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 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).
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”.
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.
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❤
@@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.
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.
(-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
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!
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.
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.
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.
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!
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!
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).
Great explanation! Thank you so much for taking time out of your very busy schedule to enlighten your audience with these beautiful intuitive examples!
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
This is the best explanation for imaginary numbers i have ever had 🙌🏻🙌🏻🙌🏻 you got a new subscriber 🙌🏻🙌🏻 , when dealing with Inductors and capacitor we used V=jwLi , i couldn't understand what j as an imaginary number was for until this video. 🙌🏻🙌🏻🙌🏻
I always thought what the heck was a complex/imaginary number, Now I think I really got the answer. I will never see Imaginary numbers the same way! Loved it!
Never tried to think why i = root(-1). Learning and visualizing it's explanation makes my grasp on the concept of complex numbers much stronger. Thanks man
I have recently found your channel and it has been my favorite ever since. You have really changed my perspective on a lot of physics and math stuff. Oh, and I'm also doing a bachelor's in physics and computer science, so I'm really loving your content. looking forward for more vids!
I'm a drop out with only elementary education and I think that these videos even though a little chaotic (you going a bit off script at times) explain a lot in very simple terms so even blockheads like me can understand. Kudos!
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
I can tell whether my professor really understands what he is teaching or he just repeating what he memorized over the years, you definitely understand what you are teaching on a deep level keep up the good work!!
Really good explanation, a great help to visualizing! Thanks! Minor typo (verbo? misspeak) at 7:58 "Take this 3 and multiply it by 1" should be "by negative 1"
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.
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.
Genuinely one of the best explanations. Math teachers just show you how to solve equations step by step but don't explain the real-world applications sometimes.
when I first saw the thumbnail, I was like.."oh! like when we plot complex numbers the imaginary axis is at 90 degrees to the real axis...so that means we turn 90 degrees when we multiply a real number with I"...and yes that is what you explained secondly, I would like to ask whether the graph that you plotted to show the switch between even and odd numbers should be that sort of a continuous one instead of lets say points because decimal numbers are neither odd nor even, the graph can only be plotted for integers. loved the video!!!
Thank you. That helped me picture what is going on better. I wish you could provide such an explanation, extended, for quaternions. IU'll watch out in case you do.
Yes please help with Fourier Transforms 🙌🏻 My E&M teacher was the first person to make any of it make sense. He calls it magic, one of my astronomy professors calls it "dark voodoo" and I hate that they do that tbh. It doesn't help make it more intuitive or understandable when my teachers call it magic and voodoo. I'm half expecting you to do the same though 😂
I got chills when I saw 'Caltech' in your shirt. It reminds me that I'm gonna apply for Caltech for the Fall 2026! Ambivalent emotion of both anxiety and happiness😌
Thank u Ali, it was eye opening. I would like to go deeper into this topic , so request u for another video on imaginary number before u jump to the fourier transform.
Thank you Dr. Ali. Your explanation of this is really cool and understandable. I was talking to my physics teacher and we were wondering if the angle in degrees is related to imaginary numbers. I felt that it might be. I'm waiting for more videos because they are really interesting. Greetings from Poland!
Angles in degrees and radians are exactly related to complex numbers. Euler's formula states it quite well: e^(iθ) = cos(θ) + i•sin(θ) I don't want to write a long comment here at the moment, but I will if you want to know more. Also, scaling and rotation by an angle... and complex numbers... are equivalent to 2×2 real matrices. (It's all the same stuff!)
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.
"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'?
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
Imagine number can be imagined but can't be felt physically like happiness can be felt but never have physical appearance in real word right??... Beautiful
You just gave an excellent explanation and i learned many concepts that was not covered in college! I can even say, that i may even be able to apply this new knowledge to my own research!
Amazing Explanation! I'd really love a video about transfer function of a system and the roots of it (poles and zeroes) and how to intuitively think about it. I'm an ECE engineer taking system dynamics course this semester.
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😢
Imaginary numbers definitely aren't imaginary to electrical engineers. We use imaginary numbers to represent associated orthogonal values. We were already using "i" for current, because we were already using "C" to represent capacitance, so we use "j" as the imaginary operator. Anode and cathode also have the opposite meanings compared to chemistry. Please don't try to understand it. "Complex number" is a more accurate description than "imaginary number". When this was first introduced early in the EE curriculum, the professor told a story, almost certainly apocryphal, of a mathematics symposium where a mathematician introduced the concept of imaginary numbers and stated with glee that finally, mathematicians have a pure mathematical construct that they could explore that couldn't be sullied by engineers, and an electrical engineer in the back of the room shouted, "I have the PERFECT application for this!"
