Imaginary numbers aren't imaginary

[@Dialga9187]

Just a dude explaining maths, exactly as the internet was intended to be used.

[@wanderson4360]

this Internet video lacked the presencee of more cats. Other than that, great video! XD

[@enervoncee5327]

​@@wanderson4360what do u mean? i dont understand. 😊

[@juliavixen176]

The use of the word "Complex" is as used in "Shopping Complex" or "Apartment Complex". It's two or more things glued together. Think of it like "Number Complex" rather than "Complex Number"... because that's what it is; numbers glued together.

[@mlab3051]

maybe composit number make sense?

[@___Truth___]

@@mlab3051 Seems bad naming convention strikes again because there already exists Composite numbers. The only thing is they aren’t really “composite” since composite stuff retain the multiple/different stuff that makes it while “composite numbers” work with the exact same quantities (natural numbers) that makes it so it works with the exact same stuff. Complex numbers are actually better suited to be called Composite since besides the extension from integers by use of -1 they retain the multiple/different stuff that makes it (integers/reals & imaginary numbers).

[@___Truth___]

I NOMINATE, that complex numbers be called COMPOUND NUMBERS! Because it’s “a quantity expressed in terms of more than one unit or denomination”

[@mlab3051]

@@___Truth___ OMG now the naming problem occurs to us all beside programmer.

[@vickyk1861]

In Greek they're called μιγαδικοί which means hybrid numbers 😮😂 I think it's better

[@Premgod-zu5jp]

I think another video on imaginary numbers as a follow up would be amazing

[@rakshitgupta2660]

Agreed

[@reginald8102]

i agree

[@محمدحازمعبدالرضا35]

I agree

[@orlandot6]

agreed

[@lnribeiro1]

Bring it on

[@HenrikMyrhaug]

YES, I have wondered for SO LONG why teachers don't teach negatives as being a (180°) _rotation_ to the opposite direction, and why i isn't then also tought as a rotation half as far (90°) as a negative rotation. It makes it MUCH easier for kids to understand why positive•negative=negative and negative•negative=positive, if only you teach them that a negative is a "flip" 180°, to the opposite direction, and so two negatives multiplied ends up flipping twice; once to negative and once more back to positive. i is an extention of the concept, by imagining half a "flip" as a 90° rotation. Because two 90° rotations in the same direction total to 180°, we can say that i•i=-1. When you see i as a rotation, the whole field of complex numbers becomes so much easier to understand, and the ways we use complex numbers to describe rotations just becomes intuitive. If you make a follow up video, you should show how a linear combintation of real and immaginary numbers can form a complex number describing a rotation of _any_ angle. You should show complex numbers graphically as a vector, and show graphically how multiplying two complex numbers together necessarily adds the rotations of the vectors. This was how I learned to intuitively understand why complex numbers are used to model rotations.

[@AlexUsername]

It's not explained that way, because a "180 degree rotation" makes no sense in terms on 1 dimensional numbers. It's still just an arbitrary "flip" rule, if you're not explaining complex numbers along with it, as much as "multiplying by negative, equals negative" is an arbitrary rule. And no, teaches will not be explaining complex numbers to children in 3rd grade.

[@HenrikMyrhaug]

@@AlexUsername I feel like I kind of got my language mixed up here. When teaching complex numbers to high schoolers/ university students, the teacher should point out that a negative can be interpreted as a or a 180deg rotation, meaning the number is in the opposite direction. This sets up an intuitive way to understand i as half a 180deg rotation. When teaching negative numbers in lower grades however, of course you aren't going to teach about complex numbers or rotations. Teaching the concept of a negative simply as a "flip" to the opposite direction is absolutely good enough.

[@toby9999]

​@HenrikMyrhaug It is taught that way. That's how it was explained when I was doing my maths degree in the 1990s, but mostly in terms of the complex plane. It's less usefull conceptually on a number line. So -1 can be considered 180 rotation and i 90, but on a number line, any rotation that is not a multiple of 180 is rather meaningless.

