Imaginary numbers aren't imaginary

[@alithedazzling]

For more chalkboard videos including the next one: kzbin.info/aero/PLZZOG63zmCLE1eWvazftEMl8kMpXvSzst

[@helpingnetwork2556]

I really enjoyed this and after only watching your two videos i decided you channel is worth subscribing. i also wanted to ask my doubt, Insha allah i can get an answer from you. timestamp of my doubt is 7:10 , when our number line got 2-Dimensional aka got an another axis (Y-axis) why we used i, 2i, 3i, and so on to scale it?

[@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. 😊

[@spacedominator7163]

Need more cats​@@enervoncee5327

[@TheOriginalManTrust]

​@@enervoncee5327there is a popular saying that the internet is made for cats lol

[@georgebush6002]

...what you think he do after, hmmm? (Avenue Q reference)

[@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

[@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

[@Dazforce]

This explanation only took thirteen minutes. How was this not explained in my 4 years of studying my engineering degree? It made so much more sense. Thank you

[@rossmau]

I think that this video made sense because your engineering background has you primed to really understand what this means in a more meaningful way. This guy is an electrical engineer, as was I and to us, i has a very practical meaning. In other disciplines I'm guessing you are exposed to it, just not to the same magnitude electrical engineers are. I'm now a teacher and often times students who had me in the past will come to me for help with another class. I explain it to them and they say "it makes so much sense, why didn't Mr. So-and-so explain it as well as you?" I explain to them that they probably did, but you just weren't ready to fully understand it. Now that they had grappled with it and let it simmer they were ready to finally understand.

[@Dazforce]

@@rossmau thanks for your comment and I think you are probably right. I studied it as part of the “engineering mathematics” subject and was applying it to control systems as part of a mechanical engineering degree. I just couldn’t make sense of what the imaginary number meant apart from if a system as oscillating or not. Now thinking of it as how far between two different states (such as odd/even in the example) makes more sense to me

[@stefano.a]

because it is wrong

[@Dazforce]

@@stefano.a is it? Probably not the place to explain why but could you link a video to a correct or better explanation?

[@Mono_Autophobic]

it took you 4 years to understand only this much?

[@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.

[@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 ".

[@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!

[@MarkMYoung]

Yep, and if you consider it in 3 dimensions, there is at least one other square root of -1 which would "rotate" the number out toward your face. [See quaternion roots i, j, and k.]

[@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.

[@hellohabibi1]

@@romanvertushkin6791 It's just a coincidence

[@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!

[@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?

[@rainerzufall42]

@@matswessling6600 Indeed. As an association for better understanding okay. But wrong. Formally you could see multiplication with i as the operator that rotates polar coordinates by 90° or π/2 or τ/4 without changing the modulus. More words, but less pitfalls and formally correct. In other words: r e^(i θ) * i = r e^{ i (θ + π/2) }. Or PC(r, θ) * i = PC(r, θ + π/2), as i = PC(1, π/2).

[@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

[@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 ❤

[@Влади́мирПу́тин-м1и]

حبيبي .. افضل شروحات في يوتيوب شروحاتك .. استمر وفالك المليون متابع ❤

[@alithedazzling]

thank you!

[@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.

[@rasheekfrederick5872]

Awesome video, first time I was able to make intuitive sense of imaginary numbers and the 3*i*I = -3 definitely landed it for me. Appreciate you taking the time to pull the rest of us up. Peace and blessings

[@rainerzufall42]

Just one tipp: Look up "polar coordinates". If you understand them, this is no longer any kind of woodoo... And yes, in my school, they were taught. Then again at university. No confusion about that.

[@rainerzufall42]

en.wikipedia.org/wiki/Polar_coordinate_system -> there's even a chapter about Complex Numbers...

[@rasheekfrederick5872]

@rainerzufall42 Appreciate the added insight and info, 🙏

[@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!

[@thechadshow6830]

You have cleared up SO MUCH of my math misunderstanding with this. Really opened my mind. Thank you

[@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

[@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!

[@curry2515]

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

[@rakshitgupta2660]

Yes

[@joseabendeck8266]

Yes pls!

[@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"

[@siavashamirhoseini5667]

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

[@fauxfirefur]

Your description of these concepts is so simple to understand. Really makes me feel happy that, despite not knowing what a negative properly implied until now, I always enjoyed writing stuff like 6--8 as either +6+(--8) or +6--(+8) when describing how to do some maths for classmates back in high school.

