I love the way he writes something down, turns to the audience and smiles. I can easily imagine his disappointment when he sees that only few students really understand why he's smiling - because the result is beautiful. Fundamentalism of this theorem is not in dealing with some stuff we've just invented and named "complex numbers", it's fundamental because we don't need any other stuff, we are at the end of arithmetics, everything we can build from these numbers has roots described by only these numbers. It's a very beautiful theorem and it deserves to be shown to everyone.
I love how he does really care about making sure everybody is seeing the board and focusing, this lecture gonna help me to prepare for my high school exams
Introduction to Complex Numbers ● [0:14]. Introduction to complex numbers. ● [0:33]. Notation for real numbers (ℝ) and complex numbers (ℂ). ● [1:12]. Definition of a complex number: z ∈ ℂ means that z = a + ib, where a and b are real and i² = -1. ● [1:42]. Real part of a complex number: Re(z) = a. ● [2:01]. Imaginary part of a complex number: Im(z) = b. Basic Operations with Complex Numbers ● [2:18]. Addition of complex numbers: (a + ib) + (c + id) = (a + c) + i(b + d). ■ The addition of complex numbers is commutative. ● [3:14]. Multiplication of complex numbers: (a + ib)(c + id) = (ac - bd) + i(ad + bc). ■ The multiplication of complex numbers is also commutative. ● [4:31]. Division of complex numbers, z = (a + ib) and w = (c + id) : ■ The conjugate of the denominator is used: (c + id)(c - id) = c² + d². ■ (a + ib) / (c + id) = [(a + ib)(c - id)] / (c² + d²). ■ The expression is simplified to obtain the real and imaginary parts of the result. ● [6:40]. Division is defined as long as the denominator is not zero (w ≠ 0). ● [7:18]. Equality of complex numbers: Two complex numbers are equal if and only if their real and imaginary parts are equal. ● [7:43]. Conjugate of a complex number: z = a + ib, then z̅ = a - ib. ■ The product of a complex number and its conjugate is a non-negative real number: zz̅ = a² + b². ● [9:29]. Real part of z: Re(z) = (z + z̅) / 2. ● [9:39]. Imaginary part of z: Im(z) = (z - z̅) / 2i. ● [10:12]. Properties of the conjugate: ■ (z + w)̅ = z̅ + w̅. ■ (zw)̅ = z̅w̅. ■ (z / w)̅ = z̅ / w̅ (if w ≠ 0). Complex Numbers and Polynomials ● [11:49]. Motivation for complex numbers: They arise from the need to find roots of polynomials that do not have real solutions. ■ Example: The polynomial x³ + 2x² + 2x + 1 has a real root at x = -1. ■ The other two roots are complex: -1/2 ± i√3 / 2. ● [14:45]. Fundamental Theorem of Algebra: Every polynomial of degree n has n complex roots (counting multiplicities). ● [18:17]. Theorem (unnamed): If all coefficients of a polynomial are real, then: ■ The polynomial can be factored into linear terms corresponding to the real roots (z - α₁, z - α₂, ..., z - αᵣ). ■ And into quadratic factors corresponding to pairs of complex conjugate roots: (z - γ₁)(z - γ̅₁), (z - γ₂)(z - γ̅₂), ..., (z - γₛ)(z - γ̅ₛ). ● [22:15]. Consequence: If a polynomial has real coefficients and γ is a complex root, then its conjugate γ̅ is also a root. Polar Form of Complex Numbers ● [23:26]. Graphical representation of complex numbers: ■ The horizontal axis represents the real part (x). ■ The vertical axis represents the imaginary part (y). ● [24:46]. Polar form: A complex number z = x + iy can also be represented in polar coordinates (r, θ). ■ r: modulus of z, denoted as |z|, is the distance from the origin to z: |z| = √(x² + y²). ■ θ: argument of z, denoted as arg(z), is the angle between the positive real axis and the line connecting the origin to z. ● [26:46]. The argument is undefined for z = 0. ● [27:22]. Special cases of the argument: ■ If y > 0 and x = 0, then arg(z) = π/2. ■ If y < 0 and x = 0, then arg(z) = 3π/2. ● [27:58]. The argument is periodic: θ, θ + 2π, θ - 2π, θ + 4π, etc., define the same argument. ● [28:59]. The principal value of the argument is in the interval 0 ≤ θ < 2π. ● [29:57]. Properties of the conjugate in polar form: ■ |z̅| = |z|. ■ arg(z̅) = -arg(z). ● [30:47]. Relationships between Cartesian and polar coordinates: ■ x = r cos(θ). ■ y = r sin(θ). ● [31:27]. Proposition: ■ |zw| = |z||w|. ■ |z/w| = |z| / |w| (if w ≠ 0). ■ |z|² = zz̅. ■ arg(zw) = arg(z) + arg(w) (if z ≠ 0 and w ≠ 0). ■ arg(z/w) = arg(z) - arg(w) (if z ≠ 0 and w ≠ 0). ● [35:35]. No information is provided about the modulus or the argument of z + w in this lesson. ● [35:49]. Triangle inequality: |z + w| ≤ |z| + |w|. Proofs of Some Properties ● [36:41]. Some proofs are omitted, but can be found in the course's online notes. ● [36:49]. Example of proof: (zw)̅ = z̅w̅. ● [39:58]. Example of proof: if a polynomial has real coefficients and γ is a root, then γ̅ is also a root.