Great video! The confusing part is that for the 180 degrees we put the minus("-") before the number (-3), and for 90 degrees we put an "i" after the number (3i). Better way would be to use another symbol like an mirrored L and put it in front of the number. Another idea would be a minus with the degrees written over the minus symbol. Would create much less confusion..:)
You need to do a bit more “hand holding” between representing the real/imaginary axis with cos and sin and how those relate to eulers equation. For many seeing e^j*theta for the first time, they need the proof. Otherwise, great video!
This is so great man. Subscribed. This has me thinking about a question. What if you and I wanted to rotate along the z axis, can we model a "z-shift" into a "3-dimensional number" using i? What would be the way to illustrate a 3-dimensional number (x, y, z) in a similar way to how you describe the shift from unidirectional to bi-directional? Or, you know what I am trying to say I think. My work is on the linguistics and cognitive sciences end of things but this is relevant to my work with dialectical and ideological functions of thought.
Quaternions. Quaternions are basically the algebraic 3D equivalent of 2D complex numbers. If you like group theory, SO(2) and U(1) are basically multiplication by complex numbers with a radius of one unit from the origin... so it's just rotation. The SO(3) group is 3D rotations, which is isomorphic to multiplicarion by quaternions one unit in distance from the origin. The complex numbers are also isomorphic to 2×2 real matrices (which also includes the split-complex numbers and dual numbers) Quaternions are isomorphic to 3×3 real matrices... which are rotations and scaling in 3D Cartesian space. Everything in Mathematics has several different names for historical reasons.
Shows explanation for how complex numbers are displayed geometrically in relation to the real number line. Though the title feels more misleading to the reason why it was called imaginary in the first place. Pretty good though.
Thanks again for these intuitive approaches and explanations For the criticizers, these videos are not meant to replace scholar courses but to give you a flavor or an interpretation or a visualization of what could be very abstracted! Ali, this is exactly the way I am explaining complex numbers to my kids and they love it! To go from 1 to -1 you have many ways: you go in one shot pi or you go in two steps same move by pi/2 (sqrt(-1)) or you go even 3 steps using cubic square root of (-1) Also to go from 1 to 1 you may make 2pi move or twice a pi (-1)^2 or in 3 steps 2pi/3 which we usually call j (j^3=1) Etc … Best
bro we see differentials in masters(econ) and as solution of phase diagrams there are complex numbers. I was thinking about a intuitive reasoning and you my dear brother just nailed it. Thank you from bottom of my heart
Ok this was great, im in engeneering school and havé trouble understanding the use of FT so it would ne Nice to hear your take on it, thx for the video
Thanks, bro. Cool stuff last month has been introduced to the complex exponential of Euler identity. Is all about what you just explained now🎉🎉🎉 thanks
I think another video on imaginary numbers as a follow up would be amazing
Agreed
i agree
I agree
agreed
Bring it on
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.
maybe composit number make sense?
@@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).
I NOMINATE, that complex numbers be called COMPOUND NUMBERS! Because it’s “a quantity expressed in terms of more than one unit or denomination”
@@___Truth___ OMG now the naming problem occurs to us all beside programmer.
In Greek they're called μιγαδικοί which means hybrid numbers 😮😂 I think it's better
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”.
Yep. The etymology of the word is basically "joined together"!
I think you've changed a large part of they way I think with this understanding
Like an apartment complex, yes
This is similar to my thought process with irrational numbers lol.
7:59 I meant to write a negative sign in front of the one :)
I noted ❤
pin this
the sign - is actually a symbol for a straight angle, i.e. 180 degrees or Pi
so -1 is 1 rotated 180 degrees
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.
That's awsome Ali! A second video about imaginary numbers would be great, especially on why they are so useful
Noted!
This gotta be one of the best if not the best explanation I've seen on imaginary numbers and math thinking in general.
i am honored!
It really was.
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❤
@@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.
@@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.
My bank balance is an imaginary number…
Bro, you deserve more attention, keep it up! This video highlights a very fine perspective which needs to be spread!