[@shreyanshthakur5754]

when teaching complex geometry and rotation they do teach that (I am a high school student)

[@ttt69420]

Probably because graphs are an irrelevant visual representation of what is actually being modeled. They do more harm than good imo.

[@11anonymous6]

My bank balance is an imaginary number…

[@LeyScar]

If you were to put your cash into a square, you'd be in debt?

[@thepattersons2031]

As is my savings

[@RehanKhan-Peace]

That would mean you don't owe to bank and neither they have to pay you but still the money is flowing. How is that possible?

[@11anonymous6]

@@LeyScar close: if I put my cash in a square it’d be a circle

[@abx_egamer4874]

But mine is real but negative

[@tehlaser]

the ah-ha moment for me was realizing (no pun intended, but I’m keeping it) that “complex” does not mean “complicated”. What it really means is “joined together”. More like “combined” than “intricate”.

[@darkseraph2009]

Yep. The etymology of the word is basically "joined together"!

[@munashemanamike4217]

I think you've changed a large part of they way I think with this understanding

[@nosuchthing8]

Like an apartment complex, yes

[@NickRoeder]

This is similar to my thought process with irrational numbers lol.

[@aym9869]

Yes, in some other languages it is clearer because it is translated from Europian language. For examble in Arabic it is translated to a word means " Combined ".

[@timkruger6520]

That's awsome Ali! A second video about imaginary numbers would be great, especially on why they are so useful

[@alithedazzling]

Noted!

[@klayed]

@@alithedazzling and get a new shirt cause that nasa shirt makes it all so very hugely unbelievable

[@NevadaMostWanted658]

​@@klayedwdym lol

[@ratharyn1834]

Absolutely amazing, none of my lecturers have even touched on why j = 90degrees. You have a new sub here!

[@matswessling6600]

i is not identical to 90 degress. Thats would be ridiculously sloppy.

[@ratharyn1834]

@@matswessling6600 well yes, obviously my takeaway from the video wasn't that j=90 degrees as, you know, i did actually watch the video. Thanks for your valuable input though, lord knows where the world would be without such clever and helpful people like you in it to enlighten us morons. Speaking of ridiculously sloppy, how is your mother doing btw?

[@larrygoodman2442]

YO!!!!! DUDE!!! @6:12 YOU GAINED ANOTHER SUBSCRIBER!!! I've been a math tutor for quite some time, and I never thought about what multiplying by a negative number actually did!!! Now I have a clear understanding!!! Thanks for adding value to my life and this tool to my tutoring repertoire!!!

[@alithedazzling]

glad to hear!

[@Morimea]

good video nice to see "learning" moved from "remembering formulas(no one understand)" to "how it actually work - knowing it - it easy to just make any formula"

[@alithedazzling]

7:59 I meant to write a negative sign in front of the one :)

[@ignaciolarco8841]

I noted ❤

[@usergafe]

pin this

[@romanvertushkin6791]

the sign - is actually a symbol for a straight angle, i.e. 180 degrees or Pi so -1 is 1 rotated 180 degrees

[@classicalmechanic8914]

Imaginary numbers are not just imaginary but also real numbers at the same time. Sqrt(i) is the proof that every imaginary number is square of the sum of real and imaginary number. And imaginary number part of sqrt(i) is also a square of the sum of real and imaginary number. This pattern repeats all the way to infinity.

[@mhm6421]

@@romanvertushkin6791 It's just a coincidence

[@АндрейСветлица]

Dude, I am a third year bachelors student, and your videos helped me so much with understanding wave equation of the electromagnetic energy, and before your videos, I was completely desperate and thought that I will never fully understand differential equations. This is exactly the way I wanted someone to explain physics to my. I was always good at just solving equations, but the lack of non vague, non formalistic explanation of what the concept is, and where in the realm of my math knowledge I shall put it, no teacher in the uni ever taught me. And that exact approach to math I try my hardest to utilize when I tutor myself. Thank you, you are the real one.

