7:00 (When very good physicist are wrong, they are not wrong for silly reasons, but they are wrong for good reasons, and we can learn from their thinking.) I love it!
Einstein was not saying that QM was wrong, it gives such accurate answers it can't be wrong. He was saying that QM was incomplete. Bohr's dominant personality resulted in history recording that Einstein lost these early arguments in the interpretation of QM but many of the questions he raised are still unanswered and need to be resolved along with incompatibilities between QM and GR.
@@iiwi758 when it was in my recommendation I thought "this is something interesting to watch" But in comment section I had already liked the top 2 comments.
Change your major. Unless you're physics or stats or social science with stats then you already in place. Continue thinking. Be the genius you were meant to be.
@Arriaga Two El Perú es un país compuesto por un crisol de razas: mestizos, nativos, blancos, negros, asiáticos, etc. PERÚ: País de cultura milenaria y de todas las sangres.
gauss hated the name imaginary, because it's confusing.He suggested to use lateral, because the complex number are represented on the lateral axis unlike all other numbers.
He sounds right, but I am not to deep in Algebra to have an opinion. Maybe Imaginary has also some point of truth? The square root of -1 is impossible, so imagine being possible. B times that imaginary square root -1. But sure for me as a student, it would be made me so much clear if it wasn't called imaginary axis.
It's impossible in the usual high school restriction to real numbers, but then people starting to study complex numbers complain, because why study something that is impossible/imaginary, it creates a bad approach, not very open minded.
i can be easily defined as the product between to points in a Cartesian plain which is (0;1).(0;1)=(0;-1) looking at an Argand Gauss plain it would look like i.i=-1, so we can basically multiply to real points and obtain minus one
Who the fuck is that Harrison Ford? Why are people so obsessed with screen clowns and disregard quantum physics? I had to google to know who that guy is and was very disappointed to find out that it's yet another random film actor :(
@@eklipsegirl Relax. Some of us that like physics, also like movies. In fact some of us like physics because we picked it up from watching movies! Twister was one of the first movies i saw as a kid...I largely believe that this movie alone is one of the things that shaped my entire life to who I am today, being both a filmmaker and a hobbyist physicist.
@@eklipsegirl chill, mate, nobody's disregarding QM here. As a physics major who minored in other things, I love cinema. "Blade Runner" starring Harrison Ford was a bold film that dared to ask thought-provoking questions. "Interstellar" is another cerebral masterpiece that also deals with existential ideas as well as Theoretical physics, mainly General Relativity and higher dimensions. It truly sparked my interests again and inspired me to retake my Relativity course. Films have influenced many great physicists and engineers in real life, most notably Hyperspace in Superstring Theory and most recently the NASA's warp drive from "Star Trek" by Alcubierre.
@@VoidFame I'm not from the States, so unfortunately I'll be stopped at the border even though I'm trying to make Friends - because I don't have Money...
@@codeisawesome369 I see now. It's not an issue of tuition, but an issue of living arrangements. I wish you the best of luck if you decide that is your pursuit.
EXSCTLY ZHE DIDNT EXPLAIN AT ALL WHY WHY DOES TJE WAVEFUNCTION have imagonary i in there to begin with...it has partly to do with not being bale to have time move backwards but he doesn't get into that at all..
We don't need complex numbers it's just a simple way of taking into account things which change with regards to the period of sin function, like ac current, or for simplifying manipulations of vectors, which can be done without complex numbers but just in a nightmarishly complex way.
Pure Mathematics don't search for Its applications. Pure Mathematicians do mathematics for fun & they get pleasure doing it . For example - Group Theory was Invented for Fun . But later other people found its uses in Computer Science & Quantum Physics
Great video. I’m reading Ruel Churchill’s book on complex numbers and applications. I like his introduction. Instead of starting with the definition of i as the square root of negative one, i is introduced as part of a function that is necessary for certain equations (an ordered pair with certain, somewhat unusual mathematical properties). As, I read it, the fact that it turns out to be the square root of negative one is more a consequence of the definition , rather than the basic definition of i. It’s a subtle point, but that explanation sits better with me. Most modern books start with “i is the square root of negative one,” and that’s harder to get my head around than the more fundamental definition.
'I' is a solution of the equation: x^2+1=0,so we could take this equation as the generator of imaginary numbers, i.e. the positive square root of '-1' is the imaginary unit i=(0,1), an ordered pair.
This is actually so damn useful. I wish more instructors/professors/reference books approached the more abstract concepts from this perspective, as the majority of learners - particularly those who don't enjoy maths - will have a better chance at getting a complete and thorough understanding.
most of the reasons it is introduced like this is because it was used by him previously in the lecture, when talking about Motion in 1D, i think. i do agree that it is a good way at looking at i, and complex numbers as a whole instead of just defining it by itself
Exactly! That's how we should learn: why was a concept created and not 'here is a useful information to remember'. Things must be learned as they arose: out of necessity not possible utility. That's why I find it fascinating to read history which renders the present necessary or in hermeneutics (e.g. psychoanalysis) to grasp things from their fundamentals.
The most enlightening way to teach complex numbers is to show the student that from N to Z to Q to R to C is merely four different quotient set extensions designed to remove the obstructions to the inverse operations: subtraction, division, logarithms, and root extraction, respectively.
@@LarryD-ul3le, no, I do not mean limits. I mean logarithms. And tell me, what obstruction to an algebraic inverse operation did the quotient set extension to form quaternions remove?
@@JavedAlam24,not all polynomials have roots in R. While every non-constant polynomial will have roots in the complex numbers (according to the Fundamental Theorem of Algebra), some polynomials with real coefficients may have complex roots that are not on the real number line (meaning they are not in the set "R")
@@xyzct Hi there ! I'm curious as to why the step from Q to R corresponds to logarithms in your mind. I'd have said limit too, considering R is the set of all limits of cauchy sequence in Q
When I studied this subject 25 years ago, back on the engineering classes, I remember I got to understand the topic quite well as it was necessary to solve circuits problems. But I never got to use that on the real world, and now it is a "complex" concept for me. Anyway, I hope someday I have the time to brush up on my advanced maths.
