Introduction to i and imaginary numbers | Imaginary and complex numbers | Precalculus | Khan Academy
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Introduction to i and imaginary numbers
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Precalculus on Khan Academy: You may think that precalculus is simply the course you take before calculus. You would be right, of course, but that definition doesn't mean anything unless you have some knowledge of what calculus is. Let's keep it simple, shall we? Calculus is a conceptual framework which provides systematic techniques for solving problems. These problems are appropriately applicable to analytic geometry and algebra. Therefore....precalculus gives you the background for the mathematical concepts, problems, issues and techniques that appear in calculus, including trigonometry, functions, complex numbers, vectors, matrices, and others. There you have it ladies and gentlemen....an introduction to precalculus!
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I better get an imaginary 100 on the test.
lol 69 likes
When u realize that i^100 is 1💀
If you just wait a little while you’ll have -100 real points.
What did you get?😂
If there are imaginary numbers I should be able to make an imaginary answer
I hope that's a joke.
Quachil Uttaus
I am Jack's fulfillment of seriousness.
+The Narrator HAHAHAHA
You win the internet today.
Lmao EXACTLY 😂😂 sometimes I feel like bullied smart people came up with all of these complicated formulas to get back at the world
khan academy= 5 minutes
teacher= 1+ hour(s)
preach, this dude actually made me like math
You spell it wrong its 10 y e a r s
@@JQBYedits You spell it wrong its 10 d e c a d e s
@@fhunnyprawidya2193You spell it wrong its 10 c e n t u r i e s
Complex numbers actually indicate rotation.
i = rotation by 90 degrees.
i^2 = rotation by 180 degrees = -1 = negative of that number.
.
i^3= rotation by 270 degrees = -i
i^4 = rotation by 360 degrees =1.
.
Take any object in your hand and rotate it by 90 degrees 4 times. You will see that it comes back to as it was before rotation.
There is a whole video on this on my channel if you would like.
With Love
Binnoy
I came here to learn and I'm feeling so attacked right now
+Kostakis VeganLove yes it's a meme
lol
lmao
I prefer quadratic equations now
i^4 could be more easily explained as i² * i², dont you think? -1 * -1 = 1 is obvious
That's how my pre-calculus professor taught me. It's a lot less confusing.
it's still easy
your right took me 5 mins to understand sal's explanation
how did you wrote the i^2 as i² ?
Could this mean that 1 is the only number with 2 or more square roots?
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.
my old math teacher in elementary had mutual friends with Salman Khan!
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.
+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.
"Aye times aye!" Sounds like Pirates math.
MG P WAS I GYM
'i' is just a way by which mathematicians could denote that the object is rotated or inclined. For example if a stick is kept horizontal its length say is '5'. Now if the same stick is kept vertical, can we measure it on the same number line? No right.
For this reason, mathematicians used complex numbers to just denote that the object is just rotated now, so they denoted it as 5i.
Hope this helps,
Binnoy
Visualizing Maths
A book beyond formulas.
i = rotation by 90 deg
-1 = rotation by 180 deg
-i = rotation by 270 deg.
-1 x i x i = rotation by (180+90+90) = rotation by 360 deg.
Also -i x i = rotation by (270 + 90) = rotation by 360 deg.
'1' means after a complete rotation, the object has now come back to start .
This happens at 0 deg, 360 deg, 720 deg, 1080 deg... so on.
Hope this helps
Binnoy
Visualizing maths
A book beyond the formulas
Why am i watching this at 3am I don't even go to school anymore
woah, i is crazy.
"i" agree that you are crazy.
Im waiting for the grammar police
@@meltagalicud1173 it has been 5 years since the original comment & no grammar police yet
josef wow hi :)
Cas Briar *am**
The numbers may be imaginary but my confusion is 100% real
thank you for having captions
Thank you for putting this up. Imaginary numbers always confused me in high school.
🤭do they still confuse you??
@@ringdingdong8377 why that emoji
@@cubingspeed7649 becuz it's been 11 years and people would think it's creepy and I'm stalking them so the emoji is there to make it fun and clarify I'm joking.
Lol I love the internet. Here we all are having multi-year conversations. The web is amazing.
@@ringdingdong8377 HAHAHAHAH
..........I stoped the video just in time before my head exploded
Carl Frederic Good job. 7 years later, this video is still exploding heads.