@@vincentvejdovsky2795 First, I provided the video to NotebookLM along with the complete transcription and timestamps. Then, I asked it to create a professional summary for a UA-cam video. Then, I provided the result to ChatGPT and asked it to correct the indentation as well as any errors in the theory. Finally, I reviewed and adjusted the timestamps better because the result isn’t perfect, as well as corrected any typographical errors.
It's sad that if you think about it, school and university lectures deprive you of the independent discovery of mathematical concepts. On the same day they introduce a problem, they reveal the solution immediately. I suppose that in the long run people learn faster when they're taught, and that the real skill isn't in finding the solutions but rather asking the questions, but still, I'd like it if the standard was so that people could get the satisfaction of realising it without being told.
I agree. This is not real math, it is just passively watching somebody else doing something. Moreover, everybody in the class, including the professor, thinks this is the way to learn. University is such a scam, paying thousands for this crap??!! And people still do it in 2024-25! Better off learning on your own. Edit: In addition, observe that profs never actually even teach you HOW to think in maths. They just mumble theory at the board and stare at their notes. I thought they were good at maths, why the need for notes?. AND once again, we foolish people pay thousands for this??!! It makes no sense!
The Imaginary axis is in units of i, i.e. 1i, 2i, 3i, etc. A distance i in the Im direction is drawn the same length as a distance 1 along the Re axis.
@@pelasgeuspelasgeus4634in the name of keeping things simple, I'm going to be a little loose with definitions and say that i = sqrt(-1) Typically it can be said that |sqrt(x)| = sqrt(|x|) In this case |-1| = 1 so |sqrt(-1)| = sqrt(|-1|) = sqrt(1) = 1 So it's not really arbitrary, just a result of its definition.
You're not studying it. You're merely memorizing what is i^2, geometric representation and basic equations. This lecture is not a study either. First lecture on a given topic serves as a reminder or introduction to establish common nomenclature.
I Blood BOILs to see him explain arguement and triangle inequality without explaining the geometrical significance of complex nos.! GO and learn that first!
Is that necessarily a plane ? Obviously , it is an assertorical object that you are symbolizing with ‘ i ‘ . There are ‘ imaginary sets ‘ that that logically generate topologies that don’t necessarily respect the metric topology of the plane .
That ‘ s one theme . There ‘ s also the consideration that it ‘ s not a vector space in particular : it ‘ s a field of scalars .. Self similarity is not a theme that ought to be butchered . Do any two nonexistent abstract objects ( by property ) define a plane ? : here , that is to say a 2 dimensional system. As an audio engineer , when I see that space of numbers , I ‘ m thinking how are we going to model sound amplitudes … It could be an ‘ invention ‘ to design a plane model . There is a question of apodicity . It ‘ s a canonical representation that imaginary numbers are required for Quantum mechanics … That ‘ s not a priori the case . What exactly is the implication with a plane ? Lines are certainly interesting now : I am not always attentive to my having represented them beneath awareness .
@solidpixel Entire mathematics is standing on axioms and assumptions. Ok, for a moment let's say it's not an assumption. What's the basis for the formulation of this number? What compulsions were there?
I think this is an excellent question that doesn't get explicitly discussed enough; this is by no means obvious. Why do we need at least a+bi? - Because we need the Real numbers to be part of the new set ( using b=0 we have this) - and of course we need the imaginary part - But what why we need to define i^2 = -1? Historically, complex numbers were introduced by Cardan as far back as the 16th century. While people did not really understand them at the time , introducing this famous notation a+b√−1 served in intermediate computations for finding explicit solutions to cubic equations (with real solutions!). Also, it turns out that they have this wonderful property that every polynomial (real or complex) can be factored into a product of complex polynomials of degree one. This means that complex numbers are enough to study any polynomial with real coefficients.
