Introduction to i and imaginary numbers
Practice this lesson yourself on KhanAcademy.org right now:
https://www.khanacademy.org/math/precalculus/imaginary_complex_precalc/i_precalc/e/imaginary_unit_powers?utm_source=YT&utm_medium=Desc&utm_campaign=Precalculus
Watch the next lesson: https://www.khanacademy.org/math/precalculus/imaginary_complex_precalc/i_precalc/v/calculating-i-raised-to-arbitrary-exponents?utm_source=YT&utm_medium=Desc&utm_campaign=Precalculus
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https://www.khanacademy.org/math/precalculus/parametric_equations/polar_coor/v/polar-coordinates-3?utm_source=YT&utm_medium=Desc&utm_campaign=Precalculus
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Text Comments (241)
## Would you like to comment?

Primal.G.Shepard (13 days ago)

million views thats a surprise

Aaron Levy (14 days ago)

Music major here. In any context as a musician, we don't need this. Ever. #prereqssuck

Caroline Williams (29 days ago)

10th

George Sealy (1 month ago)

Millennials should have no problem with i.

gazelle k (2 months ago)

Oh you had no idea how much I needed this

Rachel Holladay (2 months ago)

I've learned this three times over in the past two years, and I just sob every time I try. I can't even learn it, I just get a basic understanding enough to put the answers on the paper, and then I let it go. I can't think of a single occupation in which I will ever use this, and there are plenty of other brain stimulating and strengthening things that I can do to build cognitive pathways. I want somebody, anybody to argue why this is a REQUIRED piece of knowledge that I can't graduate without, and why I should suffer the anxiety and stress of trying to understand something that has no applicable value.

ilikepizza (1 month ago)

I major in math, but I agree with you to a large degree. When you work a job in our society, and the only math you should apply in life is taxes and paying bills, why would you EVER need to know calc or properties of i if said career isn't catered toward a math inclined job? I believe that mathematics courses should stop at 6th grade, unless of course you have a pursuit in math or are gifted. I agree completely. Mathematics isn't necessary in every career.

Replay Arena (2 months ago)

It's kinda explained in the top comments you can just read those

nandhi suguna (3 months ago)

i^18-3i^7+i^2(1+i^4)(-i)^26 solve it and give the reply

Dashock Pixle (3 months ago)

Thank for doing what Ms.larssen cant

andrien cuellar (3 months ago)

what about i to the 67th power

diamondlancer (1 month ago)

It's -i: i^67 = i^64 * i^3 = (i^4)^16 * i^3 = 1^16 * i^3 = 1 * i^3 = -i

Pablo Rodriguez (4 months ago)

You’re the goat thanks!

HiddenTalent (4 months ago)

wow.
*dials up khan academy*
Me: Excuse me, may I speak to Mr. Khan? Yes, I'm a student and I'd like o request
that he come as a guest speaker to my pre-calculous class. Why? Oh, because he's
ten times better than any math teacher at my school.

Ervin Perez (5 months ago)

Im a 10th grader and learning this smh

Horace Dennis (3 months ago)

Ervin Perez you make your parent dissapointed

Ethan Means (4 months ago)

boi im in 8th grade so calm yourself

Lutetium Luni (5 months ago)

When imaginary units introduce themselves
"i am i"

Keith B (6 months ago)

Me, myself and i

Joe Savath (6 months ago)

i to the 4 is = 1? but i to the 7 is equal to -1??
i3 = -i times that by i = 1? not -1
i 6 time i = -i time i = -i that make sense

Mesbahose Salekeen (8 months ago)

hey sal why not explain complex variable/function by digging a bit deeper. there are other stuff like branch point, branch cut, rieman surface........ all those great stuff that i dont have a complete grasp.

ohh (9 months ago)

Oh my head🤕😷😢😭. I gotta make an A on a test to get on the deans list😭

ryan holland (9 months ago)

thank you for having captions

abubakar ishtiaq (10 months ago)

Theirs a better and easier way to understand this. It always repeats the same number over and over. It goes i,-1,-i,1. It doesn't change so i^1 would be i. Then i^2 would be -1. i^3 is -i. Finally i^4 is 1. Then i^5 Or ^6 ot ^7 or ^8 just repeats agian in that order and it keeps going forever

The Dark_Speed_Ninja/ShadikkuX (11 months ago)

If x^2 = -1, It is i.
But what about sqrt(x) = -1?

Philipp Hoehn (10 months ago)

There's no solution

ishan bhardwaj (11 months ago)

But nothing to the 2nd power is negative??????
i^2=-1

Shaine MacDonald (11 months ago)

I'm confused at the 3:44 mark. How did you get (-1)? I can see the rest but not that part. I thought i^1=i. Not negative one.

