Basic complex analysis | Imaginary and complex numbers | Precalculus | Khan Academy

[@Visualizingmathandphysics]

i = rotation by 90 deg. i x i = rotation by 180 deg. i x i x i = rotation by 270 deg. Complex numbers are used to indicate rotation. Hope this helps. Binnoy Visualizing Maths A book beyond formulas Dedicated to you and all such similar minds searching for answers.

[@atefrod680]

I discoverd this myself when I was struggling what i^23 for example is. Just divide it by 4 and use the last decimales to.determing what it should be. Here that would be 0,75, which means i^3 and that is -i

[@thomasolson7447]

Want an algebraic vector to play with? [(x-1)/(x+3), (x+2-i*x)/(x+3), (x+2+i*x)/(x+3)] The distance is 1.

[@HellaPerformance]

No because if you think about the Pythagorean Theorem: c^2 = a^2 + b^2 Well, we are looking for the magnitude of the complex number, or in other words it's radius. r^2 = a^2 + b^2 Therefore r = Sqrt( a^2 + b^2 ) or |z| = Sqrt( a^2 + b^2 ) |z| = Sqrt( 3/4 + 1/4 )

[@Krabby_Patty]

I dont know how to put this, but the timing of this video is *spot* on, just before my terminal exams. Thanks!

[@HellaPerformance]

Sqrt( 1 ) = + or - 1.. The reason he chooses one in this scenario though is because he is looking for the length of the modulus. As Klatjofsky put it, lengths are always positive.

[@patrickoneillnl1966]

I have watched several of your videos, several of your videos and they are wonderful, they are wonderful.

[@mouadovic04]

this is so easy ,we want real analysis complex

[@Motivationtosuccess-qs7ih]

Thank you sir, now I'm ready for my quiz on complex numbers 💯

[@anonymousL251]

Thank you so much! I don't know how many times you've heard this but you've seriously helped us. :D

[@Visualizingmathandphysics]

Complex numbers indicate rotation. In maths, the number '1' indicates 'as it is' The number 'i' indicates 'rotated by 90 deg' The number '-1' indicates 'rotated by 180 deg'. The number '-i' indicates 'rotated by 270 deg'. Binnoy Visualizing maths A book beyond formulas Dedicated to you and such similar minds searching for answers.

[@jackfaggie9477]

basic complex

[@debajyotibhattacharjee3180]

You are great SIR. Thank you for your online lesson.

[@salaheamean]

what an amazing video. really helped me a lot thank you and as a muslim May Allah guide you to his way. :P

[@fahmylina123]

Excellent explanation of the complex number. Thank you, indeed.

[@Klatjofsky]

Yes, but remember that he's trying to figure out the length of the position vector and lengths are always positive :)

[@thetempest0]

at 10:07, why did you square the denominator from 2 to 4?

[@MrYounis26]

Just to be in the safe side, this is done in Argand diagram right?

[@Visualizingmathandphysics]

Yes, you are right, complex numbers indicated 'rotation". The comments above might help you. Binnoy Visualizing maths A book beyond formulas. Dedicated to you and such similar minds searching for answers

[@Mrbenz-dr8zu]

complex thing simply done

[@visiting31]

i thought i^2 was -1? or are you taking the absolute value of it cause of the distance thing

[@genarovillarin8810]

9 years late and got stuck on this too but I think I figured it out. We're only gonna take the real part when solving for the magnitude

[@basia2330]

@@genarovillarin8810 Yes, Khan is calculating the magnitude by square rooting the squares of the real parts (a and b). The coefficient of i is b, and is the component used in the magnitude formula.

[@koushikghosh465]

Please start a series on COMPLEX ANALYSIS.

[@FreeMarketSwine]

What exactly is happening when e is raised to the power of an imaginary number?

[@missAyzl]

thanks so much for this video~!!! nice visual presentation and made a lot of things clearer :D

[@pri7974]

This helped me a lot.Thank you so much!!

[@Aurelien4131]

Really good video, this was my first introduction to the complex analysis and it makes me want to learn a lot more about it =)

[@disgracedmilo]

Still learning about complex numbers?

[@einsteinino]

@alawrence89 I understand your reasons. In fact, a long time ago I put to myself the same question. Well, it looks like the expression "imaginary part" WAS used to denote bi, so being consistent with the word "imaginary"; later on, the i was dropped - maybe with the axiomatization of complex numbers as the set R² of ordered pairs (a, b) of real numbers? - thus leaving with just the coefficient of the "imaginary number" - that's correct, instead - bi.

[@burnsyy9119]

I'm just learning this but i'm sure it has it's uses.Like a kid counting stones wouldn't have any use for fractions,or someone measuring the length of something won't have any use for negative numbers.At our level we probably won't know of any application for imaginary numbers but later we will.

[@sirfalcon1]

Thanks Sal, one would think this would have been covered in math classes at some point before partial differential equations.

[@Seedofwinter]

I appreciate your response. I will be sure to check your channel out.