I have never seen a joke like this even real numbers are taught in grade 9 and you are talking about complex number I bet you don't even know the value of i
It still is, but usually in the first year of university they focus on making sure everyone is on the same level of knowledge before introducing more complex ideas.
@@nodeathuif those indians can understand the same thing. Lol they think they learnt this in high school and Cambridge just teaches this so they are superior in some sense, even though they are just bollocks down to the drain.
Can someone explain where is the location of the imaginary axis with respect to the 3 real axes that define our world? I mean the world has 3 real axes, right? Latitude, longitude and height.
Imaginary axis IS the y axis!! That’s the beauty of complex no.! It takes us from 1D (numbers like 1234...) to 2D world (x+iy) Maths is a language. so we can say that... vectors and complex numbers and geometry and matrix are like synonyms you know? To represent the same coordinates in diff ways
n^2 - (n-1)^2 = n^2 - (n^2 - 2n +1) = n^2 - n^2 + 2n -1 = 2n-1 where n is an element of the set of integers or alternatively 2n+1 if we start with (n+1)^2 - n^2
I have struggled to understand the why (why do we need these numbers). They say it is because certain equations can only be solved "in a different dimension". And they "lift" a 2D curve into a third dimension. But that is just a 3rd dimension.
We have that there exist any root for the complex numbers and it belongs to the complex numbers, but we don't have the same thing in any other subset: 2 is an integer and a rational, but the root is just real, so you need to extend to the reals in order to find any root of 2. Then -1 is an integer, a rational number, a real number, but the root is not real, so you just suppose to have i=sqrt(-1), meaning that i^2=-1 and that's called imaginary. You just introduce the operations with other real numbers and you get z=a+ib complex. (Complex: Consisting of interconnected or interwoven parts; composite of real and imaginary) Then it depends in which field you want to study these numbers, but in algebra you have the amazing property that allows you to write any polynomial equation as a product of complex factors of degree 1 or 0: x^2+x+1=(x+1/2-i*sqrt(3)/2)(x+1/2+i*sqrt(3)/2) This can be useful in some cases. Then there are some integrals that you can solve thanks to the complex numbers, such as the integral of sin(x)/x. You can't graduate in mathematics, physics or engineering without studying these numbers.
@@vortanoise.2625 thanks, I'm ashamed to admit I have a PhD (in immunology). However, I got obsessed with quantum mechanics, and there you ABSOLUTELY need to understand these numbers. It's not easy: "Then -1 is an integer, a rational number, a real number, but the root is not real, so you just suppose to have i=sqrt(-1), meaning that i^2=-1 and that's called imaginary." The square root of -1 simply shouldn't exist
@@jaimetorres950it is hard to understand, but it's powerful abstraction and expansion of expression. Remember, even the notion of negative numbers was strange at some point. I know qbit is described with a complex function. Also mandelbrot set, fractal, where a dimension becomes a real number, like 1.14 😂. This is what the notion of a complex number enables.
Fancy paying all that money to study at Oxford and be lectured to by this bloke who's obviously just a nervous wreck who can't even get sign rules right!
@ Had thermodynamics lectures where the lecturer would just put up page after page of derivations. What I found when I lectured is you can put up the first couple of lines of a derivation and then you might have an integral, so you can talk about what you could do to solve that integral,then proceed todo it.
If in india, you were given the introduction to complex numbers in this type, then you would be lost to see the fundamental theorem of algebra in a first introductory lecture... And c'mon, this is Oxford , and the professor knows what and how to teach!
This is probably the worst lecture I have ever witnessed. Back to the audience...reading from something that should hae been passed out ahead of time...insufficient explanation of processes...Oxford should be better than this.
People criticizing a Introduction, yes, a introduction. Straight bs comments he cant even have notes. I have meet extraordinary mathematicians that used lecture notes for their classes specially in the introductory ones since they don’t want to miss anything.
Most courses (though not this one) comprise 16 one-hour lectures. That's a lot to deliver when it might not be the subject of your research and has to be 100% spot on, hence the notes which are meticulously prepared. And it needs to tie in with the problem sheets that the students are given after the lecture. This way of teaching may not be to everyone's liking, but it does take the students through each step of the learning. The students then go on to tutorials in pairs with their tutor. Thanks for the feedback.