[@alithedazzling]

wow that is awesome to hear!!! stick around for more ;)

[@dixon1e]

Lots of +1 for the comments and praise, and one more from a life long learner that took this topic in 1979. It's never too late to really understand something. Well done, please continue your work.

[@alithedazzling]

@@dixon1e thank you very much!

[@Napso178]

Вижу ник на русском

[@Meson_s7]

This gotta be one of the best if not the best explanation I've seen on imaginary numbers and math thinking in general.

[@alithedazzling]

i am honored!

[@fredg.sanford634]

It really was.

[@savage22bolt32]

I remember the first time hearing the term 'imaginary number' in high school. Today is the second time I've heard the term 'imaginary number'. In between the first time and second time, 55 years have passed. In my life it doesn't seem like an important thing, but I definitely think this is a great video. Excellent presentation! Maybe I'll learn something❤

[@savage22bolt32]

​@@alithedazzlingwhen multiplying by "i" (or neg i), you always rotate the plot by 90°. Does that mean "i" is a constant? Basically maybe, but "i" also has other meanings? I never had a need for this in my career, but I find it fascinating in my old age.

[@fredg.sanford634]

@@savage22bolt32 - I am retired now, after being in IT for nearly 40 years, and I want to relearn the hard math that I studied in my university days.

[@markl4593]

“No one understands what the hell i is” - great line & great video, Ali - Subscribed! (a retired EE)

[@Noname-rc8uc]

But it flips the understanding of what "i" is to how it's used in physics and engineering

[@alithedazzling]

haha glad you like the casual teaching style!

[@Elizabeth-nq9ly]

I love the pace you go at. Helps stop my mind from drifting away.

[@placer7412]

dope thanks for the info I think i actually understand this jawn better now

[@pupsvids5333]

please, keep going! the more videos on all those topics would be great! ITs fascinating to see the way you think about all these things, and it really helps to bring back my interest in things i didnt know about. thank you so much!

[@mateuszsrebniak8214]

Great explanation! I would really love to see more of the geometrical perspective on complex numbers.

[@Player_is_I]

Bro, you deserve more attention, keep it up! This video highlights a very fine perspective which needs to be spread! Thanks for this ❤

[@dees2621]

This is really nice and I wish more people would teach imaginary numbers like this, awesome video! Though, I have a suggestion for how this could be approached in a more fruitful way: When you mutiply two numbers, you are multiplying their magnitudes and adding their angles with respect to the positive real number line. Ex. -3*-5 = (3∠180)*(5∠180) = (3*5∠180+180) = (15∠360) = (15∠0) = 15 or -2*2 = (2∠180)*(2∠0) = (2*2∠180+0) = (4∠180) = -4 So negative*negative=positive and negative*positive=negative, checks out. By extension, when you take the exponent of a number, you exponentiate the magnitude of the base and multiply the angle by the exponent. Ex. (-2)^2 = (2∠180)*(2∠180) = (2^2∠180*2) = (4∠360) = 4 so (-2)^3 = (2^3∠180*3) = (8∠540) = (8∠180) = -8 So exponentiating a negative number by an even number makes the output positive, and by an odd number makes it negative, also checks out. So naturally one could try to do it for fractional exponents: (-1)^(1/2) = (1^1/2∠180/2) = (1∠90) This takes us off the real number line, landing us at the number i in the complex plane, the number with a magnitude of 1 that is 90 degrees from the positive real numbers. So i^2 = (1∠90*2) = (1∠180) = -1, giving us the definition that i^2=-1. and 3*i = (3*1∠90+0) = (3∠90) = 3i and i^n = (1^n∠90n) = (1∠90n) = 0 when n=0, i when n=1, -1 when n=2, -i when n=3, and 1 when n=4. This oscillates. Checks out :) This is a very natural pathway to the complex plane that doesn't introduce the plane out of nowhere. Not at all rigorous, but very intuitive and mechanical. This also encapsulates how multiplication and exponentiation of complex numbers works. (Only exponentiation by real numbers) Then you could convert the r∠θ form to the exponential re^iθ and everything ties in together.