[@ultratribalist]

@goldstein23 i is not squared but the coefficient of i is squared.in the first example which he deleted,the coefficient is 1 and in the next example the coefficient is 1/2.hope that helps.

[@aakashkarajgikar3935]

9:00 , 10:10 Wait, but I thought you can't distribute a square root to terms inside the radical when separated by addition or subtraction. What happened to that?

[@All-Videos]

Thank you sir! Could you please find the absolute value of this complex number: √3- √(2i )

[@Rararari1]

thank youuu

[@ekoi1995]

this can be applied in adding 2d vectors :)

[@thabsor]

why is a video of precalculus referencing a calculus video for complete understanding?

[@HotPepperLala]

Basic Complex Analysis...? Will you do the advanced ones?

[@ToNmAnAyO]

Argand Diagram Argand Diagram Argand Diagram

[@gauripradhan8510]

Thank you !

[@sebzim4500]

At 10:21 shouldn't that be - 1/4?

[@Menegoth]

This is so complex!

[@MiladD25]

Thanks!

[@Visualizingmathandphysics]

Yes, it is due to some reason counterclockwise. Binnoy Visualizing Maths Amazon

[@hoaque092]

it's very useful. thank you.^^

[@michaeldenero1728]

how did i power 2 become a positive number instead of a negative number

[@MrDhanesh2nair]

Yes how did that happen. Square of i is -1.

[@ElizaberthUndEugen]

yes, it is an error.

[@jphare1958]

No, it is not an error. You don't square the I. You only square the 1. We are trying to find the magnitude, and the value in the I plane is 1. The I is only there to indicate which part of the complex number is being defined.

[@davidcossa7970]

(i^2)=-1 not i=-1

[@axemurdererrl]

thanks so much, im taking a n extension course at uni where they assume u know this (idon;t XD) :( very clear much appreciated.

[@experthe9574]

just a correction, Euler is pronounced like "Oiler"

[@norwayte]

Good. Keep on going.

[@isabellenicholls6528]

Phi is not necessarily the arctan of b/a if the tan of phi equals b/a...

[@haveuseenme3990]

argand*

[@zoranhacker]

How come?

[@einsteinino]

@alawrence89 Nope, sorry. You are wrong. The "imaginary part of a complex number" is the real coefficient multiplying the imaginary unit i.

[@harrytuttle777]

This lecture should be labeled Complex Analysis for USAians. See Eurpopeans already know everything about all things. Only USAians (or Americans as we are sometimes misidentified) are ignorant about complex analysis. The only thing we could ever analyze are Big Macs and Cheeseburgers.

[@goldstein23]

Did Sal forget the negative signs when squaring i around 9:00 and 10:10? If he didn't then i'm quite confused.

[@erezklein5769]

Are negative numbers real numbers? And yes, by the diagram shown only 1/4 of the area is used. Replaying 13 years later.

[@einsteinino]

Another correction: the imaginary part of a complex number is just b, not bi.

[@Aranda-frtn]

muahhhhhhh Te AMO rrrrrready for my test,

[@isaacsechslingloff8008]

In the precalculus class you did not introduce these taylor series or "e" in terms of sin and cos yet.

[@nora_8080]

in the formula z=a+bi, can b=0 and if so, don't we gain a real number?

[@dannyboy12357]

@experthe i heard it pronounced either way, i like the Youler pronounciation my self

[@Ynno2]

lol are we supposed to have watched the calculus playlist before watching the precalculus playlist?

[@creatorsense2674]

why are u in versing the tan in 4:51

[@xloserlove123]

remember this is all counterclockwise

[@Vlaxirious]

sorry i have a question how cos something + i sin something related to exponential form?

[@ridium7721]

until now I thought a, b is in R (real number line)

[@vsenderov]

What software does he use to prepare the videos?

[@s.aslahahmadfaizi4687]

Can you please teach me how to write like that with mouse.

[@itsklol]

He probably uses a tablet

[@k333r]

It's a drawing tablet...

[@shoyoishida5901]

How did the sqrt of 3/2 became 3/4?

[@ABeardedDad]

if we didn't have 'e' the universe would implode.

[@SegaFanClub]

@Menegoth You're just imagining it

[@Adinzo95Studios]

Isnt the square root of 1, +1 or -1? Either way though, thanks

[@arvetis]

yooler

[@Jenae-yb3mv]

how have i watched this 4 times and still don't understand. Welp ig the perks of dyslexia and Dyscalculia yayyyyy :0

[@ahappyimago]

What part don’t you get

[@lionnick1849]

What is e here ? Is it equal to 2.71 ? Or is it another thing ? Who can tell it me

[@Dadarkxy]

No, that's YOU... (Do you just know what a "FUNCTION" is ??)

[@shinfu8190]

Assalamu Alaikum wa Rahmatullah. Sir Khan I didn't get the part how u converted under root 3/2 to 3/4 . plz guide me to a video of yours so I can revise this

[@LizDcruzzie]

he squared it. He used Pythagorus theorem to get the modulus of Z

[@shinfu8190]

+LizDcruzzie thank uuuuuuuu😊

[@ilyxb00]

My teacher told me to watch this guy. lol

[@soludsnak]

0:00

[@P0PH0USE]

"Yooler" is like fingernails on a blackboard for a mathematician.