I am an 11ᵗʰ grade student but hearing this necessity of complex numbers I loved it and it is also more understandable than taken by school teachers. It is my life dream to study in mit.
What a great introductory video. The professor is comfortably understandable and thorough.. Fantastic, short introduction to complex numbers and their importance. Thanks for posting!.
4:24.. yes we can work and get it.. McLaurin's series is one best way to use and prove that e^ix = cos x + i sin x.. U can enjoy proving it bcoz it gives a detailed and satisfying proof. I've done many times.. it's interesting.. 😊😊😊
@@RangerCaptain11A Yes, exactly, and that goes deeper than just needing them to solve equations. Complex numbers, "complete" the real numbers, in a strong sense. Even as needing them to solve equations opens the door to their existence. Fred
@@ffggddss 'Complete' or 'Completeness' Is it something empirically defined or more of a feeling ? Does Integer complete whole number ? The reason one part of it is called is called imaginary is because well it's really imaginary and came into being as a notational convenience for mathematicians. It's a great imagination and opens door to solve equations which could not be solved before.
@@pandit-jee-bihar Well, no, it wasn't meant in any formal sense. And yes, integers (ℤ) could equally well be said to "complete" counting (aka, natural) numbers (ℕ); as do rational numbers (ℚ) for integers; as do real numbers (ℝ) for rationals. And in each case, there's an in-built operation in the original system, that generates the extended one: • subtraction (inverse operation of addition) extends ℕ → ℤ • division (inverse operation of multiplication) extends ℤ → ℚ • limits of convergent sequences extends ℚ → ℝ • exponentiation extends ℝ → ℂ ( [-1]^½ , e.g.) The same could not be said for the quaternions, e.g. Perhaps the most compelling case for complex numbers is that, on the real line, not every differentiable function is analytic; in the complex plane, a function can't be differentiable without being analytic. Basically, in the complex plane, the constraints imposed by differentiability suddenly become much more stringent than those on the real line. Fred
I am now 30 years old and finally feel comfortable with and appreciate imaginary numbers. I hit a wall in impedance circuit analysis when I realized I didn’t understand imaginary numbers. Went back, relearned it and wow. So incredible that imaginary number not only express oscillations but quantum mechanics probabilities as well. It’s never too late to go back and relearn the fundamentals!
It is a tragedy that the terms _real_ and _imaginary_ were adopted to classify these numbers since: a) The origin of the terms was actually meant to be used as an insult to certain mathematicians (more in a moment), and b) It confuses students learning math who, through no fault of their own, assume the lay, or common, definition of imaginary, that being something that is “fantasy”, “make-believe” or “made up”, leading to a student’s understandable conclusion: _how can something that “doesn’t exist” be in any way useful? _ Good question! Origin and usefulness to follow, but first . . . . A side track in nomenclature . . . . In physics, there is a fundamental particle called a quark. There are 6 types of quark. These types are called flavors. The flavors of quarks are: up, down, bottom, top, strange and charmed. Why is one strange and the other charmed? Can you really taste them if they are called flavors? Nope - they are just names whose origins come from the imaginations of the physicists involved. (en.wikipedia.org/wiki/Quark#Etymology) *The origin of the terms Real Number and Imaginary Number* In the late 16th to early 17th century, when some mathematicians began developing the idea of the square root of negative numbers, other mathematicians were not too impressed. One prominent mathematician (and naysayer) of the day was Rene Descartes, who wrote, scathingly, "_These people play with their imaginary numbers while we mathematicians work with real numbers_." Herein lies the origin of both terms real and imaginary. Yes, before Descartes remark, the numbers we now call real numbers were not called real numbers by mathematicians, they were just called numbers!
This is excellent. One of my degrees is in Physics. I have a lot of math in my background. Complex Numbers were a necessary subject in order to do the math. The problem was that the concept of mapping complex numbers to a Cartesian Plane was just presented as a given, with absolutely no explanation why. "That's just the way it is." Dr. Zwiebach does a much better job of presenting the "why" than most professors. But the ultimate understanding for me occurred when I stopped and read the history of Rene' Descartes, one of the greatest mathematicians ever, and the reason we call this plane representation "Cartesian". If you get an understanding of Descartes's thought process and where the concept of Complex numbers comes from, you can think like a mathematician and not just depend on memorization.
complex numbers are fundamental in electrical engineering and pretty much anything that deals with waves because that angle gives you a way to represent the phase of the wave.
A great professor hints at things beyond what are being taught. @2:30, "It's actually zz*, a very fundamental equation". And with year's of math under my belt now, I'm like, "Oh, man, that is a huge deal." That you can use the multiplication of a complex number with it's conjugate to get a real number that is a squared norm and generates a measure on the space of C. Mind still blown (even though I know this stuff well). But the professor just moves on and leaves it lying there. Quietly acknowledging the importance, but knowing that it's a distraction from what needs to be taught. Bravo.
Fantastic, short introduction to complex numbers and their importance. Thanks for posting!. What a great introductory video. The professor is comfortably understandable and thorough..
I really love it we need this type of teacher in india because I always think why this is required to study and he had a very clear point of it with examples i love it never in my life i had this my clear view to any chapter a lots of love from india.
Because of this lecture, I now understand the foundation of cos x + i sin x, and also how "i" came to exist and it's usefulness. Never saw these explanations before.
Honestly, this was probably the best introduction to quantum mechanics i'ver ever heared. Before you get to this whole superposition shit and stuff, first explaining the fundamental maths behind it, which by all means isnt that hard to not teach it to students. Great job.