[@gianpaolozanconato5012]

(-1)^(1/2) = (1∠90) because i =(1∠90) and (1∠90) ^2 = (1∠90) *(1∠90) = (1*1∠90+90) = (1∠180) = -1 this is the only thing I would write differently than yours

[@sciencedaemon]

No. That idea of adding angle/direction means you must be using vectors, not real numbers alone. This is indeed what should have been explained using geometric algebra to show what i really is, a bivector formed of two basis vectors. All that complex stuff with i is limited to 2D, whereas bivectors are available in any dimensional space of 2 or greater.

[@pulsartsai7776]

I believe it should be i^n = (1^n∠90n) = (1∠90n) = 1 when n=0, not 0.

[@JoaoVitor-ib9ip]

Good one Ali. Greetings from Brazil

[@SuzeArizoon]

You are a genius! Showing concepts in such an intuitive and visual way, and making sense of things, is truly respectable. I remember that I was trying to understand 𝑝^-1=1/𝑝 in an intuitive way, but it seemed impossible at first. After days of breaking my mind over it, I thought about the basic principles: the number line, neutral elements, and the multiplicative or additive inverse. Eventually, I gave myself an explanation, and your video, idk , brought back that experience! How cool!

[@ambassador_in_training]

Great explanation! Thank you so much for taking time out of your very busy schedule to enlighten your audience with these beautiful intuitive examples!

[@swatt862]

Great video. Just outstanding. The comprehension of a vector is a life changer.

[@celestianeon4301]

This is madness that this example wasn’t taught nor illustrated my entire school life absolutely absurd. Please continue the series. Not too long ago I learned that math is geometry and there’s no math without geometry. It’s crazy they teach people letters without the sounds. Yet expect students to speak read and comprehend the language. Pure insanity if not evil. I suspect the evil aspect as this Riggs of an ill and intentionality behind this.

[@darinheight6293]

I’m a teacher and we do teach kids letters along with the letter sounds they make. Not sure where you got your information

[@Chrisratata]

What do you mean letters without the sounds?

[@DJamz-xp7dc]

​@@darinheight6293 He is referring to math as a language not regular school language

[@rahatkhan4446]

@@darinheight6293 That was an analogy.

[@FreakGUY-007]

I already thought about maths is geometry after taking evolutionary and developmental biology class...

[@neji7713]

this was such a beautiful explanation , thank you

[@parinose6163]

Very, very interested in the Fourier Transform! Thx in advance...

[@fuzailkhan9701]

Yes i want a more deeper understanding of complex numbers and please continue this series.

[@farzbod]

kzbin.info/www/bejne/q6bRhHyVYpt4a9k

[@sciencedaemon]

You actually want geometric algebra then. It encompasses complex numbers, vectors, and multidimensional objects (e.g. multivectors). Goes way beyond limited 2D complex numbers using i.

[@curry2515]

Hey Ali, it would be great to have another video on "imaginary" numbers !! ( 13:38 )

[@rakshitgupta2660]

Yes

[@joseabendeck8266]

Yes pls!

[@Tade.G.B]

As always you are legend. im really excited to watch your next video fourier transform /frequency analysis

[@enetniss]

Thank you! This was super helpful. I would really appreciate a part 2 that dives deeper into complex numbers.

[@farzbod]

kzbin.info/www/bejne/q6bRhHyVYpt4a9k

[@williamwolfe9335]

Awesome video! Great way of thinking about it. I kinda had a light bulb moment at 7:40 cause I was thinking, "Well this sounds interesting but how does it relate to the square root of negative one." And then when you said to multiply I by I, it just clicked for me. I love finding new ways of thinking about numbers. I wasn't even looking for anything about this. KZbin just knew I'd like it so it appeared in my feed.

