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.
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”.
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.
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 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.
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!
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
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.
I really liked this video.. But i have a question that.. " As complex no. [i] make it rotate by 90 so can there be another kind of no. Which can make it rotate by 60? "
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!
Always had questions about topics like this. But my teachers would always tell me i wasn't ready to grasp the full concept, and that what we learn in highschool is enough to make us go by
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!!
if it is a rotation by 90degrees, do we need another set of complexity for 45 degrees, for example. If we can work with just imaginary numbers, ie, 90degree rotations, to get to the 45 degree rotation, why cant we just work with negative numbers to get to 90 degree rotation since negative numbers are 180 degree rotations?
With only positive and negative numbers, you have a 1-dimensional number plane, so the only angles you can use are 0° and 180°, but when you add imaginary numbers and their combinations with real numbers, you now have a 2-dimensional plane where you can find all possible angles between 0° and 360°. So the answer lies in how many dimensions you can reach with the number system in use
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 😂
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
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.
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
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!)
Today I needed to pick up an elective for spring of 25. It was either engineering ethics, patent law, or intro to systems engineering. The choice is obvious. Only problem? It starts at 8 AM🥶
Imaginary numbers are not consistent. All numbers are proven corelates of each other if you can place them all on the same axis since they share a factor, which is possible if you allow imaginary numbers, which are not always real. This is impossible, since coprime numbers must be shown to exist for consistent phenomenas such as magnetism or electricity.
Roots are not the only problem with the reals. Irrational numbers are part of the reals but nowhere to be pinpointed on the reals. It could only be approximated to some degree of accuracy, but never could they be precisely represented. Mathematicians live inside the brain world until some reality hits them hard. They live by definitions and very strict when it comes to closure. They don’t like surprises. So when they define an operation on a set, they make sure that the product is closed under the operation. When they defined roots of real numbers, they were stunned to find that the results didn’t land them on the island of the reals. The square root of negative numbers didn’t land on the real line, so to them, the mathematicians, that was the equivalent of a ghostly act until someone pointed out the concept of rotation, that is, numbers could be represented not only on a straight line, but also in more expanded directions, which give rise to vector spaces. Had mathematicians first started with that numbers are quantities that posses both magnitude and direction, they would have not fallen into the trap of “imaginary or complex” numbers. However, they were so used to their field axioms model that they couldn’t fit the surprise on a straight line. The concept that there are infinite count of numbers that are equal in magnitude but different in orientation is a more pleasant concept philosophically. While equal somehow, they are nevertheless different! Defining a metric on the space in which such numbers exist would have solved the problem and we perhaps didn’t have to smirk at the “imaginary or complex” notion of numbers. Without Euler, the mystery of rotation would have not been unraveled.
This is definitely a cool way to look at it especially from an engineering perspective. To me the name imaginary unit is still a great name because all other numbers are tangible things in the real world, however the square root of -1 is not tangible at all yet it has a solution and so i have to imagine this number i
ّI think of it as a vector with real numbers as an x axis and imaginary number as an y axis. But we can only measure the x axis which is why it is real. But thinking of i as a 90 degree feels more natural with the multiplication. Nice Video! Thx!
Yes, plotting i (the square root of -1) on the vertical axis while the horizontal axis is reserved for Real numbers is a way of making sense of the square root of -1. This view opens up a large and very consequential branch of mathematics, namely complex analysis. But, to me at least, it's also perfectly reasonable to call i an imaginary number, since logically -1 does not have a REAL square root. Many of the commenters seem to feel that the explanation presented here suddenly makes sense of an area of mathematics that they previously considered indecipherable. I remember this exact same explanation being presented by my teacher at a small country high school in southern Ohio in the early 1960's. Well-presented then and now. Dazzling? Maybe not.
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.
Once it is realised that the minus sign in front of a number (for a negative number) is actually a mathematical OPERATION, (i.e. times -1) on the number tings become much clearer. In the same way, "i" or "j" in a number represents an OPERATION on the number, and the only way to get it back into the real world is to use the operation on it a second time.