[@BenGreen1980]

I get that you're not a fan of i being sqrt(-1) but your explanation of what you called "the math" was deliberately bad in a way that isn't very honest or fair. In particular, the way you dismissively reduce the mathematical justification for i = sqrt(-1) to "everybody claps" right before incorrectly attributing the very process by which mathematicians extended the concept of numbers into new territory to engineers and mathematics. This video is a good example of how engineers' dismissiveness and condescension toward the subject matter can lead them to mess up explanations of literally grade school math. Multiplying by -1 isn't a "rotation by 180 degrees" because that doesn't make sense in a one dimensional system. You're jumping to using operations that require 2+ dimensional space because you want to arrive at a conclusion but don't actually care how the mathematicians who came up with the idea you're attributing to physics and engineering justified it, so you just barrel over it with sloppy nonsense. If you look at how mathematicians originally justified 5 x -1 = -5 - or how it's justified today - it has nothing to do with a rotation because it's premature to do so.

[@chelmano0]

Oooooh spicy! Tell more.

[@Elizabeth-nq9ly]

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

[@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.

[@placer7412]

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

[@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...

[@rsaforjm]

Absolutely fantastic. Great explanation and an eye opener. Just understood what I learnt 50 years ago. Whoever decided to call it imaginary number did a disservice to all of us. That word imaginary created a mental block which made it hard to understand. Negative numbers are also just a concept nothing physical and we take it for granted, these are just a further extension. I am looking forward to your explanation of Fourier transform.

[@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.

[@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.

[@JoaoVitor-ib9ip]

Good one Ali. Greetings from Brazil

[@kevangibbs5819]

6:00 And this right here is the moment so much just clicked for me. Sometimes, it's the simple explanations that go the longest way. Imaginary numbers are out of my depth. Applying them? Gonna be a while before I get there, if ever. But the way you just summed up a very complex subject in laymans terms just put a lot together for me. Thank you for this!

[@parinose6163]

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

[@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.

[@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 💪

[@Intelligence3]

That was amazing!!! The first minute or so you spoke so quickly that i was worried i’d fall behind but not at all!

[@alithedazzling]

Glad you were able to keep up!

[@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!

[@rainerzufall42]

Guys, don't bother a physician with this... for him mathematics is not useful in "the real world"...

[@Loopify3D]

I just discovered this channel. I am already subscribed to your first channel. You are doing an awesome job. From now on i will be thinking of numbers as 2 dimensional it makes more sense. Thank you so much.

[@unacomn]

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

[@Skynet_15]

Omg يسلمُ إديك the best explanation so far

[@Knottz]

4:58 no need to remind me 😭

[@maxfuentes5435]

Your seasonal analogy is top notch, since the ecliptic charting the solar position over the year forms a circle with a cross just like the complex plane

[@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

[@mayskikot3823]

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

[@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.

[@neji7713]

this was such a beautiful explanation , thank you

[@Valerius123]

Gauss was right. They should be called lateral numbers.

[@MarkMYoung]

Yes, this has been a part of my understanding since I delved into complex numbers and quaternions. When negative numbers were created, imaginary numbers became necessary.

[@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

[@larsnystrom6698]

Yes it would be a solution to x^2 = --1. But it would conflict with other things. Multiplying with a complex number adds the angle (in the complex plane) to the other number. If we square --i by using that, we do get --1. But the angle is now more than 2 pi, and we usually don't like that. It works, but we prefer angles less than 360° when working with complex numbers. (We probably treat them as an equivalence class, where those with 360° are equivalent, and choose the principal one.)

[@johnnyjoestar7769]

🤓

[@rainerzufall42]

@@NinjasOfOrca You are absolutely right. The problem here is, that √x² is not x, it is |x|. Thus x = ± i.

[@Tade.G.B]

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

[@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

[@Abhishekkumar-w7c2g]

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

[@davidpo5517]

So...are there 3 dimensional numbers on a z-axis? Asking for a friend...

[@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

[@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

[@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

[@AwesomeTheAsim]

This was incredibly insightful, 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)

[@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.

[@TechBlueprint-V]

Dude, you explain things better than my math teacher.

[@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.

[@Claudialupper]

A chalkboard. Wow.

[@albertlevert2988]

I hate chalkboards and chalks.

[@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!