@@someweirdo 8 years later*
10 YEARS LATER
This was i-opening!
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?
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.
i'm in 7th grade and my teacher taught us about real numbers
i asked what a non-real number would look like and she told me "you'll learn that next year"
but curiosity killed me and i looked it up
my brain is turning inside out right now
Hol up you’re learning imaginary numbers in 8th grade? My teacher told me it’ll be in 10th grade! Can’t wait for it!
@@Firefly256 I’m learning in 9th grade :) good luck! ❤❤
@@Firefly256i learned them in 6th grade 😭
This stuff is so easy thanks to you khan...u simplify it
Lets say you are facing towards the East.
Now suddenly you shift by 90 degees and start facing the North.
You just term this position as "i" which indicates rotation by 90 degrees from the start point.
Now lets say again you rotate by 90 degree. You are now facing the west. You term this as -1 since your direction is negative as compared to the start point.Hence i*i = -1 which means rotating twice by 90 degrees makes you point in the opposite direction.
Binnoy
Visualizing Maths.
Oh khanacademy math man you are a god send
Thanks a million! I got it at least❤
Oh my goodness this is so cool!! Thank you Khan Academy
I am currently having a full on can't stop crying emotional break trying to study for my exams :,,,,,,(
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.
Oh my head🤕😷😢😭. I gotta make an A on a test to get on the deans list😭
How'd it go?
😳🥰
He's a genius.
ready for my imaginary A!!
thank you for everything😍
I’m in seventh grade and my mom expected me to know this, so here I am
Imagine a surface of water. Now if you hit this surface with one hand, the impact is (impact)^1.
If the surface is hit with two hands, the impact will be (impact)^2.
With three hands, the impact = (impact)^3.
But with no hands, there is no impact. So now the surface of water remains 'as it is'. In maths 'as it is' is denoted by the number '1'.
Hence (impact)^0=1.
Thus (anything)^0 actually denotes absence of that "anything", thus keeping the surroundings "as it is".
Binnoy
Visualizing maths.
Amazing dude, thanks
Oh you had no idea how much I needed this
I hope you know how you always save me. Thank you so much❤️
Perfect. Very helpful.
Thanks, at least i understood 1 or √4 things
i^4 is better described as (i^2) ^ 2. That way its : i^4 = -1 * -1 = 1
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)
I watch and still confused , i flunked geometry 4 times and almost didnt make it out of high school
This video teaches better than my in person teacher
Hi! I’m learning this in 12th grade lol idk why I’m here I just got done catching up my work and notes on this.
This isn't even on the SATs but I'm still studying it ha
It's on the ACT.
Thank for doing what Ms.larssen cant
i is my favorite number.
Thank you too much, I understand now!
I subscribed, you're very helpful! thanks!
It just keeps folding over itself, like a lasagna noodle; imaginary numbers are logic lasagna!
Thanks a lot much help was needed
Thank you for this video.
dont ever want to hear the 9th letter of the alphabet ever again
can I make up my own number?
w= the number of 0's you have to add to get 1.
0*infinity =/= w.
+Re Tend wat
w = 1/0
No since technically, "w=the number of 0's you have to add to get 1" is false.
Great. Now my eye is twitching.
So basically it’s like a stand in to a theorem ? Or a principal?
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.
It's kinda explained in the top comments you can just read those
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.
Have you ever heard of "Imaginary Time", Rachel?
Stephen Hawkins discovered it. It is a higher dimension of our universe, and we need to use imaginary numbers to calculate coordinates in it.
If you are in High school now, then at some time *in your lifetime* , we could be using imaginary numbers to do calculations to travel through what we now think of as "hyperspace".
Am I exaggerating? You tell me.
. . . And let's not forget someday when _your kids_ come home with _their_ Calculus homework, and need _you to help them_ understand it!
very interesting
Awesome.
how we can plot a graphical representation of "i" to the power "i" in complex plan ????
Thank you
Can you show me imaginary number representations with apples, oranges and bananas?
Hope you can think simple things deep.
1 imaginary apple
@@danieladowell5025 HAHAHAHA
I will never look at "I" the same ever again.
Thanks for that. I have no clue what Sal is doing when it comes to i4, he says -i x i, is positive 1, but then says it's the same as -1 x i x i. Where the does that last part come from?