Itsoo1 (11 months ago)

The expression was i*(-i). He got the (-1) factor from (-i) in the expression. so i*(-i)=(-1)*i*i

LPSpineapple inc (11 months ago)

I am currently having a full on can't stop crying emotional break trying to study for my exams :,,,,,,(

Afrah Syeda (5 months ago)

literally what happened to me also last year in exams- when i m really stressed- and have a lot of tests and quizzes i start to get so overwhelmed. And final exams- alwayYs make me get exhausted and burn out. But seeing how you put this comment six months ago- how did u do? Did U do well. And did all your hard work pay off.

I eat bowls of cigarettes for breakfast (1 year ago)

Can we please just call the imaginary number nil.
Well.. that'd be a problem... but it's something similar to "Idk if I am true or false" (i)
So, why does ixi = -1? What if the imaginary number is made known, then it wouldn't be i, it wouldn't be -1 if timesd itself..
so what purpose would the -1 answer if you don't need a number that doesn't exist why does it have a value??????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????whoa ... that gltch where the comment box is pushed all the way to right of screen haepfsna hagain .,..waaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa

Maybelin Herrera (1 year ago)

Thank you too much, I understand now!

Norville Rogers (1 year ago)

i did i to the power of i in my graphing calculator and tried to graph it and the graph just stayed blank

Afzal Sayed (1 year ago)

Also watch complex numbers made simple https://youtu.be/P-jW3Cd3yn4

Jimmy Tremblay (1 year ago)

And what about i to the i?!

AngelFire550 (1 year ago)

I prefer quadratic equations now

Veronica Henley (1 year ago)

Did mines different got the same thing

FLAMYFREDDY (1 year ago)

Does this work with every square root and not just 2

MakParis Supreme (1 year ago)

when am i ever gonna use this in life

Ampwich (1 year ago)

So i times itself twice is -1, while i times itself 0 times is 1? HUH

Ampwich (1 year ago)

WHAT IN THE WORLD. Who thought imaginary numbers were a good idea?? lol WHAT ANSWER WOULD THERE BE FOR THESE??

b o n e l e s s s h r i m p (1 year ago)

aye

SirBanana (1 year ago)

I'm here because of MAP testing.

Marianne Koech (1 year ago)

Thank you

Angel Lomagdong (1 year ago)

what if its on fraction like 1/i?

Damiyuhh (1 year ago)

i * i * i = -i? Seems legit

Xx_ShADoW-FALLzz (1 year ago)

So confusing .-.

MEMEBOY ADVANCE (1 year ago)

i don't understand.

Aksangel _ (1 year ago)

I doesn't even sound like a sound anymore

SNoCappidona (1 year ago)

yea but if i = the square root of negative 1, and i squared = -1, then what does -i = ?

Mohna Khan (1 year ago)

how we can plot a graphical representation of "i" to the power "i" in complex plan ????

amara shah (1 year ago)

This isn't even on the SATs but I'm still studying it ha

CaptHayfever (1 month ago)

It's on the ACT.

Sheridan Milestone (1 year ago)

love this

The Processionist (1 year ago)

These videos help me a lot. thank you. I do have a question, on this one.
What if the exponent is a real number? What happens between the integer exponents?

diamondlancer (1 month ago)

You can visualize these numbers on an xy-plane. Say the x-axis represents real numbers and the y-axis represents these imaginary numbers.
i^0 = 1 is the point (1,0) on your xy-plane
i^1 = i is the point (0,1) on your xy-plane (since the y-axis measures these imaginary numbers)
i^2 = -1 is the point (-1,0), and
i^3 = -i is the point (0,-1), and
i^4 = 1 gets us back to (1,0).
Drawing the circle that connects these dots will help you (or someone else reading) understand what goes on between integer exponents.
For instance, i^(0.5) would be on your circle at roughly the point (0.707, 0.707)--geometrically this would get you at a 45 degree angle from the x-axis on your circle.
So i^(0.5) is roughly 0.707 + 0.707i.
Likewise i^(1.5) is roughly -0.707 + 0.707i, and so on.

Green Alien (1 year ago)

Is this supposed to have sound? Because it doesn't.

Kenney (2 years ago)

I just learned about i two days ago in my algebra class, but am still wondering why it exists. My instructor answered my question with "It's just something we memorize" Can someone explain it to me please?