Thanks for this ❤
I love the pace you go at. Helps stop my mind from drifting away.
Absolutely amazing, none of my lecturers have even touched on why j = 90degrees. You have a new sub here!
i is not identical to 90 degress. Thats would be ridiculously sloppy.
Hey Ali, it would be great to have another video on "imaginary" numbers !! ( 13:38 )
Yes
Yes pls!
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.
(-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
petition to rename complex numbers to composite numbers
That’s already taken
aren’t those the opposite of primes?
@@iwanadai3065 yes
@@iwanadai3065yeah
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!
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.
I’m a teacher and we do teach kids letters along with the letter sounds they make. Not sure where you got your information
What do you mean letters without the sounds?
@@darinheight6293 He is referring to math as a language not regular school language
@@darinheight6293 That was an analogy.
I already thought about maths is geometry after taking evolutionary and developmental biology class...
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.
wow that is awesome to hear!!! stick around for more ;)
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.
@@dixon1e thank you very much!
Great explanation! I would really love to see more of the geometrical perspective on complex numbers.
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!
❤ brilliant. Some of the most profound things are under our nose but it takes a special person to point it out.
Thank you.
Subscribed.
Thank you! This was super helpful. I would really appreciate a part 2 that dives deeper into complex numbers.
4:58 no need to remind me 😭
Yes i want a more deeper understanding of complex numbers and please continue this series.
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!
Yes they're just numbers, so simple but so powerful!
Complex as in "Shopping Complex" or "Apartment Complex". It's two or more things stuck together.
I am a first year student in Electronics and really like the way you see concepts.
Thanks Ali this video was awesome! Definitely do a more deep dive into complex numbers
“No one understands what the hell i is” - great line & great video, Ali - Subscribed! (a retired EE)
But it flips the understanding of what "i" is to how it's used in physics and engineering
haha glad you like the casual teaching style!
Good one Ali. Greetings from Brazil
This video is more than Math, literally Mind Opener
Outstanding! Simply phenomenal. I have been looking for a reasonable explanation for years, decades really, and finally found one. Thank you.
Exactly what I needed to understand Circuit Analysis 2😂..thanks Ali
I'm glad it helped!
You're astronomically lucky, my friend.
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).
Great explanation! Thank you so much for taking time out of your very busy schedule to enlighten your audience with these beautiful intuitive examples!
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
yasss more complex numbers pleassee and fourier transforms
so excited for the next vids
may Allah bless you brother.
Bro, it is very important what you do, keep going
This is the best explanation for imaginary numbers i have ever had 🙌🏻🙌🏻🙌🏻 you got a new subscriber 🙌🏻🙌🏻 , when dealing with Inductors and capacitor we used V=jwLi , i couldn't understand what j as an imaginary number was for until this video. 🙌🏻🙌🏻🙌🏻
Awesome! I'm glad it finally made sense!
I always thought what the heck was a complex/imaginary number, Now I think I really got the answer.
I will never see Imaginary numbers the same way!
Loved it!
Never tried to think why i = root(-1). Learning and visualizing it's explanation makes my grasp on the concept of complex numbers much stronger. Thanks man
I have recently found your channel and it has been my favorite ever since. You have really changed my perspective on a lot of physics and math stuff. Oh, and I'm also doing a bachelor's in physics and computer science, so I'm really loving your content. looking forward for more vids!
That's awesome to hear! Keep an eye out for more videos!
Pretty neat, thanks. The relationship between 1D and 2D was nice, but I’d appreciate a deeper dive.
I'm a drop out with only elementary education and I think that these videos even though a little chaotic (you going a bit off script at times) explain a lot in very simple terms so even blockheads like me can understand. Kudos!
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
This is a very good explanation for what complex numbers are. Now I understand their relation to the Cartesian plane better. Thanks from Kenya
Discovered your channel, and subscribed ! Keep up !
Welcome aboard!
Awesome explanation and great perspective. I agree the terminology makes it seem more complicated than it is
This was absolutely amazing. Beautiful description.🎉🎉🎉
Glad you liked it!
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
Great video very helpful. Definitely would enjoy a deeper dive onto complex numbers
I can tell whether my professor really understands what he is teaching or he just repeating what he memorized over the years, you definitely understand what you are teaching on a deep level keep up the good work!!