[@Knottz]

4:58 no need to remind me 😭

[@iloxix4483]

This was amazing! excellent work I never in all my university heard it explained this way which is obviously the right way to see it.

[@Entroprox]

Im an engineering student, and i had only 2 high school tuition teachers who taught in a similar way. You're no 3. Keep up 👍 the good work 💪

[@adastra123]

❤ brilliant. Some of the most profound things are under our nose but it takes a special person to point it out. Thank you. Subscribed.

[@MubashirAhmed-i9r]

Thanks Ali I am currently learning AC circuit analysis in my electrical engineering major and there is a lot of imaginary number equations to solve

[@SathnimBandara]

Thanks Ali this video was awesome! Definitely do a more deep dive into complex numbers

[@siavashamirhoseini5667]

I am a first year student in Electronics and really like the way you see concepts.

[@Abhishekkumar-w7c2g]

That's really cool how these things are making sense, awesome!!

[@exo-580]

yes it was helpful ,i learnt this in my first year at uni and i used to think complex really meant complex but it's just simple algebra.i love it !! keep it up Ali ,so cool!

[@alithedazzling]

Yes they're just numbers, so simple but so powerful!

[@juliavixen176]

Complex as in "Shopping Complex" or "Apartment Complex". It's two or more things stuck together.

[@junderhill1107]

This was absolutely amazing. Beautiful description.🎉🎉🎉

[@alithedazzling]

Glad you liked it!

[@unacomn]

Well, this would have made things clearer 25 years ago.

[@jessemcelroy5266]

This was the best video on imaginary numbers I've ever seen. I actually understand it now. Thanks

[@diogeneslaertius3365]

Rotations on the complex plane are just a consequence of z = a + sqrt(-1)b. Obviously, complex numbers are "two-dimensional", each complex number is isomorphic to a linear transformation on R^2, i.e. 2x2 real matrix. A complex number like i = sqrt(-1) has |i| = 1 making it isomorphic to an SO2 matrix, hence the rotations you observe. The connection manifests itself clearly in Lie groups theory (and Lie algebras).

[@retrogameenthusiast4703]

Love it! As a recovering math/physics major and current actuary I remember thinking of imaginary numbers as 2 dimensional and playing around with the idea of whole numbers with a dimensional component and trying to work out a ring that could help me understand E&M. Long story short, it was fruitless, a waste of precious study time but totally worth it 😬

[@angeldude101]

Yup! A complex number is indistinguishable from a 2D scaled rotation matrix!

[@jamesharmon4994]

I love this explanation! Before this video, I had no trouble "understanding" where this video was going to "go." However, this video was so intuitive that it made it much easier for me to conjure this in my mind.

[@thebiomark716]

Exactly what I needed to understand Circuit Analysis 2😂..thanks Ali

[@alithedazzling]

I'm glad it helped!

[@gbemigabolajoko1403]

You're astronomically lucky, my friend.

[@aiwanano6507]

This video is awesome. This helped me understand imaginary numbers a bit more. Thank you.

[@ramymedhattobia]

petition to rename complex numbers to composite numbers

[@lucasm4299]

That’s already taken

[@iwanadai3065]

aren’t those the opposite of primes?

[@lucasm4299]

@@iwanadai3065 yes

[@Ibreatheoxygen]

@@iwanadai3065yeah

[@lpi3]

No way

[@whophd]

Whoa you are on FIRE with these last few weeks of videos. This one is the best explanation of “i” I’ve ever had … and I was a top 0.5% maths student 2/3 of my lifetime ago. Instant subscribe, and I don’t give out many Likes on KZbin but you get one 👍

[@NinjasOfOrca]

around 2:35, you write that x^2 = -1, then x = i; but to be complete, x = + / - i, as (-i)^2 will also give you -1

[@Hindu_hain]

Don't pick type mistake.... Go for learning

[@NinjasOfOrca]

Yes we’re all here to learn. And if x^2 = -1, and we’re told by the “expert” that x = i (omitting half the solutions), we have learned the wrong thing. If i had a channel i would make sure I’m coming correct

[@humanmoradi4230]

Just standing and clapping for you for this video! 👏🏻

[@TechBlueprint-V]

Dude, you explain things better than my math teacher.