[@ahappyimago]

It’s pronounced YOU-ler

[@rbmarik6082]

Ok so can anyone tell me what exactly is an argument of a complex number... I know how to find the argument and these stuffs BUT WHAT EXACTLY IS AN ARGUMENT?? Why is range always -pi to pi??

[@AriaHarmony]

Draw a line from the origin (0,0) to the complex number (ie, x+yi), the **angle** between the positive real axis and that line is what we call the argument of x+yi. Note: the argument is the angle θ theta itself and not a function of it like cos(θ)/sin(θ)/etc. The range -pi to pi is called the principal argument, technically you can keep rotating full circle and get infinitely many angles to represent the same complex number and they will all be correct. According to Wikipedia we restrict it like that when "a well-defined function is required" and "Some authors define the range of the principal value as being in the closed-open interval [0, 2π)". So looks like -pi to pi is just the common convention.

[@ghilmandilshad9495]

a small stupid question why we are not considering the ' i ' with the value of b in a.square + b.square

[@experthe9574]

@viewer280 I see that I made a mistake by using my other account to comment on this... but I assure you I'm the same person you sent the other comment to.

[@Pr0x1mo]

What i don't get is why someone thought of to initially raise the entire thing by e?

[@0726-o9b]

I can't believe Salman Khan pronounced Euler as "Yuu ler"

[@VndNvwYvvSvv]

He isn't really that good, especially when he repeats himself.. he repeats himself.. repeeeats.. him... self... seeellllf... repeats himself like this.. like... like this... thisssss.

[@jamesleem.d.7442]

(1/2 i) when squared gives (1/4 i squared) or MINUS 1/4 not 1/4 as you showed inside the square root sign

[@jamesleem.d.7442]

I retract my comment. I got it now: Our b value (2) is all that is squared here

[@ibbyizaG]

can't have a negative modulus.

[@Visualizingmathandphysics]

Just as there is east, north , west and south. In maths ,there is 1, i,-1 and -i. I have some videos over it on my channel. Binnoy Visualizing Maths A book beyond formulas, dedicated to you and all such similar minds searching for answers.

[@supertren]

@3809ault Perhaps you could do some videos like that, and put it together free in Internet without repeat yourself!.

[@HellaPerformance]

Because you're the one with the misunderstanding, not him.

[@Seedofwinter]

I don't understand the point of complex numbers. I wish some folks could describe some applications.

[@yuvraj7214]

Mark Taronji Its been 5 years, but still if this helps someone else...why did you learn real numbers, because you liked to know about it right?? That's the point of math, you just make up something a d work on that. Also there are many applications...some examples are transformation and quantum physics.

[@sayardutta1525]

they have applications in electrical and electronics and comm. engineering extensively too, since im from that field

[@I77AGIC]

excellent video! just try to stop repeating yourself repeating yourself so much

[@richardaversa7128]

I77AGIC that's actually how Khan is so successful, people being introduced to a complex topic for the first time tend to not really notice the repetition - it is require for effective e learning, and fills in the holes of what they muss while they are thinking and processing the new information.

[@flawns]

lol basic complex analysis. ... like saying this is simple hard math

[@GetUpAndWatchVideos]

The views to like Ratio....LOL

[@kevinclaypool6345]

I hated this lesson because nowhere do you teach how to use identities to solve where the radian or degree lies on the quadrants. "Using the identity 'tan(x+-theta)' , we know that the following is also a solution of the equation." That quote tells me nothing, regarding how to choose which identity to use.

[@Futacy]

My brain hurts.

[@Luxcium]

Futacy yeah I know because he said i^2 = 1 instead of -1

[@richardaversa7128]

oh god, Sal please tell me you're not pronouncing the great mathematician "yooler"...

[@thiebes]

Please stop mispronouncing Euler so I don't end up throwing a rock through my monitor.

[@SlyMaelstrom]

Avernus Inferno I don't know what the problem is... he posed a pretty simple statement. The commentator is pronouncing Euler as YOU-LER when it is, in fact, pronounced OI-LER.

[@janvermaak87]

I will now mispronounce Euler in your memory from this day forward just to spite you English-elitist jackoffs.

[@SlyMaelstrom]

Yatsi Kaamrev Pronunciation is so elitist.

[@TwistyTrav]

Stop turning a Math problem into an English problem. Lol

[@SirHiggalot]

But Euler is not even an English name.

[@Ghaiyst]

ah! now I understand why they taught us that graph in algebra 2 trig, even though im in calculus 2 now. God its important.

[@Ghaiyst]

so 1=e^itheta where ln1=i(pie/6) and 0=i(pie/6) in which case i is the square root of negative 1, 0 or hiding beyond logic's sense? yea I can imagine that.

[@Ghaiyst]

so wait is r=z?

[@mastersamnonk8361]

thank you so much..