[@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

[@kiera-0]

As a high school student, I have always wondered how and why imaginary numbers came into existence. The way teachers explain it always leaves questions in my mind. I'm eagerly waiting for new videos.

[@ignaciolarco8841]

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

[@alithedazzling]

you're very welcome

[@dell2693]

Love it! I lost my vibe for math in my final year of high school where my math teacher introduced the 'boogeyman' imaginary numbers. Really appreciate that in >15 minutes I've now got the concept and have had it visualized. Thanks

[@TheVirtualArena24]

4:54 students loans so infamous 😂

[@DD-gi6kx]

there's videos like this and then there's people proving the earth is flat, a strange world

[@Knowledge2Intelligence]

This video is more than Math, literally Mind Opener

[@lastknight7071]

Extremely fantastic video Hoping for more in future Already subscribed

[@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.

[@solsav]

I think your explanation and the direction you took in making (for the lack of better word) imaginary numbers more intuitive, is great! I also think that if you give some further structure to your explanation and provide more streamlined examples, it can be one of the best videos out there for people who want to understand the intuition behind sqrt(-1).

[@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.

[@TwasBrilligandtheSliveyToves]

Had this same issue. First let me say I’m very much an amateur at this. With that in mind I am aware graphical representations of number systems are far from new and as a way of “visualising” these issues they are very useful. They have helped me. If we stop there then this is probably fine (ie it’s just a handy creative tool for learning and not a deeper representation of the maths). However when the discussion proceeds to say that “i” is equivalent to a 90 degree rotation and then we equate i*i =-1 and 90 degree + 90 degree =180 degrees then there seems to be some underlying issues as clearly these two cannot be simply related as one involves multiplication and the other addition. It’s not 2*90 degrees as the comparative it’s 90 degrees squared. So we must then either have a more complex formulation of the axiom between the two or leave the relationship as purely visual tool.

[@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!

[@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."

[@swatt862]

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

[@dukeofabq]

I can have 4 apples, but I can't have i apples. It's not rocket science to see why they called them imaginary numbers. It's great that you're trying to show how multiplication by i (and complex numbers in general) results in rotation and scaling in the complex plane. Why do you insist this is *the* way to interpret them and to dismiss the imaginary unit's definition as the principle square root of -1? And what on Earth was that even odd oscillation ridiculousness? How even or odd is 4.35? Engineers 🙄

[@null6482]

But we have i phone

[@seanc.2439]

You can’t have -1 apples. Are negatives imaginary then, too?

[@dukeofabq]

@@seanc.2439 you could easily think of -1 apples as losing one apple or having a debt of one apple. What would i apples mean?

[@IamProFish]

This comment is insanely stupid

[@daymenpollet4202]

There is a good reason. Because 'i' doesn't mean really anything but the representation of degrees instead of 'i' is useful and used in real world applications.

[@satyabhangt]

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

[@onurbole7921]

All numbers are imaginary

[@glenliesegang233]

All numbers are symbols

[@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.

[@thomassutrina7469]

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

[@eitherrideordie]

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

[@ProfeARios]

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

[@junderhill1107]

This was absolutely amazing. Beautiful description.🎉🎉🎉

[@alithedazzling]

Glad you liked it!

[@vinayk7]

Amazing insight that positive and negative signs are just direction , in and out

[@SuperRealhigh]

Very good explanation Ali, shukran!

[@Mslaquazia]

I wish they taught us about "imaginary" numbers this way from the beginning. Not only would I have a deeper understanding but I believe I might've better understood trigonometry and complex analysis when I took them😢

[@mode1charlie170]

This was awesome! I need a few days to digest this information.

[@alithedazzling]

It takes time to get used to the concept of imaginary numbers, but it's so worth it!

[@brianpastore7354]

25 years ago, I took Signals and Systems in engineering. I got my degree but switched careers. This video somehow made sense of confusions I hadn't even thought of in decades. If you had been my professor, I'd still be an engineer (and not a lawyer). We need more teachers like you.

[@endlesswick]

This is the best explanation for imaginary numbers I have ever seen. You also give a good explanation of negative numbers. The age old question: a negative times a negative is a positive...why? I can finally answer that question.

[@BiJoy2925]

Videos like this prove time and time again that the educational system needs inspired teachers to bring out the best in the future.

[@user-ug2vw9vb2v]

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

[@TheOlderSoldier]

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

[@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