[@BackStab1988]

The best explanation ever! ❤

[@TheVirtualArena24]

4:54 students loans so infamous 😂

[@mayskikot3823]

Outstanding! Simply phenomenal. I have been looking for a reasonable explanation for years, decades really, and finally found one. Thank you.

[@user-ct1ns6zw4z]

There are stretchy numbers and there are spinny numbers, and complex numbers do both.

[@alithedazzling]

is that the queen is dead album? i love the smiths!!!

[@user-ct1ns6zw4z]

@@alithedazzling Yes it is! Definitely my favorite album of theirs :D

[@williamcompitello2302]

Complex numbers have properties of both rubber and gum- jk.

[@user-ct1ns6zw4z]

@@williamcompitello2302 there's a reason why "rubber sheet geometry" (topology) and complex analysis are so connected (pun unintentional)

[@FF7EverCrisis]

This is so cool. Thank you very much. I love the chalk and board method. Very educational

[@miloyall]

For me, the moment this really clicked was when I typed in “i^3.5” and then “sin(45°)” back to back, realizing they were the same value.

[@FAK_CHEKR]

I’m trying to understand this. unless I am mistaken - i^3.5 = sqrt(-i) What am I doing wrong? I don’t understand sqrt(i) or sqrt(-i), or how this relates to 1/sqrt(2).

[@wiczus6102]

But it's not the same value. You're ignoring either the imaginary or the real part. With this reasoning sin(45) = sin(315) or sin(135) depending on what you ignore. Just because you see a coincidental root of two divided by two doesn't mean it's the same value. This is the actual reason why it has this result: (i+1)^2 = i^2 +2i + 1 = 2i (2^0.5*i/2 + 2^0.5/2) = i^0.5

[@user-ug2vw9vb2v]

Amazing. I never thought of complex numbers and negative numbers in this way. Thanks!

[@zika9688]

I'm almost in tears from how beautiful this is, I feel like I can actually love learning again. Thank you so much for your thoughtful and thorough videos!

[@MrSidney9]

Bro you're such a drama queen 😂

[@zika9688]

@MrSidney9 just came at s time when i was weak haha

[@MrSidney9]

@@zika9688 No shame in it bro. Stay blessed

[@Chickon273]

This is not meant to be a hate comment because you seem very clearly passionate about what you do and about educating and that is always a great thing to see. Personally, as someone who loves math particularly algebra, has my degree in pure math, and is interested in philosophy of mathematics, I have just never really agreed with this point. One reason is simply personal bias that I think “imaginary” is a fun mysterious word and I am in the minority that actually really likes it. But also to reference a comment someone online left in a different discussion about imaginary numbers “Santa Clause has real world applications in that it measurably gets children to behave. That doesn’t make him any more real.” And I feel similarly about imaginary numbers. That being said, I think no numbers truly “exist” so that certainly includes complex ones. And of course the nature of what it means for something to exist is constantly debated. In terms of whether the video gives a solid intuition of complex numbers, this is one of those situations where I think a lot of people in the comments either are already familiar with them or have at least heard of them before. It is admittedly hard for me to imagine a newcomer seeing this and following along in any meaningful sense especially with the e^i(theta) identity thrown in, I can imagine would be really intimidating for someone who hasn’t come across it in any context. From a pedagogical perspective, I was a little confused that you seemingly dismissed the “classic” way of explaining what i is where an algebra teacher will say “it’s the square root of -1”, when you proceed to introduce the imaginary axis as a solution to this rotate-by-other-angles problem, and explain that it makes sense to draw this axis this way because “i times i is negative one so it fits with our picture”. So I feel like in either instance you and the teacher are simply just out of nowhere saying “we have a number that squares to -1 because we say so” except yours is supposed to be more grounded in reality and less abstract when I’m sure a new student could name more uses for finding roots of a polynomial from examples they’ve seen in school than the uses of needing to rotate a number. I also don’t know how convincing the examples are to motivate complex numbers being any more useful than real numbers given that the chalk length example could be parameterized in the real xy-plane and fourier transforms have formulas using explicitly real numbers as well. Again, as someone who deeply loves algebra, I love imaginary numbers too and I think they are useful and convenient and great. I just don’t know if I’m convinced a skeptical student would buy the explanations in this specific context. Or that they would be able to really follow along. I also understand this isn’t supposed to be a lecture level of comprehensive detail so I would be interested to see what that would look like from you. In any case, I am happy to see more people making videos about math online and reaching out to get people interested.