A great story telling. If taken too far you find an emperor with no clothes. Transverse waves, polarisation, 'double cover', etc. It was Heaviside who introduced complex number representations to EM theory, which was then applied to photon-atom interactions to get QM, etc. Lot's of conventions hiding tricky stepping stones to higher levels of understanding.
I'm Sido Rodrigues Brazil I really like Quantum Physics Classes. Very important to know quantum physics. Teach everything the universe knows and you gain self-knowledge about everything. Great series of really useful lectures on quantum mechanics. I am also very grateful to MIT OpenCourseWare and Barton Zwiebach... etc...
Complex numbers play a crucial role in quantum mechanics, serving as an indispensable mathematical framework that helps describe the behavior and properties of quantum systems. The state of a quantum system is represented as a vector in a complex Hilbert space, where the complex coefficients encode probabilities and amplitudes essential for making predictions about measurement outcomes. This formalism allows for the superposition of states, capturing the inherently probabilistic nature of quantum mechanics. Moreover, the use of complex numbers facilitates the application of linear algebra and differential equations, essential tools for analyzing quantum phenomena. The inclusion of complex wave functions and operators leads to a more elegant and comprehensive understanding of phenomena such as interference and entanglement, highlighting how complex numbers are not merely a mathematical convenience but rather a fundamental aspect of the quantum realm.
I agree, but perhaps it is the case that thinking of them in maybe a new more pure way as described by the professor gets some of the clutter and possible confusion about them out of mind.
At my school, introductory mathematics is a prerequisite for quantum mechanics, but I think complex numbers was still very briefly discussed. Always good to quickly put everyone on the same page to follow discussion, and emphasize some important math concepts that will connect to physics concepts later on.
@@flowerwithamachinegun2692 nah dude, i'm just wondering why these lectures videos keep appears in my reccommendation, yeah sometimes i do watch 3b1b/blackpenredpen/welch labs/ocw but most of the time, i watches family guy/simpsons/futurama/one piece/the last airbender, i just like to do it, i'm just not very sure why this video keeps appearing in my reccommendation
That’s the correct order of learning mathematics, congratulations. Once the idea that only positive number has a root square has been internalised it really is difficult to understand complex numbers and complex numbers are another level of mathematics and that is a real magic.
@@stephenbeck7222 Yes. Complex numbers were invented to ensure the *closure* property for solving quadratic polynomial equations, ensuring any of them will necessarily have 2 complex roots (which can be real numbers or not).
🙋🏻♂️ 5:00 are you talking about a block or cube with multiple vectors that when adding pressure or force and calculating the loss or back pressure, like calculating the inside vectors of a dice. Only 1-3-5 connect and everything else is a multiple of a possible angle, direction or distance of vector 💁🏻♂️
I like your observation. It's about the information density of this prof.'s words. An involved receiver is also a requirement. Eminem videos have more views.
All my life from ~ age 10, when in the presence of a good teacher I have always felt on the verge of understanding math, but have never had it quite click into place. Next year I turn 70, and am still trying, albeit with declining hope.
The "norm" is otherwise also denoted as the magnitude of the complex number vector. May I respectfully add that Z = cos(theta) + i * sin(theta) only if magnitude(Z) = 1. Complex number are used and have been used for a very long time in AC circuit theory. We can indeed very well measure complex numbers by simply measuring amplitude and phase of voltages, currents, field vectors.
I thought I was going to sleep, and here I am watching Harrison Ford talking about the importance of complex numbers in quantum mechanics. Sometimes life is strange
Many people who had bad or mediocre teachers turned out really well. Your future lies in your own hands, not in the hands of some teacher. All the tools are available; you choose to pick them up or walk away. If the "1999" part in your name reflects you year or birth, you are a measly 20 years old. Your life has hardly begun. You have all the opportunities ahead of you to "do well" in your life. Or, you know, you could just sulk on in the internet and blame others for not following your hopes and dreams.
thicc lumber who the fuck are you to judge someone as a failure, though? There are n factors that may cause one to get unmotivated and bad teachers is surely one of them.
Decent lecture, but it begs the question of the title of the video. He just states that they’re necessary, meanders around a few examples of how we’d be lost without imaginary numbers, but other than this necessity for their existence, doesn’t explain them.
the answer to your question lies in Geometric Algebra. The result of the work of Grassmann, Clifford, Hamilton and sort of rediscovered by Hestenes. It gives you a geometric interpretation for the equation i^2= -1. It can be associated with some plane in physical space.
@4:38 "Complex numbers, you used them in electromagnetism, you sometimes used them in classical mechanics, but you always used them in an _auxiliary_ way. It was not directly relevant because the electric field is real, the position is real, the velocity is real, everything is real. And the _equations_ are real. On the other hand, in quantum mechanics the equation *already has an i* . So in quantum mechanics, psi is a complex number. _Necessary_ . *It has to be* ." This lecture is from a course on quantum mechanics. The title of the video is apposite, concise, and absolutely correct.
I am first time meeting with Walter Levin in IIT Bombay and this time I see that professors is no difference between that!!! I love tham very much in this time I am in harverd in us I am very happy too
Adrien If you mean that the coordinates of z = 0 + i in a complex plane are (0,1) then you are right, but a graphical representation is not equal to the definition.
@@NLGeebee The way complex numbers are built is by defining two operations on the set R^2 : one addition and one multiplication. i is a notation for the couple (0,1) because it is easier two manipulate this notation. It really is just that. You can represent all complex numbers in R^2 because they are just R^2 with two operations !
Does anyone else feels like you understand these things way down in your career and it just went above your head when you were actually learning in college 😂?
It's no surprise. We hardly have any time in university to actually read the damn textbook because each professor assigns so many homework assignments each week plus the 3-4 lab reports to complete.