Love the video format. You explain things how i would explain it to someone. Just quick schetches, perfect way to explain.
An amazing start, I would love a little more breakdown before moving on. Regardless I greatly appreciate the video
Thank you ! I really like this but would love to see one more video on imaginary and complex numbers with more real-time examples 😁
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 💪
your teaching skills are amazing . love from italy
Really good explanation, a great help to visualizing! Thanks!
Minor typo (verbo? misspeak) at 7:58 "Take this 3 and multiply it by 1" should be "by negative 1"
yep check pinned comment!
wow! i thought i understoof complex numbers already but this really opened my mind. I'd love another video! subscribed to stay notified
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.
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.
Great video. Great topic. I will check out more of your videos.
Thanks for watching!
What a wonderfully insightful thumbnail! You've immediately clarified the way I think about i
Thanks for making these videos they're super helpful
Genuinely one of the best explanations.
Math teachers just show you how to solve equations step by step but don't explain the real-world applications sometimes.
when I first saw the thumbnail, I was like.."oh! like when we plot complex numbers the imaginary axis is at 90 degrees to the real axis...so that means we turn 90 degrees when we multiply a real number with I"...and yes that is what you explained
secondly, I would like to ask whether the graph that you plotted to show the switch between even and odd numbers should be that sort of a continuous one instead of lets say points because decimal numbers are neither odd nor even, the graph can only be plotted for integers.
loved the video!!!
Dude, you explain things better than my math teacher.
Thank you. That helped me picture what is going on better. I wish you could provide such an explanation, extended, for quaternions. IU'll watch out in case you do.
Yes please help with Fourier Transforms 🙌🏻
My E&M teacher was the first person to make any of it make sense. He calls it magic, one of my astronomy professors calls it "dark voodoo" and I hate that they do that tbh. It doesn't help make it more intuitive or understandable when my teachers call it magic and voodoo. I'm half expecting you to do the same though 😂
I got chills when I saw 'Caltech' in your shirt. It reminds me that I'm gonna apply for Caltech for the Fall 2026! Ambivalent emotion of both anxiety and happiness😌
best of luck! Caltech is awesome
Thank u Ali, it was eye opening. I would like to go deeper into this topic , so request u for another video on imaginary number before u jump to the fourier transform.
Thank you Dr. Ali. Your explanation of this is really cool and understandable. I was talking to my physics teacher and we were wondering if the angle in degrees is related to imaginary numbers. I felt that it might be. I'm waiting for more videos because they are really interesting. Greetings from Poland!
I love poland!! I've been to poznan, amazing food :)
@alithedazzling wow I'm from Poznań! What food do you have in mind?
@@RivloGruby pickle soup, goose leg, ox cheek, and of course the potato and cabbage perrogis! :)
Angles in degrees and radians are exactly related to complex numbers. Euler's formula states it quite well:
e^(iθ) = cos(θ) + i•sin(θ)
I don't want to write a long comment here at the moment, but I will if you want to know more.
Also, scaling and rotation by an angle... and complex numbers... are equivalent to 2×2 real matrices. (It's all the same stuff!)
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.
"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'?
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
@GMGMGMGMGMGMGMGMGMGM you've tied yourself in knot and have nothing to show for it.
We’re not reading your essay
@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
Imagine number can be imagined but can't be felt physically like happiness can be felt but never have physical appearance in real word right??... Beautiful
You just gave an excellent explanation and i learned many concepts that was not covered in college! I can even say, that i may even be able to apply this new knowledge to my own research!
Awesome to hear! That's why I make these videos!
Amazing Explanation!
I'd really love a video about transfer function of a system and the roots of it (poles and zeroes) and how to intuitively think about it. I'm an ECE engineer taking system dynamics course this semester.
Excellent ❤ I think we need a video series about stupid naming in math and physics .
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😢
your explanations are always awesome sir, i appreciate it.
Glad you like them!
holy shit, i understand imaginary numbers now. Thank you for this!! clicked much more as to why the concept of i is wanted/ needed in mathmatics
Really nice video! Please do more on complex number 😊 like complex numbers in AC systems 😅
Imaginary numbers definitely aren't imaginary to electrical engineers. We use imaginary numbers to represent associated orthogonal values. We were already using "i" for current, because we were already using "C" to represent capacitance, so we use "j" as the imaginary operator. Anode and cathode also have the opposite meanings compared to chemistry. Please don't try to understand it. "Complex number" is a more accurate description than "imaginary number". When this was first introduced early in the EE curriculum, the professor told a story, almost certainly apocryphal, of a mathematics symposium where a mathematician introduced the concept of imaginary numbers and stated with glee that finally, mathematicians have a pure mathematical construct that they could explore that couldn't be sullied by engineers, and an electrical engineer in the back of the room shouted, "I have the PERFECT application for this!"