Johnny Martinez (1 year ago)

Kenney no

rahul (2 years ago)

hi sir i wana ask
what is arbitrary number and arbitrary 0. sir response as soon as possible i am completely confuse about this.

damislav (2 years ago)

what the fuck?

Thomas Cowley (2 years ago)

I'm guessing you'll use base 4 to find the value of i to the large number.

Mira Love (2 years ago)

The oscar goes to......

Luis Hernandez (2 years ago)

khan academy= 5 minutes
teacher= 1+ hour(s)

RIGENSTEIN (1 year ago)

preach, this dude actually made me like math

Justin Wheeler (2 years ago)

I'm glad to see you explain this in such an approachable way. Surely this topic invites some seriously complicated topics but I feel like you explain it in a way that anyone could understand. Does anyone else feel like this needs to be taught to kids much earlier so that they have more time to sort through the topic and come to terms with it? I tell my younger relatives all the time that they need to be taking calculus in high school, but admittedly I never mention that they need to understand this topic as well. I hope in the coming years we see some real attention given to such an important topic for budding engineers.

Jon (2 years ago)

+Justin Wheeler Definitely. Phrases like imaginary units and complex numbers are quite intimidating to a layman. Covering them in school even at a basic level might encourage more kids to carry on to study it further.

NUKE (2 years ago)

+Khan Academy Why do you start your lecture about i without introducing the REASON we need i . and that is to rotate points in 2 dimensional space. That's what i is. i is a VERY ELEGANT way to rotate in 2d space. How? You have to be introduced into complex numbers to learn that. That's what i is all about... Nothing spooky, nothing "imaginary".
The whole "imaginary" bullshit is totally wrong. This "number" (which is not even a number) is a sighn (like a road sign) that points 90 degrees in angle and its purpose is to point you 90 degrees into the 2nd dimension.
so 5+4i means... move 5 units in the right direction and 4 units in the upwards direction.
That's what i is... upward directional movement. But because we use the same numbers (5,4) we have to make a way to represent this movement.
So dear Khan Academy, don't confuse people with this imaginary bullshit. Introduce them properly and we will be glad
(and you know what is the saddest thing? If you go to wikipedia page, and search for i. IT DOESN'T TELL YOU THAT! it only tells you all these bullshit, but it doesn't tell you the real purpose of i... why?)

Bin Wei (1 year ago)

Thank you sooooooo much. I hate those bullshit explanations.You are the best, you must make a Youtube video about this because there are so many confusions among people.

Evan Wang (1 year ago)

What is I? (existential crisis deepens)

Jon (2 years ago)

+Quachil Uttaus It's one thing to understand how to work with i, it's another to understand why you work with i.
Perfect analogy.
Differentiation.
Teachers often emphasise the "quick" way to derive a term (nX^n-1). But they don't get taught why or how this is. There's barely any mention of first principles or the real world application of differentiation. It can get in the way of the real understanding of math.
Sure you don't need to understand to pass your High School Calculus exam, but when you get to college, the understanding is going to help you a lot.

Quachil Uttaus (2 years ago)

+CornerrecordZ We can't percieve i. But you don't need to percieve it. You just need to apply rules of algebra to it and it'll all be fine. Why were you so hung up on what i is? Of course your shitty teachers didn't help, but I'm honestly not curious at all about perceiving i.
Peace out, nigga.

NUKE (2 years ago)

1st my comment is a mess, I apologise for that :/ but anyway, I believe that students need to know what i is, or at least how it can be used, where it is usefull to us. Otherwise they will get confused. I can still remember the confusion that I had when I was introduced in imaginary and complex numbers back in highschool because I was looking for answeres and nobody seem to have them. All I was getting was, " look buddy, this is what it is and if you dont like it, we dont care" basicaly...
students need to be introduced to imaginary numbers in a way that they understand what they are. And if my comment doesnt cover what imaginary numbers then this video should have covered that in the 1st place...

John Miles (3 years ago)

dose this system only work in the decimal system? would it look different in the dozenal system? and if so, what about any number of number systems
I'm really curious but not smart enough to work it out myself

Krebs (3 years ago)

i works in any number system, as it is the imaginary equivalent of 1, and 1 is in every base system (except unary, but is unary really even a thing?)

Zero Point (3 years ago)

Duh!!!!!!!!!!!!!!!!

Cara Lee (3 years ago)

why does the double negative rule in the exponent 4 not apply in the 7th exponent equation?

sirawesome hat (3 years ago)

Why is i special? Isn't this the same for any square root of a negative number?

Unicorn Slayer (3 years ago)

Yeah, I know. I just had a dumb moment.

sirawesome hat (3 years ago)

+Krystal Colinco
No, the number.