Do not confuse notation with deeper meaning. The equation can stated as a set of two equations of Re and Im parts and complex numbers do not have to be invoked. So, no complex numbers are not necessary but they simplify notation.
Still not really any closer as to why they're relevant other than describing a result that has no real solution. I've heard they have applications in electrical engineering, but I haven't studied them, sadly. Something I'll have to do in future.
That's what you're doing right now. Many of the big universities have full classes on youtube now and it's quite possible that you'll get more from those courses than the freshmen who pay top dollar to attend in person.
In Geometric Algebra (which is a development of Clifford Algebra), the unit imaginary is given a geometric interpretation that is extremely useful in formulating and solving mathematical problems that arise in a broad range of fields, including quantum mechanics. Our channel is mainly for lower-level users of GA, but some of the members of our associated LinkedIn group are GA experts, and will be happy to direct interested viewers to sources of additional information.
7:00 (When very good physicist are wrong, they are not wrong for silly reasons, but they are wrong for good reasons, and we can learn from their thinking.)
I love it!
Booom
Love it too. Very inspiring!
When something sounds right but is also actually right, great quotes are made.
stfu brainlet
Einstein was not saying that QM was wrong, it gives such accurate answers it can't be wrong. He was saying that QM was incomplete. Bohr's dominant personality resulted in history recording that Einstein lost these early arguments in the interpretation of QM but many of the questions he raised are still unanswered and need to be resolved along with incompatibilities between QM and GR.
See you all next year when the algorithm brings us back
Andrew Chen Hahaha had to laugh so hard!
Hi bud
It actually did. Interesting.
I will be reliving this comment in 6 months time again
@@iiwi758 when it was in my recommendation I thought "this is something interesting to watch"
But in comment section I had already liked the top 2 comments.
I am not mathematics major. But whenever I watched videos regarding math it brings Peace in heart. I don't know why
That makes you a mathematician
That's because mathematics is a gift from god
Change your major. Unless you're physics or stats or social science with stats then you already in place. Continue thinking. Be the genius you were meant to be.
Music of the Spheres...
Well you feel nostalgia, that's why.
Barton Zwiebach is peruvian. He was born and studied school and electical engineering in Lima, Peru. As a peruvian I feel so proud of him.
Que orgullo, saludos desde chile
Wow, increíble que haya llegado tan lejos!
Peruvian flake cocaine 👍
Ya me di cuenta de que su acento al hablar inglés es de hispanohablante, pero ni su nombre ni su apellido son de hispanohablante...
@Arriaga Two El Perú es un país compuesto por un crisol de razas: mestizos, nativos, blancos, negros, asiáticos, etc. PERÚ: País de cultura milenaria y de todas las sangres.
I had to learn this by distance education (1992) before the internet and always struggled. Barton makes it seem so easy. What a fabulous lecturer.
ruclips.net/user/shortssixxTjNBYoU?si=UDpyjWGKfGCqZ7Ic
Up
gauss hated the name imaginary, because it's confusing.He suggested to use lateral, because the complex number are represented on the lateral axis unlike all other numbers.
He sounds right, but I am not to deep in Algebra to have an opinion. Maybe Imaginary has also some point of truth? The square root of -1 is impossible, so imagine being possible. B times that imaginary square root -1.
But sure for me as a student, it would be made me so much clear if it wasn't called imaginary axis.
It's impossible in the usual high school restriction to real numbers, but then people starting to study complex numbers complain, because why study something that is impossible/imaginary, it creates a bad approach, not very open minded.
@@ridovercascade4551 'imaginary' numbers are as imaginary as negative numbers.
i can be easily defined as the product between to points in a Cartesian plain which is (0;1).(0;1)=(0;-1) looking at an Argand Gauss plain it would look like i.i=-1, so we can basically multiply to real points and obtain minus one
@@ridovercascade4551 Actually imaginary numbers are real, so it's better call them "lateral"
As a peruvian I feel proud of Barton, he is the best student of the National Engineering University in Lima Peru
Who cares.
@@sciencedaemon tf someone asked for your opinion cpp
@@sciencedaemon a tu mujer le importa
@@luismarcelosantacruzmujica3130 If I write in English why are you replying in another language, fool?
It cares to your mom silly@@sciencedaemon
I clicked because I thought he was a young Harrison Ford.
Now I know how complex numbers are crucial part of wavefunctions in quantum.
Who the fuck is that Harrison Ford? Why are people so obsessed with screen clowns and disregard quantum physics? I had to google to know who that guy is and was very disappointed to find out that it's yet another random film actor :(
@@eklipsegirl yeah useless actor
@@eklipsegirl Relax. Some of us that like physics, also like movies. In fact some of us like physics because we picked it up from watching movies!
Twister was one of the first movies i saw as a kid...I largely believe that this movie alone is one of the things that shaped my entire life to who I am today, being both a filmmaker and a hobbyist physicist.
@@eklipsegirl chill, mate, nobody's disregarding QM here. As a physics major who minored in other things, I love cinema. "Blade Runner" starring Harrison Ford was a bold film that dared to ask thought-provoking questions. "Interstellar" is another cerebral masterpiece that also deals with existential ideas as well as Theoretical physics, mainly General Relativity and higher dimensions. It truly sparked my interests again and inspired me to retake my Relativity course. Films have influenced many great physicists and engineers in real life, most notably Hyperspace in Superstring Theory and most recently the NASA's warp drive from "Star Trek" by Alcubierre.
@@eklipsegirl calm the fuck down you pseudo intellect
If I ever strike it rich, I would love to go to MIT to study physics at leisure with amazing teachers like this...
Tbh you could just go for free and not get credit. Make friends not money :p
@@VoidFame I'm not from the States, so unfortunately I'll be stopped at the border even though I'm trying to make Friends - because I don't have Money...