For any number x, -1 * x = -x. So if you have -i * i, since -i is the same thing as -1 * i, that means -i * i is the same thing as (-1 * i) * i, or -1 * i * i.
I is actually very simple in engineering. You take pi 3.14 and you split a lollypop 369 degrees and count backwards hold your breath 2.7 secs and theres your answer.
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
so shouldn't this be listed as imaginary number? seeing as it is can be a single digit as well as multiple digits
Tysm
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
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.
ily who is reading
yo just found something pretty cool, 1/i = -i :o
What would happen if Kahn Academy used a white background for 1 video ..?
the world would collapse
T-THIS IS PRECAL? WELL IM IN ADV. ALGEBRA...EYE-
sAME SAME
are u dumb.. a huge chunk of precalc is literally review of alg 2 ofc this vid would be labeled under precalc ..
gay4vivi Well, I didn’t know that. That doesn’t automatically mean me or the commenter is dumb though, there’s no need to be rude.
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.
The expression was i*(-i). He got the (-1) factor from (-i) in the expression. so i*(-i)=(-1)*i*i
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).
So is it just an infinite loop or does it change randomly at a certain point?
The laws of math: "the square root of -1 cannot exist,"
other people: "lets just make it another number anyway"
It's pseudomath at this point. The moment they use "i", most mainstream science is pseudoscience.
There is only ONE real sqrt Matrix[-1].
It's rotation.
Matrix is the only way to sqrt(-1).
sqrt(
[-1 0]
[ 0 -1])
=
[ 0 1]
[-1 0]
Counterclockwise rotation, inductive.
[ 0 -1]
[ 1 0]
Clockwise rotation, capacitive.
@@lukiepoole9254 it can be useful for factoring some equations that can’t be which I think is it’s main purpose
@@youtubeuser3766 "i" is pseudomath. Anything that use "i" is pseudoscience such as quantum quackery.
Hello Dear.
how about if you do that,
i4=i2xi2= -1x-1=+1
and continue.
I feel so stupid cause everyone in the comments gets it but I just can’t
Im literally watching any videos on the subject before my test lmao
Me, myself and i
@khanacademy it's hard to understand the reason and applications of this - even after reading Imaginary_number#Applications_of_imaginary_numbers from wikipedia, i still can't understand - what for a so 'biased' stuff like this is, and how is it used on "signal processing, control theory, electromagnetism, fluid dynamics, quantum mechanics, cartography, and vibration analysis" - please explain how can this be used there, and why isn't this imaginary number so useless i think it is... - thanks!
why does the double negative rule in the exponent 4 not apply in the 7th exponent equation?
0:04 ROMAN NUMERALS
√-1=?
My teacher: That's illegal.
Me, an intellectual: *i*
What is the name of the software u using
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.
I, Giorno Giovanna, will never use this after college
Hi, contact me on my WhatsApp
+91 9167921556 for any kind of help like questions solutions etc in any of the subjects.
Try this, it's really helpful.
Immediately I was lost as he said I am introducing you to the number i
I found a pattern where every odd or even exponent alternatively switch from positive to negative. i^2 goes negative, i^4 goes positive, i^6 goes negative and so on. i^1 goes positive, i^3 goes negative, i^5 goes positive and so on.
Hi, contact me on my WhatsApp
+91 9167921556 for any kind of help like questions solutions etc in any of the subjects.
Try this, it's really helpful.
Square root of negative numbers is error
oh god i have a test tomorrow and i'm doomed
عاش فشخ يالا
What about fractional powers? :)
That’s basically square roots
What work fields needs this
Computing.
Physics, astronomy, and engineering. And Mathematics, of course.
Diana Pham Fractal geometry :) Of course that is not a work field, but fun to mention ;p
Music major here. In any context as a musician, we don't need this. Ever. #prereqssuck
Are you sure... ? Let's not forget someday when your kids come home with their Pre-Calculus homework, and need you to help them understand it!
Why do you have to multiply each one with ' i ' ? Like on the ' i ' to the 3rd power example
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
thats kinda the whole point of imaganary numbers
If x^2 = -1, It is i.
But what about sqrt(x) = -1?
There's no solution
Also watch complex numbers made simple th-cam.com/video/P-jW3Cd3yn4/w-d-xo.html