Great video! The confusing part is that for the 180 degrees we put the minus("-") before the number (-3), and for 90 degrees we put an "i" after the number (3i). Better way would be to use another symbol like an mirrored L and put it in front of the number. Another idea would be a minus with the degrees written over the minus symbol. Would create much less confusion..:)
Thanks mate. Clear, concise and calculated. 🙏🏾
Quite amazing and new perspective. Thank you so much for sharing
I wish i had a teacher like you i am currently going through complex variable and circuit analysis course your videos are very helpful.. Jazakallah 😊
yes bro I never imagined numbers in this manner. A follow up video on imaginary numbers would be awesome
I like the way you explain these things! Fourier transform would be great.
You need to do a bit more “hand holding” between representing the real/imaginary axis with cos and sin and how those relate to eulers equation. For many seeing e^j*theta for the first time, they need the proof. Otherwise, great video!
yep agreed, will include that in my next video!
This is so great man. Subscribed. This has me thinking about a question. What if you and I wanted to rotate along the z axis, can we model a "z-shift" into a "3-dimensional number" using i? What would be the way to illustrate a 3-dimensional number (x, y, z) in a similar way to how you describe the shift from unidirectional to bi-directional? Or, you know what I am trying to say I think. My work is on the linguistics and cognitive sciences end of things but this is relevant to my work with dialectical and ideological functions of thought.
Quaternions.
Quaternions are basically the algebraic 3D equivalent of 2D complex numbers. If you like group theory, SO(2) and U(1) are basically multiplication by complex numbers with a radius of one unit from the origin... so it's just rotation. The SO(3) group is 3D rotations, which is isomorphic to multiplicarion by quaternions one unit in distance from the origin.
The complex numbers are also isomorphic to 2×2 real matrices (which also includes the split-complex numbers and dual numbers)
Quaternions are isomorphic to 3×3 real matrices... which are rotations and scaling in 3D Cartesian space.
Everything in Mathematics has several different names for historical reasons.
Awesome video Ali :)
Thank you for explaining the application of imaginary numbers:)
i have never seen a more intuitive explanation, thanks
Glad it was helpful!
Shows explanation for how complex numbers are displayed geometrically in relation to the real number line.
Though the title feels more misleading to the reason why it was called imaginary in the first place.
Pretty good though.
love your channel!
Appreciate the love!
There are stretchy numbers and there are spinny numbers, and complex numbers do both.
is that the queen is dead album? i love the smiths!!!
@@alithedazzling Yes it is! Definitely my favorite album of theirs :D
Thanks again for these intuitive approaches and explanations
For the criticizers, these videos are not meant to replace scholar courses but to give you a flavor or an interpretation or a visualization of what could be very abstracted!
Ali, this is exactly the way I am explaining complex numbers to my kids and they love it!
To go from 1 to -1 you have many ways: you go in one shot pi or you go in two steps same move by pi/2 (sqrt(-1)) or you go even 3 steps using cubic square root of (-1)
Also to go from 1 to 1 you may make 2pi move or twice a pi (-1)^2 or in 3 steps 2pi/3 which we usually call j (j^3=1)
Etc …
Best
your kids are lucky to have such a thoughtful parent!
The way your brain works fit mine, so that your explanations always resonate and I understand concepts I never could before.
bro we see differentials in masters(econ) and as solution of phase diagrams there are complex numbers. I was thinking about a intuitive reasoning and you my dear brother just nailed it. Thank you from bottom of my heart
im a tenth grade student, and this explanation made lots of sense to me. thanks for the video
Ok this was great, im in engeneering school and havé trouble understanding the use of FT so it would ne Nice to hear your take on it, thx for the video
Thanks, bro. Cool stuff last month has been introduced to the complex exponential of Euler identity. Is all about what you just explained now🎉🎉🎉 thanks
🤯🤯🤯... Best explanation... thank you
Glad it helped!