@@codeisawesome369 I see now. It's not an issue of tuition, but an issue of living arrangements. I wish you the best of luck if you decide that is your pursuit.
@@VoidFame Thank you! :-) Have a great week ahead.
@@VoidFame What! You can?
I’m actually an engineer but this is the first time I understand why we really need the complex numbers
Thank you sir !
He didn't really get to a full explanation but it was a good start...a couple of identity equations doesn't explain at least for me
EXSCTLY ZHE DIDNT EXPLAIN AT ALL WHY WHY DOES TJE WAVEFUNCTION have imagonary i in there to begin with...it has partly to do with not being bale to have time move backwards but he doesn't get into that at all..
We don't need complex numbers it's just a simple way of taking into account things which change with regards to the period of sin function, like ac current, or for simplifying manipulations of vectors, which can be done without complex numbers but just in a nightmarishly complex way.
Started with x^2 + 1 =0. People were not used to with these kind of equations
Pure Mathematics don't search for Its applications. Pure Mathematicians do mathematics for fun & they get pleasure doing it . For example - Group Theory was Invented for Fun . But later other people found its uses in Computer Science & Quantum Physics
I am a teacher at the beginning of my career. That was a very inspiring explanation.
you teach in university?
Teach geometric algebra then, not complex numbers.
This brings me back to the good old days of engineering school. Ironically I miss it. I felt so sharp in my mathematical skills.
damn bruh u said that 5 yrs ago and im js now in gr12 tryna make it into engineering in uni
MIT has the best blackboard erasers, that big boy is a dream.
Great video. I’m reading Ruel Churchill’s book on complex numbers and applications. I like his introduction. Instead of starting with the definition of i as the square root of negative one, i is introduced as part of a function that is necessary for certain equations (an ordered pair with certain, somewhat unusual mathematical properties). As, I read it, the fact that it turns out to be the square root of negative one is more a consequence of the definition , rather than the basic definition of i. It’s a subtle point, but that explanation sits better with me. Most modern books start with “i is the square root of negative one,” and that’s harder to get my head around than the more fundamental definition.
'I' is a solution of the equation: x^2+1=0,so we could take this equation as the generator of imaginary numbers, i.e. the positive square root of '-1' is the imaginary unit i=(0,1), an ordered pair.
This is actually so damn useful. I wish more instructors/professors/reference books approached the more abstract concepts from this perspective, as the majority of learners - particularly those who don't enjoy maths - will have a better chance at getting a complete and thorough understanding.
most of the reasons it is introduced like this is because it was used by him previously in the lecture, when talking about Motion in 1D, i think. i do agree that it is a good way at looking at i, and complex numbers as a whole instead of just defining it by itself
Agreed!
Exactly! That's how we should learn: why was a concept created and not 'here is a useful information to remember'. Things must be learned as they arose: out of necessity not possible utility. That's why I find it fascinating to read history which renders the present necessary or in hermeneutics (e.g. psychoanalysis) to grasp things from their fundamentals.
The most enlightening way to teach complex numbers is to show the student that from N to Z to Q to R to C is merely four different quotient set extensions designed to remove the obstructions to the inverse operations: subtraction, division, logarithms, and root extraction, respectively.
You mean limits, not logarithms. And you should really continue on to quaternions.
@@LarryD-ul3le, no, I do not mean limits. I mean logarithms.
And tell me, what obstruction to an algebraic inverse operation did the quotient set extension to form quaternions remove?
@@xyzctisn't root extraction applicable to both R and C though?
@@JavedAlam24,not all polynomials have roots in R. While every non-constant polynomial will have roots in the complex numbers (according to the Fundamental Theorem of Algebra), some polynomials with real coefficients may have complex roots that are not on the real number line (meaning they are not in the set "R")
@@xyzct Hi there ! I'm curious as to why the step from Q to R corresponds to logarithms in your mind. I'd have said limit too, considering R is the set of all limits of cauchy sequence in Q
Fantastic, short introduction to complex numbers and their importance. Thanks for posting!
When I studied this subject 25 years ago, back on the engineering classes, I remember I got to understand the topic quite well as it was necessary to solve circuits problems. But I never got to use that on the real world, and now it is a "complex" concept for me. Anyway, I hope someday I have the time to brush up on my advanced maths.
Oh jee aspirant!
I am an 11ᵗʰ grade student but hearing this necessity of complex numbers I loved it and it is also more understandable than taken by school teachers. It is my life dream to study in mit.
I like his writing, elegant
What a great introductory video. The professor is comfortably understandable and thorough.. Fantastic, short introduction to complex numbers and their importance. Thanks for posting!.
MIT is MIT. It is always absolute. Thank you, MIT
ruclips.net/video/XQIbn27dOjE/видео.html 💐💐💐
4:24.. yes we can work and get it..
McLaurin's series is one best way to use and prove that
e^ix = cos x + i sin x..
U can enjoy proving it bcoz it gives a detailed and satisfying proof.
I've done many times.. it's interesting.. 😊😊😊
I thought that was Harrison Ford in disguise
Yeah right?
Harrison Ford + Benedict Cumberbatch
ROFL
Lol I had the same thought "wtf is Harrison Ford doing at MIT????"
Dude! Tot's! 🤣
He summed it all up with the statement that "complex number was needed to solve equations". That's it!
That's a shit reason my nigga
शास्त्र ध्वनि - Recorded Scriptures
right, imaginary numbers are for completeness. that is a huge reason.
@@RangerCaptain11A Yes, exactly, and that goes deeper than just needing them to solve equations. Complex numbers, "complete" the real numbers, in a strong sense.
Even as needing them to solve equations opens the door to their existence.
Fred
@@ffggddss
'Complete' or 'Completeness' Is it something empirically defined or more of a feeling ?
Does Integer complete whole number ?