Unicorn Slayer (3 years ago)

+sirawesome hat Reading the first sentence of your comment, I thought you were using incorrect grammar. "Why is i special?" Sort of a dumb moment.

aden li (3 years ago)

what grade is this usually taught in?

Harmony (5 months ago)

11th grade for me

ツSupersaine (5 months ago)

10th ish

Doctor Luzbubs (8 months ago)

i grade

Kyle Yee (8 months ago)

5th

Dusty Mack (9 months ago)

We are currently doing real numbers but my teacher went off lesone so we are kinda learning this now for the week

Rose B (3 years ago)

what is the purpose of imaginary numbers?

Hoo Dini (3 years ago)

+ShowBox Bacon Not really solve. We just substitute sqrt(-1) with i

ShowBox Bacon (3 years ago)

+Rose B i think the answer is: it allows us to solve negative square roots,

Blake Chreene (3 years ago)

Cell phones

Anastasia Penklis (3 years ago)

Awesome.

Re Trend (3 years ago)

can I make up my own number?
w= the number of 0's you have to add to get 1.
0*infinity =/= w.

Matheus Schuindt (2 years ago)

w = 1/0

Hoo Dini (3 years ago)

+Re Tend wat

Kierra Freeman (3 years ago)

Why do you have to multiply each one with i ?

Kierra Freeman (3 years ago)

Why do you have to multiply each one with ' i ' ? Like on the ' i ' to the 3rd power example

John Smith (3 years ago)

x to the power of n is basically x to the power of (n-1) times x. So 2^4 = 2^3 * 2. Same with i: i^3 = i^2 * i

ethan sauer (4 years ago)

Thanks a lot much help was needed

prettytwistedd (4 years ago)

I came here to learn and I'm feeling so attacked right now

vivian mamboleo (8 months ago)

lol

prettytwistedd (3 years ago)

+Kostakis VeganLove yes it's a meme

Mais Computers (4 years ago)

This makes no sense. Dumbest thing I've ever seen. Why not just tell us what the number is? Or at lest tell us what #'s it's in between? If you can't, mabe it's all just in your head; intellectual masturbation.....

Ryan McCormack (3 years ago)

+TheShockerKnocker Why do you feel the urge to say that? You are probably some distressed school student who is in confusion, and, instead of learning, blames his failure on the math itself.

Sam Chapman (4 years ago)

The imaginary unit is used to solve for imaginary zeroes in polynomial functions. It is very useful in engineering, astrophysics, computing, and physics. You only discredit it because of its name, "Imaginary." It makes perfect sense, and it applies to much of maths, like in Euler's Formula. Just because you don't understand it, that doesn't make it the dumbest thing ever. It just shows you inability to understand.

Derpetron3000 (4 years ago)

What is i^i?

Derpetron3000 (4 years ago)

Wow, thanks!

Sam Chapman (4 years ago)

+Sultan Khalid Wrong. i^i = e^(-pi/2), or approximately 0.208

Suly - Clash & Other Games (4 years ago)

-1

chubby walruss (4 years ago)

I hate this stuff who cares about a number that may or may not exist ;_;

CaptHayfever (1 month ago)

I, too, hate when I'm forced to watch a YouTube video about things I'm not interested in. Oh wait, that never happens; I can just not watch the video if I don't care about its subject.

Sana Ullah. (8 months ago)

qpeiitskfhcb

MrChipelo (1 year ago)

Nubtom if they can be applied to help us, then I am on board with learning it, but if it's not philosophical then it's not for me

Perpetual Onion (1 year ago)

I just like this stuff because it's where math gets almost philosophical

Dyl Pikkle (1 year ago)

Nubtom I'll keep that in mind when i need to buy groceries at the market or fill up my car

The Narrator (4 years ago)

If there are imaginary numbers I should be able to make an imaginary answer

Sashimi is bad (6 days ago)

The Narrator lol that’s funny

Sana Ullah. (8 months ago)

wtiogl

Tremaine Dawson (1 year ago)

Lmao EXACTLY 😂😂 sometimes I feel like bullied smart people came up with all of these complicated formulas to get back at the world

Morrocco Mole (1 year ago)

You win the internet today.

Forever monster (2 years ago)

+The Narrator HAHAHAHA

Seneca (4 years ago)

why does -1^2 = -1 then? My calculator produced that, an i'm not sure if that's right. Does anyone know why this is?