Negative numbers make much more sense as vectors... because they are vectors. (+1) and (-1) are unit vectors, which you then scale with a counting number. This is also why adding two negative numbers gives you a negative "number" as the answer, but multiplying two negative numbers gives you a _positive_ "number". It's because vector multiplication is rotation. Negative (so-called) numbers are really a subset of the complex numbers. It's not the other way around, like how it gets taught in school. The big revelation to me was when I learned group theory and realized that the "number line" is isomorphic to the rotations of a two sided polygon... "+1" and "-1" being the two symmetries of a line. (And the complex numbers are isomorphic to the symmetric rotation of a four sided polygon (you know, like a square)). Also, you know how a slide ruler works? Logarithms and exponents transform multiplication and addition problems into addition and subtraction (and vice versa). That's why multiplication of Complex Numbers is equivalent to adding together the angles between the two Complex Numbers and the real axis. (n^a)×(n^b) = n^(a+b) remember that? i•i•i•i=i⁴=1=90°+90°+90°+90°=0° or 360° i•i=i²=-1=90°+90°=180° i¹=1×90° i²=2×90° i³=3×90° i⁴=4×90° This is why everyone's favorite equation 1+e^(πi)=0 or more generally, e^θi = cos(θ) +i•sin(θ) works.
I always used to wonder that imaginary numbers are really imaginary as its name tells or do they exist in real world and if it is imaginary then why we are learning this but then i learned that they do exist the actual confusion comes because of its name. By the way brilliant explanation.
Why can't we just use a regular grid to plot everything like for the example given of the chalk, the points given between even and odd numbers can just be represented on a regular grid can it not?
just a small thing: 7:20 I don’t think you need to write the i on the axis. Sicne that axis represent Imaginary part, you only need to write the real part there and the axis implies it is multiply with i.
My understanding of complex numbers (which is not that great) is that there are some legitimate roots that you can't solve without an i. So what I thought is these usually involve a greater than second degree function with an inversion in trend. As per the idea that complex numbers describe oscillators, yes they do, and thus they are used in quantum mechanics.
Ali, to fully understand the 2d rotational aspect of imaginary numbers, doesn't one have to factor in a trigonometric component into the i-rotational feature on the field? I don't know if "field" is the correct word to describe the backdrop. You are an excellent teacher.
I like your videos, but your audience is at a higher level than your level of explanation. You could have done the first 5 minutes of this video is 1 min. I like your stuff and am subscribed. Keep it comin'
What? Do you think every engineering student intuitively knows those things? A lot of engineers learn stuff they never truly understand. This kind of video realigns fundamental knowledge, which then solidifies and makes all of the knowledge built on that base exponentially easier to contextualize.
Amazing -- to think you could map a DateTime value to a 4-valued Season value if you embed the Season in a complex domain. I wonder if there's a free complex domain you could use, like the degrees on a circle idea.
I think you could continue on y = x^{2} + 1, because I feel left over. Here's my thought, if we define x as real number, we can conclude that there is no root from this equation. however, if we define x as complex number, now you have saddle function. (too bad I can't attach image here) when you looking for real number solution or im(x) = 0, the function not intersect with zero plane but if you allow complex solution in another word, allow x to be complex, the are some point the y = x^2 + 1 intersect with zero plane. BTW in your video make me think that we can intrinsicly interplete that root of x^{2} = -1 us i. that very impressive as I never doubt it before.
This is terrible. He dropped the Y in his algebraic explanation of i, and his jump into engineering from algebra made no sense whatsoever. And lastly, that's not how are number line or negative numbers work.
Also imaginary number is connected to "negative areas" on geometry as this Veritasium video suggests: ua-cam.com/video/cUzklzVXJwo/v-deo.html It's amazing to understand what are those mathematical concepts are really about.
Hey Ali since real-imaginary axis is 2 dimensional is there a way to have imaginary-imaginary as a 3rd axis. I searched online and found it wasnt possible but thats where quarternions come into play (4d). Im currently a physics student but i’d enjoy a video on explaining what these are
I think another video on imaginary numbers as a follow up would be amazing
Agreed
i agree
I agree
agreed
Bring it on
7:59 I meant to write a negative sign in front of the one :)
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!