The reason one part of it is called is called imaginary is because well it's really imaginary and came into being as a notational convenience for mathematicians.
It's a great imagination and opens door to solve equations which could not be solved before.
@@pandit-jee-bihar Well, no, it wasn't meant in any formal sense.
And yes, integers (ℤ) could equally well be said to "complete" counting (aka, natural) numbers (ℕ); as do rational numbers (ℚ) for integers; as do real numbers (ℝ) for rationals.
And in each case, there's an in-built operation in the original system, that generates the extended one:
• subtraction (inverse operation of addition) extends ℕ → ℤ
• division (inverse operation of multiplication) extends ℤ → ℚ
• limits of convergent sequences extends ℚ → ℝ
• exponentiation extends ℝ → ℂ ( [-1]^½ , e.g.)
The same could not be said for the quaternions, e.g.
Perhaps the most compelling case for complex numbers is that, on the real line, not every differentiable function is analytic; in the complex plane, a function can't be differentiable without being analytic.
Basically, in the complex plane, the constraints imposed by differentiability suddenly become much more stringent than those on the real line.
Fred
Well, finally, on the 27th year of my life I realized the physical sense of the complex number :)
I am now 30 years old and finally feel comfortable with and appreciate imaginary numbers. I hit a wall in impedance circuit analysis when I realized I didn’t understand imaginary numbers. Went back, relearned it and wow. So incredible that imaginary number not only express oscillations but quantum mechanics probabilities as well. It’s never too late to go back and relearn the fundamentals!
It is a tragedy that the terms _real_ and _imaginary_ were adopted to classify these numbers since:
a) The origin of the terms was actually meant to be used as an insult to certain mathematicians (more in a moment), and
b) It confuses students learning math who, through no fault of their own, assume the lay, or common, definition of imaginary, that being something that is “fantasy”, “make-believe” or “made up”, leading to a student’s understandable conclusion: _how can something that “doesn’t exist” be in any way useful? _
Good question! Origin and usefulness to follow, but first . . . . A side track in nomenclature . . . .
In physics, there is a fundamental particle called a quark. There are 6 types of quark. These types are called flavors. The flavors of quarks are: up, down, bottom, top, strange and charmed.
Why is one strange and the other charmed?
Can you really taste them if they are called flavors?
Nope - they are just names whose origins come from the imaginations of the physicists involved. (en.wikipedia.org/wiki/Quark#Etymology)
*The origin of the terms Real Number and Imaginary Number*
In the late 16th to early 17th century, when some mathematicians began developing the idea of the square root of negative numbers, other mathematicians were not too impressed. One prominent mathematician (and naysayer) of the day was Rene Descartes, who wrote, scathingly, "_These people play with their imaginary numbers while we mathematicians work with real numbers_." Herein lies the origin of both terms real and imaginary. Yes, before Descartes remark, the numbers we now call real numbers were not called real numbers by mathematicians, they were just called numbers!
ruclips.net/video/XQIbn27dOjE/видео.html 💐💐💐
interesting as heck.
This is excellent. One of my degrees is in Physics. I have a lot of math in my background. Complex Numbers were a necessary subject in order to do the math. The problem was that the concept of mapping complex numbers to a Cartesian Plane was just presented as a given, with absolutely no explanation why. "That's just the way it is." Dr. Zwiebach does a much better job of presenting the "why" than most professors. But the ultimate understanding for me occurred when I stopped and read the history of Rene' Descartes, one of the greatest mathematicians ever, and the reason we call this plane representation "Cartesian". If you get an understanding of Descartes's thought process and where the concept of Complex numbers comes from, you can think like a mathematician and not just depend on memorization.
complex numbers are fundamental in electrical engineering and pretty much anything that deals with waves because that angle gives you a way to represent the phase of the wave.
Very nice short note on complex numbers. A great professor tells you much more than just writing down those dry equations.
A great professor hints at things beyond what are being taught. @2:30, "It's actually zz*, a very fundamental equation". And with year's of math under my belt now, I'm like, "Oh, man, that is a huge deal." That you can use the multiplication of a complex number with it's conjugate to get a real number that is a squared norm and generates a measure on the space of C. Mind still blown (even though I know this stuff well). But the professor just moves on and leaves it lying there. Quietly acknowledging the importance, but knowing that it's a distraction from what needs to be taught. Bravo.
Very pure very clear very quality lecture series on QM and QFT...!❤💜❤
Fantastic, short introduction to complex numbers and their importance. Thanks for posting!. What a great introductory video. The professor is comfortably understandable and thorough..
Men of curiosity ,we meet again
sure
🎉🎉🎉awesome 🎉🎉🎉
absolutely
I really love it we need this type of teacher in india because I always think why this is required to study and he had a very clear point of it with examples i love it never in my life i had this my clear view to any chapter a lots of love from india.
ruclips.net/video/XQIbn27dOjE/видео.html 💐💐
Finally i know the use of complex no. ❤️❤️❤️ From India, Bihar.
Jai Bharat!!!!🇮🇳🇮🇳🇮🇳🇮🇳🇮🇳🇮🇳🇮🇳🇮🇳
Because of this lecture, I now understand the foundation of cos x + i sin x, and also how "i" came to exist and it's usefulness. Never saw these explanations before.
In a rational K-12 education system this would be 10th or 11th grade material, latest.
El legendario Barton de la UNI
I hope students today appreciate how lucky they are to be sitting in a classroom learning this.
From Bangladesh, very Helpful class😊❤️❤️
Best chalkboard handwriting I have ever seen
How couldn't I thumbs on this lecture. Thanks professor
Such good quality education to millions around the globe... *Claps claps*
Honestly, this was probably the best introduction to quantum mechanics i'ver ever heared. Before you get to this whole superposition shit and stuff, first explaining the fundamental maths behind it, which by all means isnt that hard to not teach it to students. Great job.