Insta Moue (4 years ago)

+Shane H Calculator reads it as -(1^2)*

Shane H (4 years ago)

I think its because exponents come before subtraction on the of the order of operations. You have to write it as (-1)^2, otherwise the calculator will read it as 0-1^2.

thadan99 (4 years ago)

i just shat.
But I get it

V8 monster (4 years ago)

big like

SandwhichBread (4 years ago)

i is my favorite number.

Zaya2499 (4 years ago)

What work fields needs this

Lyubomir Stoychev (4 years ago)

+Diana Pham Fractal geometry :) Of course that is not a work field, but fun to mention ;p

Diana Pham (4 years ago)

Physics, astronomy, and engineering. And Mathematics, of course.

SandwhichBread (4 years ago)

Computing.

Tony Tomov (4 years ago)

anyone know the difference between numbers and operations on numbers?
definition of i contrary to the laws of logic. it is stupid that set of number and
operation on this number, is equivalent to number.
by the same logic we can define j^2=-2 and ect. then the life of "scientists" will become much more difficult. Who has defined this nonsense? - nobody knows

[ IFDIFGIF ] (4 years ago)

j^666=-2562478560/i aahecke afs u thfW2

Cas Briar (4 years ago)

woah, i is crazy.

Evan Wang (1 year ago)

"i" agree that you are crazy.

AlchemistOfNirnroot (4 years ago)

Has Khan Academy done a video on finding both values of X using the quadratic equation where x=-b^2+-(sqrt) if a negative number/2a? Example x=(-5( sqrt -50))/(2*10).

LiberatedMind (4 years ago)

i^4 is better described as (i^2) ^ 2. That way its : i^4 = -1 * -1 = 1

jcoronet2000 (4 years ago)

Is there a number that equals (the square root of -2)? or the square root of all negative numbers?

Ferran Manyà (4 years ago)

sqrt(-2) = sqrt(2)*i

Sevi (4 years ago)

i^4 could be more easily explained as i² * i², dont you think? -1 * -1 = 1 is obvious

Ferran Manyà (4 years ago)

it's still easy

David Moran (4 years ago)

That's how my pre-calculus professor taught me. It's a lot less confusing.

zonne vlam (5 years ago)

+4girlsmomjd Ah just shut it you clam, no one gives the slightest fuck about you or what you think. have a nice day

Evan Wang (1 year ago)

lol u dont even know how to reply to a comment.

love (5 years ago)

Amazing dude, thanks

MG P (5 years ago)

"Aye times aye!" Sounds like Pirates math.

Bjo Bjoic (5 years ago)

You shouldn't try to think of i as some arcane value that has weird properties. Just like how negatives are just flipped positives, imaginarys are rotated reals.

Richie Whalen (5 years ago)

mrs lambert brought me here

ANWAAR BUKHARI (5 years ago)

Hello Dear.
how about if you do that,
i4=i2xi2= -1x-1=+1
and continue.

Richyrich66 (5 years ago)

I subscribed, you're very helpful! thanks!

Brycen Wershing (5 years ago)

Which if you find the angle of theta in the complex plane, theta would be 135 degrees, but a complex number is WAAAAAAAAAAAAAAAAAAYYYYYYY more useful

Brycen Wershing (5 years ago)

that would be i^3 times i^(1/2) or -i times (1+i)/sqrt(2), which simplifies to (i-1)/sqrt(2), or in correct complex form -1/sqrt(2) + i/sqrt(2)

F22Acura (5 years ago)

What if a positive number by a positive number is a negative. Imagine all the damage that would be done.

InfiniteUniverse88 (5 years ago)

Something about this just seems wrong. I understand that this concept is applied to electricity. I guess the logic is that electrons can't create matter i.e you can't get more than 1 or -1. Yet, didn't the Large Hadron Collider disprove this? A vacuum can create matter, because energy is required to create a vacuum.

Painx161 (5 years ago)

Ugh im such a loser, i dont get this, but im not giving up!
although im sorry this video still makes no sense i may need to refer to other resources, i feel left behind -.-
but like i said, Im not giving up

ohioSPIRIT (5 years ago)

Perfect. Very helpful.

Matthew Tesfatsion (5 years ago)

Thank you for this video.

Alexandria n (5 years ago)

Oh my goodness this is so cool!! Thank you Khan Academy

BINNOY PANICKER (5 years ago)

If you could just give an example of a fractional power. Is it like i^(3/2)?
Okay that means first imagine a stick rotated 3 times by 90 degrees. 90 *3 = 270. Now imagine the stick stopped at midway of this 270 degree rotation. 270/2= 135. Thus i^(3/2) = rotation by 135 degrees.

Antony Tolstunov (5 years ago)

This stuff is so easy thanks to you khan...u simplify it

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