[@GMGMGMGMGMGMGMGMGMGM]

"That being said, I think no numbers truly “exist”". I LOVE IT when people try to make this argument. "are there *1* of you, or are there *many* of you? Is there a distinction between *none* of you and *not-none* of you? Are these distinctions *real or notreal*? thus, does the 'set of things which are real / not notreal' contain 'numbers'?

[@Seagaltalk]

This is definitely more intuitive way of expanlaning it and it's much easier to see it's uses. Every number is in a sense imaginary but calling it imaginary just makes students see it as more obscure and not useful in real-world applications and seem to be more of an estoric math concept

[@Seagaltalk]

​@GMGMGMGMGMGMGMGMGMGM you've tied yourself in knot and have nothing to show for it.

[@lucasm4299]

We’re not reading your essay

[@Seagaltalk]

@lucasm4299 of course not, but telling him will just make him depressed. It is better to smile and nod and send a "head-pat" emoji

[@lastknight7071]

Extremely fantastic video Hoping for more in future Already subscribed

[@Claudialupperocd]

A chalkboard. Wow.

[@albertlevert2988]

I hate chalkboards and chalks.

[@kaushalendraraomannava4578]

Stupendous! This is THE BEST (the only?) rational explanation I have come across in my life! Kudos to this gentlemen 👏👏 How i wish this perspective was offered in my school/ college!

[@DigitalTiger101]

Holy crap dude. I am a TA for differential equations and I understand the mathematical operations of imaginary numbers, but yours is the first video that actually made it intuitively click for me with this concept of angles.

[@maxwell8866]

Great now get back to arbitrarily grading papers

[@Benlpau]

you are the best❤ i never have seen such a direct and intuitive explanation. teachers dont even understand it that clearly. but the dust everytime you wipe triggers my breathing reflexes😂

[@Knowledge2Intelligence]

This video is more than Math, literally Mind Opener

[@rubenysuifon]

GREAT EXPLANATION. I wish someone had explained it to me this way at school or uni

[@ignaciolarco8841]

Thank you Ali for putting flesh to the bones of these entities called imaginary and complex numbers.

[@alithedazzling]

you're very welcome

[@TheOlderSoldier]

Idk how KZbin knew I needed this, but I did! Great video!

[@AkselSterling]

- "Zero doesn't exist, because Zero is nothing and 'nothing' cannot exist." - "Zero is not 'nothing', is the numerical representation of nothing. 'Nothing' cannot exist but it can be represented by using Zero."

[@jlpsinde]

So good, please more on complex numbers!

[@Delan1994]

Thanks Dr Ali . I’m just doing imaginary numbers in electrical engineering class

[@alithedazzling]

Good luck with your class!

[@DoombringerDad]

@Delan1994 If you treat imaginary numbers the way they are treated in this video, you will flunk out of engineering school and be forced to settle with at best, a poli-sci or business degree. Sqrt(-1) = i. If you want a geometric interpretation, you need the complex plane and complex algebra.

[@mightyoak11111]

I was able to follow what you were saying until the last minute of the video. Very interesting perspective on i and imaginary numbers. Thank you for sharing.