A great story telling. If taken too far you find an emperor with no clothes.
Transverse waves, polarisation, 'double cover', etc. It was Heaviside who introduced complex number representations to EM theory, which was then applied to photon-atom interactions to get QM, etc.
Lot's of conventions hiding tricky stepping stones to higher levels of understanding.
I'm Sido Rodrigues Brazil I really like Quantum Physics Classes. Very important to know quantum physics. Teach everything the universe knows and you gain self-knowledge about everything. Great series of really useful lectures on quantum mechanics. I am also very grateful to MIT OpenCourseWare and Barton Zwiebach... etc...
Complex numbers play a crucial role in quantum mechanics, serving as an indispensable mathematical framework that helps describe the behavior and properties of quantum systems. The state of a quantum system is represented as a vector in a complex Hilbert space, where the complex coefficients encode probabilities and amplitudes essential for making predictions about measurement outcomes. This formalism allows for the superposition of states, capturing the inherently probabilistic nature of quantum mechanics. Moreover, the use of complex numbers facilitates the application of linear algebra and differential equations, essential tools for analyzing quantum phenomena. The inclusion of complex wave functions and operators leads to a more elegant and comprehensive understanding of phenomena such as interference and entanglement, highlighting how complex numbers are not merely a mathematical convenience but rather a fundamental aspect of the quantum realm.
Sometimes you are wrong for the right reasons. Life of a scientist!
Sir, you don't know how grateful i am to you ! May the One True God bless you.
What a great introductory video. The professor is comfortably understandable and thorough.
Utube algorithm is so confusing it's suggesting complex numbers video and stand up comedy all together
Obviously MIT students already know what are Complex Numbers
you'd be surprised.
Yes
@Non sum dignus I had to 'cause it's freaking MIT!!!!
I agree, but perhaps it is the case that thinking of them in maybe a new more pure way as described by the professor gets some of the clutter and possible confusion about them out of mind.
At my school, introductory mathematics is a prerequisite for quantum mechanics, but I think complex numbers was still very briefly discussed. Always good to quickly put everyone on the same page to follow discussion, and emphasize some important math concepts that will connect to physics concepts later on.
Watching this amazing explanation in year 2024. Love from India 🇮🇳
i should really stop binge watching science videos when i keep saying "alright just one more yt videos, and i will sleep after this"
r/iamverysmart ?
@@flowerwithamachinegun2692 nah dude, i'm just wondering why these lectures videos keep appears in my reccommendation, yeah sometimes i do watch 3b1b/blackpenredpen/welch labs/ocw but most of the time, i watches family guy/simpsons/futurama/one piece/the last airbender, i just like to do it, i'm just not very sure why this video keeps appearing in my reccommendation
To show the physical quantity of a complex number you find the norm. How enlightening!
These are cool studies. The professor has a nice clear-cut way of explaining without overemphasizing the simpler parts of the mathematics.
Totally agree. Clear and concise. He knows his stuff.
Oh that was very good. Brought a lot of stuff together nicely.
I took a lesson on complex numbers before I took any trig, Calc. I didn't know you could use i to solve polynomials. That's incredible...
That’s the correct order of learning mathematics, congratulations. Once the idea that only positive number has a root square has been internalised it really is difficult to understand complex numbers and complex numbers are another level of mathematics and that is a real magic.
The primary purpose of complex numbers in algebra is to solve polynomials. How do you solve x^2+1=0 without i?
@@stephenbeck7222 Yes. Complex numbers were invented to ensure the *closure* property for solving quadratic polynomial equations, ensuring any of them will necessarily have 2 complex roots (which can be real numbers or not).
After all those years and progress, chalk and black board are still the most efficient and effective way to present, brainstrorm, teach... hilarious.
I clicked because i thought it was harrison ford teaching mit class
🙋🏻♂️ 5:00 are you talking about a block or cube with multiple vectors that when adding pressure or force and calculating the loss or back pressure, like calculating the inside vectors of a dice. Only 1-3-5 connect and everything else is a multiple of a possible angle, direction or distance of vector 💁🏻♂️
Man, this professor is good!
Thanks for the contribution. Make remember My times in Electrical Circuits with Samer Teacher.
When you're smart like these professors you can convey as much information speaking slowly as eminem rapping
I like your observation. It's about the information density of this prof.'s words.
An involved receiver is also a requirement.
Eminem videos have more views.
@@dozog Exactly, and very true.
@@jaacobb123 If the old adagium that repetition is the mother of learning is true, then rappers may still be great teachers.
All my life from ~ age 10, when in the presence of a good teacher I have always felt on the verge of understanding math, but have never had it quite click into place. Next year I turn 70, and am still trying, albeit with declining hope.
Why are you telling us that you never had the intention of paying attention to mathematics? ;-)
Life is complex. It has real and imaginary components.
Reading Spengler and his explication of the various "mathematics" of different Cultures. This helps
Wow , such an amazing explanation, thanks lot
The "norm" is otherwise also denoted as the magnitude of the complex number vector. May I respectfully add that Z = cos(theta) + i * sin(theta) only if magnitude(Z) = 1. Complex number are used and have been used for a very long time in AC circuit theory. We can indeed very well measure complex numbers by simply measuring amplitude and phase of voltages, currents, field vectors.
he said the unit circle
Who says physics is boring...It's actually amazing...just see these lectures.!!absolutely Amazing.!"
I thought I was going to sleep, and here I am watching Harrison Ford talking about the importance of complex numbers in quantum mechanics. Sometimes life is strange
I love how the chalk boards move up and down like window sashes.
You know it's MIT when every blackboard moves up and down like a window sash. That's reeel quality there.
The shear comedy that he starts by pointing out i which is right next to h bar is incredible.
if i had teachers like these, maybe i would not have hated math so much and actually done well in my life.