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.
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.
Exactly what I needed to understand Circuit Analysis 2😂..thanks Ali
I'm glad it helped!
That's awsome Ali! A second video about imaginary numbers would be great, especially on why they are so useful
Noted!
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”.
Good one Ali. Greetings from Brazil
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.
Thank you! This was super helpful. I would really appreciate a part 2 that dives deeper into complex numbers.
Dude, you explain things better than my math teacher.
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
Hey Ali, it would be great to have another video on "imaginary" numbers !! ( 13:38 )
Yes
Yes pls!
Thanks Ali this video was awesome! Definitely do a more deep dive into complex numbers
Thanks Dr Ali . I’m just doing imaginary numbers in electrical engineering class
Good luck with your class!
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.
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.
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
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
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.
Pretty neat, thanks. The relationship between 1D and 2D was nice, but I’d appreciate a deeper dive.
I really liked this video.. But i have a question that.. " As complex no. [i] make it rotate by 90 so can there be another kind of no. Which can make it rotate by 60? "
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
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!
Thank you for explaining the application of imaginary numbers:)
Quaternions finally make sense
What a wonderfully insightful thumbnail! You've immediately clarified the way I think about i
If something is named "imaginary" it must be imaginary.
Always had questions about topics like this. But my teachers would always tell me i wasn't ready to grasp the full concept, and that what we learn in highschool is enough to make us go by
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!!
Great video very helpful. Definitely would enjoy a deeper dive onto complex numbers
Love the video format. You explain things how i would explain it to someone. Just quick schetches, perfect way to explain.
if it is a rotation by 90degrees, do we need another set of complexity for 45 degrees, for example. If we can work with just imaginary numbers, ie, 90degree rotations, to get to the 45 degree rotation, why cant we just work with negative numbers to get to 90 degree rotation since negative numbers are 180 degree rotations?
great video btw, very helpful, would love if you could answer my question!!
With only positive and negative numbers, you have a 1-dimensional number plane, so the only angles you can use are 0° and 180°, but when you add imaginary numbers and their combinations with real numbers, you now have a 2-dimensional plane where you can find all possible angles between 0° and 360°. So the answer lies in how many dimensions you can reach with the number system in use
An amazing start, I would love a little more breakdown before moving on. Regardless I greatly appreciate the video
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 😂
Happy because I understood everything without pausing
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 like how 2/3 of our subatomic structure is heading one direction and 1/3 is heading the other direction.
HOLY SCHNITZLE !! You're putting every single math teacher I've ever had to shame.
This was an ok math video for young people who are curious about engineering and applied maths
A video about imagibary number will be amazing
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.
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
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!)
A great adaptive application, thanks.
Great explanation! Little bit more on complex numbers before moving to Fourier Transformations
Today I needed to pick up an elective for spring of 25. It was either engineering ethics, patent law, or intro to systems engineering. The choice is obvious. Only problem? It starts at 8 AM🥶
So, would a 45° rotation equivalent to i/2?
It kinda makes sense until you realize that 2i isn't actually equivalent to 180°
so meybe √i will do the job for ya.. idk
Imaginary numbers are not consistent. All numbers are proven corelates of each other if you can place them all on the same axis since they share a factor, which is possible if you allow imaginary numbers, which are not always real. This is impossible, since coprime numbers must be shown to exist for consistent phenomenas such as magnetism or electricity.