[@Mitochondria527]

Dear Ali: 180 degrees is 90 + 90 and NOT 90x90. So two rotations is not what you say as (90x90), it is rather 90+90...... It seems that your analysis is wrong because of that.

[@Teacher_Albert-kd4ky]

Nah, when you multiply by -1, you rotate 180 degrees, but when you multiply by i, you rotate 90 degrees and then if you want to rotate another 90 degrees you multiply again by i and that's i² which equal to the sum of two rotation of 90 degrees each. Or you can think of it as 90×2 and not 90×90, because it's a sum, not a product. Hope I made it clear 🙏

[@christopherventer6391]

So, if i is the square root of -1, that means that i squared is -1. Multiplication by -1 is a 180 degree rotation. Therefore, multiplication by i is half of that rotation, so it is 90 degrees. It makes a lot of sense to think of it this way when talking about phase, for example. If two signals are 180 degrees out of phase, adding them together results in 0. This means that one signal has to be equivalent to the other one exactly, but inverted (multiplied by -1). But signals can also be only partially out of phase with one another as well, which can be represented as a rotation or as having an imaginary component.

[@Rama-zs5zv]

Man this is so cool OMG 😭😭💗✨

[@unlomtrash]

Quaternions finally make sense

[@nrudy]

Your enthusiasm for the ideas you're teaching really comes through, this is really well done. I haven't had to think of Radians in a long time but this would have been really helpful to understand in calculus.

[@onurbole7921]

All numbers are imaginary

[@glenliesegang233]

All numbers are symbols

[@robertgivens2586]

Im 67 with a 9th grade education and I love this stuff I love numbers and the challenge from them. Thanks for the simplicity approach

[@rafacosta_x_]

Amazing explanation! A thing that I love to do is relearn math through the internet. It's completely different when you understand what and why you are doing stuff. Got a new subscriber!

[@ahmadfahim7283]

An excellent explanation! I knew about these stuff and I am using them often. Many thanks!

[@misterinterrupt]

wow, this is an amazing intuition, thanks!!

[@thescreamingellens9616]

This is by far the best explanation of imaginary and complex numbers that I have ever heard… as someone with a pre-calculus understanding of math. Amazing😊

[@eitherrideordie]

Bruhhhh what! That makes it make so much more sense! I absolutely can't wait to run through the Fourier transform video

[@wisdomokoro8898]

Brooo i am about to do systems engineering with applications in Aerospace Engineering and Control and you are a number one source of inspiration for that

[@bedirakhalil8154]

Great video keep it up mate

[@mdidavi]

You are truly amazing. I have plugged and chugged complex numbers through all my math classes but no professor explained this as clearly as you just did. Thanks.

[@alithedazzling]

Glad it helped!

[@ProfeARios]

Than you so much for sharing. Greetings from Panama 🇵🇦

[@bigfrankgaming2423]

Man, please keep doing videos like this! Even if i swapped from electrical to computer engineering this is still very usefull and fascinating information :D

[@Matthewmellow2k]

Absolutely amazing video. It’s difficult to get through math without the understanding of where/how certain things are used. No one has ever explained the purpose of “i”. Thank you!!

[@andremercante8451]

I would like to see more imaginary number theory! Loved the video btw!

[@niyazbekberdykulov5047]

Super! You explained the topic in a very accessible and understandable way. Thank you!

[@last-life]

I've been learning for most of my adult life. You are a good teacher with a unique ability to frame topics that the masses can understand. You are doing science good service

[@satyabhangt]

Awesome explanation and great perspective. I agree the terminology makes it seem more complicated than it is

[@i_trymy3est]

Wow, I think I've gotten a glimpse of understanding that topic for the first time. Very good explanation for me, thank you very much.

[@thomassutrina7469]

Great explanation as a mechanical engineer that worked with electronics so had to understand both works.

[@Shankaray4]

Thank you! Great video and great explanation on imaginary numbers. This is the first time I understand what they are and how they work. Please make another that goes more in depth.

[@markmattes9918]

Thanks for making these videos they're super helpful