Even now it's not too late.
Many people who had bad or mediocre teachers turned out really well. Your future lies in your own hands, not in the hands of some teacher. All the tools are available; you choose to pick them up or walk away.
If the "1999" part in your name reflects you year or birth, you are a measly 20 years old. Your life has hardly begun. You have all the opportunities ahead of you to "do well" in your life.
Or, you know, you could just sulk on in the internet and blame others for not following your hopes and dreams.
Very encouraging :)
Tomahawk1999 ur a failure because of urself and nobody else, but sure blame ur teacher if it makes u feel better
thicc lumber who the fuck are you to judge someone as a failure, though? There are n factors that may cause one to get unmotivated and bad teachers is surely one of them.
The necessity begins at 4.40 in this video. Simple and great.
ruclips.net/video/XQIbn27dOjE/видео.html 💐💐💐
4:40
Decent lecture, but it begs the question of the title of the video. He just states that they’re necessary, meanders around a few examples of how we’d be lost without imaginary numbers, but other than this necessity for their existence, doesn’t explain them.
"Other than the necessity for their existence" that is the title of the video
probably a 1.5 hour class, so most of the content is missing.
You're asking for something that's not within the context of the title of the video...
the answer to your question lies in Geometric Algebra. The result of the work of Grassmann, Clifford, Hamilton and sort of rediscovered by Hestenes. It gives you a geometric interpretation for the equation i^2= -1. It can be associated with some plane in physical space.
@4:38 "Complex numbers, you used them in electromagnetism, you sometimes used them in classical mechanics, but you always used them in an _auxiliary_ way. It was not directly relevant because the electric field is real, the position is real, the velocity is real, everything is real. And the _equations_ are real.
On the other hand, in quantum mechanics the equation *already has an i* . So in quantum mechanics, psi is a complex number. _Necessary_ . *It has to be* ."
This lecture is from a course on quantum mechanics. The title of the video is apposite, concise, and absolutely correct.
Good teacher!!! Barton Zwiebach
"Once you invent i you don't need more numbers."
Quaternion: "Am I a joke to you?"
Steven Van Hulle ... to solve any polynomial equation
I am first time meeting with Walter Levin in IIT Bombay and this time I see that professors is no difference between that!!! I love tham very much in this time I am in harverd in us I am very happy too
Except that i ≠ √-1.
By definition i² = -1, so if anything i = ±√-1.
No by definition i=(0,1)
Adrien If you mean that the coordinates of z = 0 + i in a complex plane are (0,1) then you are right, but a graphical representation is not equal to the definition.
@@NLGeebee The way complex numbers are built is by defining two operations on the set R^2 : one addition and one multiplication. i is a notation for the couple (0,1) because it is easier two manipulate this notation. It really is just that. You can represent all complex numbers in R^2 because they are just R^2 with two operations !
Adrien I believe I was tought that if the complex number z = i, or z = 0 +1i, then that number z is represented in R² as the point (0,1).
@@NLGeebee That is indeed true !
In electrical engineering we use 'j', we call that cartesian graph an s-plane, and Z is impedance. Cool stuff!
Does anyone else feels like you understand these things way down in your career and it just went above your head when you were actually learning in college 😂?
It's no surprise. We hardly have any time in university to actually read the damn textbook because each professor assigns so many homework assignments each week plus the 3-4 lab reports to complete.
Do not confuse notation with deeper meaning. The equation can stated as a set of two equations of Re and Im parts and complex numbers do not have to be invoked. So, no complex numbers are not necessary but they simplify notation.
He just derived de moivre's theorem!!! Holy cow I never noticed that.
No, he didn't...
He derived Euler's formula.
@@SuperSaltyFries He derived nothing. What's wrong with u people..?
G.O.A.T. explanation of complex numbers!
what kind of harrison ford is this?
The one that is starring in the upcoming math adventure film “Raiders of the Lost Quark”
Indiana Jones at University
Imaginary
So complex numbers are necessary because the equation has i in it. Great explanation...
Still not really any closer as to why they're relevant other than describing a result that has no real solution. I've heard they have applications in electrical engineering, but I haven't studied them, sadly. Something I'll have to do in future.
Knowledge should for knowledge only but not exami orientation The teacher is teaching very nicely
This is really well presented! This guy ought to be teaching at MIT or the like.
Oh, wait, ...
Fred
Flfoilm
obrigado por legendarem esse vídeo, gostei demais.
While the smartest of the smart get into schools like MIT, Harvard, etc., I’d love to go there I would sit in on lectures like this for fun.
If you work hard enough you can get into a good uni in the us with comparable education although you may have a large debt to loans
That's what you're doing right now. Many of the big universities have full classes on youtube now and it's quite possible that you'll get more from those courses than the freshmen who pay top dollar to attend in person.
He is speaking slowly and clearly so people can connect a lot of dots beyond mere computation.
Good refreshing course... I make living on I... I call it "j" part of a number ...for me I stands for current, I m an electrical engineer.
the imaginary unit is denoted *j* in electrical engineering since *i* denotes current
Good prefessors are the world most valuable resources, at lease before large language models
"GET TO THE CHOPPER !!!" 😂😂😂😂
When I'm not learning math and physics, I study classical Greek. So I love that he pronounces the "P" in Ψ.
ok?
Technically, it's #Π
In Geometric Algebra (which is a development of Clifford Algebra), the unit imaginary is given a geometric interpretation that is extremely useful in formulating and solving mathematical problems that arise in a broad range of fields, including quantum mechanics. Our channel is mainly for lower-level users of GA, but some of the members of our associated LinkedIn group are GA experts, and will be happy to direct interested viewers to sources of additional information.
الراجل ده عالي فشخ ❤️
What am I doing here? I can barely sum.
Great video in every respect, please keep them comming.