4:54 students loans so infamous 😂
U DEFINED i VERY INTERESTINGLY( i mean real world thinkin and all )
BUT ALL U DID IS EULER FORM OF i
Truly an excellent explanation!…a big thanks and a sub…
Roots are not the only problem with the reals. Irrational numbers are part of the reals but nowhere to be pinpointed on the reals. It could only be approximated to some degree of accuracy, but never could they be precisely represented. Mathematicians live inside the brain world until some reality hits them hard. They live by definitions and very strict when it comes to closure. They don’t like surprises. So when they define an operation on a set, they make sure that the product is closed under the operation. When they defined roots of real numbers, they were stunned to find that the results didn’t land them on the island of the reals. The square root of negative numbers didn’t land on the real line, so to them, the mathematicians, that was the equivalent of a ghostly act until someone pointed out the concept of rotation, that is, numbers could be represented not only on a straight line, but also in more expanded directions, which give rise to vector spaces. Had mathematicians first started with that numbers are quantities that posses both magnitude and direction, they would have not fallen into the trap of “imaginary or complex” numbers. However, they were so used to their field axioms model that they couldn’t fit the surprise on a straight line. The concept that there are infinite count of numbers that are equal in magnitude but different in orientation is a more pleasant concept philosophically. While equal somehow, they are nevertheless different! Defining a metric on the space in which such numbers exist would have solved the problem and we perhaps didn’t have to smirk at the “imaginary or complex” notion of numbers. Without Euler, the mystery of rotation would have not been unraveled.
beautiful 🎉
This is definitely a cool way to look at it especially from an engineering perspective. To me the name imaginary unit is still a great name because all other numbers are tangible things in the real world, however the square root of -1 is not tangible at all yet it has a solution and so i have to imagine this number i
love your channel!
Appreciate the love!
I am simple man.
i see ali video, i click.
“everyone kind of claps and moves on and, like, nobody understands what the hell” yeah, that’s math class in a nutshell
ّI think of it as a vector with real numbers as an x axis and imaginary number as an y axis.
But we can only measure the x axis which is why it is real.
But thinking of i as a 90 degree feels more natural with the multiplication.
Nice Video! Thx!
مفيد جدا
عاشت ايدك
كثرلنا هيج مفاهيم 🤍
you got it boss
I cant wait for your fourier transform video
Yes, plotting i (the square root of -1) on the vertical axis while the horizontal axis is reserved for Real numbers is a way of making sense of the square root of -1. This view opens up a large and very consequential branch of mathematics, namely complex analysis. But, to me at least, it's also perfectly reasonable to call i an imaginary number, since logically -1 does not have a REAL square root. Many of the commenters seem to feel that the explanation presented here suddenly makes sense of an area of mathematics that they previously considered indecipherable. I remember this exact same explanation being presented by my teacher at a small country high school in southern Ohio in the early 1960's. Well-presented then and now. Dazzling? Maybe not.
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.
Once it is realised that the minus sign in front of a number (for a negative number) is actually a mathematical OPERATION, (i.e. times -1) on the number tings become much clearer. In the same way, "i" or "j" in a number represents an OPERATION on the number, and the only way to get it back into the real world is to use the operation on it a second time.
Negative numbers make much more sense as vectors... because they are vectors. (+1) and (-1) are unit vectors, which you then scale with a counting number.
This is also why adding two negative numbers gives you a negative "number" as the answer, but multiplying two negative numbers gives you a _positive_ "number". It's because vector multiplication is rotation.
Negative (so-called) numbers are really a subset of the complex numbers. It's not the other way around, like how it gets taught in school.
The big revelation to me was when I learned group theory and realized that the "number line" is isomorphic to the rotations of a two sided polygon... "+1" and "-1" being the two symmetries of a line. (And the complex numbers are isomorphic to the symmetric rotation of a four sided polygon (you know, like a square)).
Also, you know how a slide ruler works? Logarithms and exponents transform multiplication and addition problems into addition and subtraction (and vice versa). That's why multiplication of Complex Numbers is equivalent to adding together the angles between the two Complex Numbers and the real axis. (n^a)×(n^b) = n^(a+b) remember that?
i•i•i•i=i⁴=1=90°+90°+90°+90°=0° or 360°
i•i=i²=-1=90°+90°=180°
i¹=1×90°
i²=2×90°
i³=3×90°
i⁴=4×90°
This is why everyone's favorite equation 1+e^(πi)=0 or more generally, e^θi = cos(θ) +i•sin(θ) works.
Pretty insightful
i has just now shown up as a complex permitivity and complex refractive index in optics. Crazy how it just shows up everywhere.
I always used to wonder that imaginary numbers are really imaginary as its name tells or do they exist in real world and if it is imaginary then why we are learning this but then i learned that they do exist the actual confusion comes because of its name. By the way brilliant explanation.
thanks bro, i learned something new
Why can't we just use a regular grid to plot everything like for the example given of the chalk, the points given between even and odd numbers can just be represented on a regular grid can it not?
