I cannot pay attention in class, this is the solution to my problem.

[@dhinas9444]

Well, I am not in a hurry or before a test, but i find this also nice. The explanations are so organic. Nothing stale...

[@Hazit90]

this guy makes me enjoy maths. my teacher can do one.lol

[@boxxer221]

Probably my favorite piece of maths. All maths should be this nice.

[@yakashgoyal]

To find the roots just use the eqn Cosx + isinx = e^ix

[@paulwood3460]

Complex exponentials are truly amazing 👏👏👍👍

[@ItsGazareth]

He calculates the cosine and sine terms from the line above.

[@TheNetkrot]

this was nothing but great Mr Khan ... thank you so much.

[@P_S_42]

thank you

[@adduun]

Thank you T. Sal

[@miltonweinhold6859]

thank you sir

[@PandorasBox2]

man you are miracle worker i was so confused about this before thank you you got it all covered in less than 12 mins lol tnX again

[@AstroKedde]

thanks i get it now :) my text book is not very good at passing on its knowledge to me and i "missed "the lecture my professors gave me on the subject..

[@KrunalRindani]

A Level Pure Maths in two days and I'm here seeing this for the first time. Anyone else in the same boat as me?

[@panthar1]

Not seeing it for the first time, I learned some of this in calc 2, but I must say, I did not learn this as a precalc thing.

[@clinteldrickpetilla9646]

Thank God

[@Phagocytosis]

That requires trigonometry... soh cah toa etc. Look at his trigonometry videos for explanation on that. Also, the triangles you get in this case are simple 30-60-90 triangles, so Sal just knew what the sides of that were.

[@somebody3398]

it helps get the message through

[@lucasm4299]

At 2:44 where did he get "e"? Why is it here?

[@ms2333]

+Lucas M It comes from Euler's formula, which is e^ix=cos(x)+i*sin(x). You can imagine you can use triangles and trig functions to do this, but it ends up being simpler to let x=the angle in radians and use the e^ix form instead. This also makes it obvious why the e^i*pi=-1, which is the famous Euler's Identity.

[@lucasm4299]

+ms2333 Thanks. 4 months ago I didn't know about it. Now I have already learned about Taylor Series, and Euler's formula. Currently working on Fourier Series. I still need help with Fourier. I am also trying to understand the gradient. \/f(x,y)=

[@erikdahl95]

Also doing fourier right now, any tips on great videos or sites for learning?

[@Selorm123]

Please can you do that of x^4=1?

[@rismarksvej]

I know it is Euler's magic number. However, where is the proof and evident source of it? You could say its lim(1+1/n)^n but please wait...there would be another book for that.

[@johnnewmann2628]

-pi

[@lllUSlll]

Thank You for elaborating this.

[@HoshikawaHikari]

Somehow this is clearer than my paid lecture

[@ingthoringason6900]

god bless you

[@xx_faze_rolf_xxl4029]

complex numbers are such cancer i legit wanna smack my head onto the table untill my skull caves in

[@sgurdmeal662]

Complex numbers allow us to do so many things, not only that, they're extremely beutiful and interesting on their own.

[@TheCaptainshem]

Hi from I am from the Philippines!

[@mohammadalam3021]

Very helpful thanks

[@hqatran23]

I think I know why that happens, it's because the brain has trouble multi-tasking. He is solving the problem AND explaining at the same time, and when he tries to solve the problem in his mind, it's quite hard to talk to your audience whilst thinking!

[@chanelbandit]

Bless you

[@phillies4eva]

@xScuzzleButtx they are like place holders in an imaginary plane, for example, kinetic energy to potential energy in a spring machine. But it's most common in electrical engineering to show the relationship between potential difference (voltage) and current, in a signal.

[@Kiyohay1]

Ugh, arrows in both directions on the imaginary axis hurts my eyes

[@siofratowe6810]

what is e D:

[@Nikifuj908]

Yeah, 2 pi / 3 radians is just 1/3 of the way around the circle, 4 pi / 3 radians is 2/3 around, and so on.

[@ZedMusic]

what if it was x^3+1?

[@jonangarcia5229]

How 'bout sin(e^i)?

[@gatlatwal8499]

Please help me calculate this Given that (√3-i) is a square root of the equation Z^9+16(1+i)z^3+a+ib=0 What is the value of a and b?

[@bridgeunwort61]

what if the exponent isnt a whole real number? how would you come up with i , e, 3/2 solutions if it was x^i=1, x^e=1, x^3/2=1

[@kyrilcouda]

Can I somehow use this to factor the equation into a multiplication of smaller polynomials?

[@dogeness]

Can’t believe I was never taught this before... I’m taking differential equations lol...

[@randeshdasanayaka257]

hey how did u find the angle, hence the degree from 4pie/3 and from the other 2 hence what happened to the "i"

[@Jemiiiimmaaa]

bless u

[@ashw730]

when do you apply 2Pik??

[@B900]

Amazing

[@YuckFoutube85]

where is the demoivre's theorem video

[@orvinal2883]

no replies in 6 years

[@wreynolds1995]

Why are you telling him to shut up? You're giving him the money so that he can speak! xD

[@NSAwatchesME]

thanks but i dont see where the -1/2 comes from :S

[@DrMudarris]

it's the value of x coordinate in the complex plane

[@dragonmemories]

Im doing the DeMoivre's theorem, Is this it? I lacked behind in class and got out off place. I have a test tommorow and Im trying to do my h/w (which I have no Idea how to do) and study for a test all in one day. Is this the DeMoivre's theorem?If not where can I find it? -sorry im rushing

[@callumgilfedder9097]

Where the hell did the x2=cos(2pi/3)+isin(2pi/3) at 8:51 come from???

[@Jemiiiimmaaa]

Euler's formula: I can't prove it for you however whenever you see an exponential of the form e^(ix) it can be written as cos(x) + i (sin x) (There are several websites online that have the rulers formulas proof!) In this context, he's trying to find the roots of a complex number.

[@ZedMusic]

what if it was x^3+1?

[@jackthompson3048]

does e refer to Euler's number in this case?

[@williamwright4813]

Yes.

[@davidkim1340]

so good/bad middle

[@evangraham1180]

i such likes fisics

[@keenankharrat9769]

I don't even care

[@Gutesmorgen]

How can Khan say its √3/2 ? Plz HELP ME!

[@NSAwatchesME]

@ 9:35 how does he get -1/2 + sqrt(3)/2 ??

[@harshita1639]

It's the value of cos 120°[90+30] sin120°[90+30] I suggest u see trigo first if u still find it confusing 💜

[@rgoldschmid]

Yeah, you should probably apologize for how annoying your voice is in the free math videos you make in your spare time for the benefit of random students you'll never see or hear from

[@xScuzzleButtx]

Why are complex roots important?

[@orvinal2883]

shoulda used tau

[@JANJAN-vw3km]

SHUT UP

[@duhboss1]

Your videos are pretty helpful, but it's really annoying when you repeat yourself. Seriously, you say a lot of your sentences, or sentence fragments, at least two or three times before you move on.

[@vbhaskar68]

pm

[@rithviknama]

i love you my man but that didnt help at all

[@astolfo9377]

OH GOD 3 SECONDS IN IM BORED ALREADY

[@richardaversa7128]

Degrees... smh.

[@vanessaquevedo6851]

He may stutter but he is sure is a lot smarter than you are : )

[@jamesdunworth-crompton3259]

just shut up and take my money

[@pragyarajput8703]

lol wats dat.... u'r too slow.. umm i guess each student(indian) will feel d same,,, omg haaah

[@972727277]

Awesome

[@vbhaskar68]

pm