It’s “3 x -1” for 180 degree rotation, at 8:01.
just a small thing:
7:20 I don’t think you need to write the i on the axis. Sicne that axis represent Imaginary part, you only need to write the real part there and the axis implies it is multiply with i.
Another video on complex numbers before going into forrier transform
More complex numbers coming soon!
We need more
Amazing, let's delve in it more, the get to Fourier
Learnt something new,. Thank you
Glad you found it helpful!
Deep dive into complex numbers please
My understanding of complex numbers (which is not that great) is that there are some legitimate roots that you can't solve without an i. So what I thought is these usually involve a greater than second degree function with an inversion in trend. As per the idea that complex numbers describe oscillators, yes they do, and thus they are used in quantum mechanics.
If i = 90 degrees does that mean -1 = 180 degrees? No, this is a very casual way of looking at numbers.
sorry man, not trying to be creepy but you are a handsome ass dude. 🙈
Ali, to fully understand the 2d rotational aspect of imaginary numbers, doesn't one have to factor in a trigonometric component into the i-rotational feature on the field? I don't know if "field" is the correct word to describe the backdrop. You are an excellent teacher.
I like your videos, but your audience is at a higher level than your level of explanation. You could have done the first 5 minutes of this video is 1 min. I like your stuff and am subscribed. Keep it comin'
even if only 20% are not at high level, id rather build from the ground up so those guys get it as well -- but thanks for the feedback!
What? Do you think every engineering student intuitively knows those things? A lot of engineers learn stuff they never truly understand. This kind of video realigns fundamental knowledge, which then solidifies and makes all of the knowledge built on that base exponentially easier to contextualize.
deeper dive (:
if we can visualize math in 2 dimension's with the help of "i" then what do we need to visualize math in 3 dimension's ?
Quaternions, _i, j, k_ or you can just use a 3×3 matrix. (You can use a 2×2 matrix in place of _i_ in a complex number too.)
lol- all numbers are imaginary. if you disagree tell us how much mass the number 5 has.
That a different way to think of complex numbers. You just taught part of complex analysis in simpler form.
Amazing -- to think you could map a DateTime value to a 4-valued Season value if you embed the Season in a complex domain. I wonder if there's a free complex domain you could use, like the degrees on a circle idea.
Dude this is so cool
Great video. However, the "math way" is way cooler...😅😅😅
I think you could continue on y = x^{2} + 1, because I feel left over.
Here's my thought,
if we define x as real number, we can conclude that there is no root from this equation.
however, if we define x as complex number, now you have saddle function. (too bad I can't attach image here)
when you looking for real number solution or im(x) = 0, the function not intersect with zero plane but if you allow complex solution in another word, allow x to be complex, the are some point the y = x^2 + 1 intersect with zero plane.
BTW in your video make me think that we can intrinsicly interplete that root of x^{2} = -1 us i. that very impressive as I never doubt it before.
for the unit circle in a complex plane (x^2 + y^2 = 1), one usually sees y=i when x=0. shouldn't y=i^2 given that i^4=1.
Phenomenal 👏🏽
Another video before fourier, sorry but you lost me at 11:08, like maybe it's because example but how do you go from 9 to 7.5 on circle/why invoked?
Yes, go for complex first and then F. T.
This is terrible. He dropped the Y in his algebraic explanation of i, and his jump into engineering from algebra made no sense whatsoever. And lastly, that's not how are number line or negative numbers work.
abs(i)=1 with arg(i)=pi/2
Also imaginary number is connected to "negative areas" on geometry as this Veritasium video suggests: ua-cam.com/video/cUzklzVXJwo/v-deo.html
It's amazing to understand what are those mathematical concepts are really about.
Hey Ali since real-imaginary axis is 2 dimensional is there a way to have imaginary-imaginary as a 3rd axis. I searched online and found it wasnt possible but thats where quarternions come into play (4d). Im currently a physics student but i’d enjoy a video on explaining what these are
Well done!