@@OxfordMathematics ooops ... not his research😅 But seriously, I believe it is important to present lectures without any notes in absolute clarity (or as close as you can get). It shows the students that it is not a regurgitation but a logical flow of ideas and concepts. In my own experience, looking at lecture notes means that you have turned off your brain and you are now the puppet of your notes. Anyway, I think people are just surprised to see this from a good university
You'd think that he wouldn't have to read this lecture from notes given that it's all such basic stuff......I'm pissing off to Cambridge to do maths.....to hell with Oxford!!!
Lost me after a minute. Like complex or simple numbers where your not actually using a number but letters confuses me. I’m sure he’s right but whats the actual practical use for this in the real world? Other than to justify a lecturers salary?
It's practically essential for many fields of engineering (largely because of their use in the study of aerodynamics, and fluid dynamics in general). Electrical theory uses them. Generally, physics relies heavily on them. Plus, they're useful in statistics and analytic finance.
It's a recap in the first week. We have students from high-schools all over the world. This is to make sure they are all up to speed. There are over 100 more lectures on our UA-cam Channel. Thanks for the feedback.
@@PauloConstantino167 Not every topic is tested to its full extent in the final examinations for many examination boards. By getting everyone up to speed, everyone is starting on the same level for this topic.
Bad « introduction » because all the historic, pragmatic, geometric and theoric justification of « i^2=-1 » is totally skipped. That’s the core, the heart, the spirit. The rest, more or less presented here is « trivial » consequences, that should not be exposed before the existence and pragmatic feeling of « the core thing » is made clear. It’s a perfect example of upside down « pedagogy », i.e. anti pedagogy
i kind of agree, although that should have been taught in highschool. nonetheless, i am currently 2nd year in mecatronics engineering, almost learning delayed differential equations, and to be honest, i have no clue of the usefulness of complex numbers... maybe you can share some videos/documentaries? or even written documents, i haven't found anything related to the "origin" (i.e. historic pragmatic etc justification...) thanks :)
@@rAdiAntOn How on Earth is that possible! « Complex numbers » have slowly emerged from Cardan initial discovery of an « illegal » arithmetic/algebraic « trick » useful to solve some cubics that he couldn’t solve otherwise. He noticed indeed that even if sqrt(-121), which sprung out « naturally » or « naively » in one of those, had no apparent arithmetic « legality », it was nevertheless useful as a intermediary « trick », since in the final result, such « monstrous » illegal quantity disappeared anyway by « conjugation » of the type : (a+sqrt(-121))(a-sqrt(-121)) = a^2 + 121. Such trick allowed him to crack cubics that couldn’t be solved without! That was the first hint and piece of the puzzle. But this puzzle took three century to be fully understood, by gradual steps, even though very soon after Cardan, the Italian Bombelli makes extensive use of it in algebra and equation solving, like quadratics whith negative determinant. But things remained very obscured in the « closed » arena of Algebra. The decisive step out, was to shift from algebra and equations, to (planar) geometry and transformations. Argan had the genius idea that this weird arithmetic « extraterestrial » and somehow still « imaginary » illegal object that was long written sqrt(-1), before Euler named it « i » (for « imaginary », which shows how it was seen for long), had in fact the same property as a half turn planar rotation, which square makes a central flip. That was the breakthrough since what it actually meant is that people opened their mind and start understanding that this -1 which equals to i^2, might not be a « usual number », or at least that it could be seen as something else than a « number » : namely a 180° planar ROTATION. Thus as a GEOMETRIC TRANSFORMATION, instead, or complementary to a « number » opposite to unity! That was the flash of lightening in Yeats poem : « The waste land »! The mysterious egg was cracked. The rest was the history of his actual birth in human mind with further steps of maturation. And finally Hamilton fully understood that what had been there discovered, was in fact a hidden multiplication on vectors that remained hidden from the « night of time », but that was there from the very beginning, and which is a omnipresent pattern used daily by Mother Nature : ROTATION. This for long « mysterious i » (of « Shiva »), was in some deep regards a positive quarter turn. And by such the building brick of any planar rotation. But to fully grab that the « exponential map » was useful. And from that, complex numbers, which are in general, rotations of points on the unit circle, composed with dilation/contraction, became the natural algebraic tool to model any waves, any circular motion, any cyclic phenomenon. It’s thus everywhere in Physics and engineering : electricity, optic, EM, electronic, mechanical, QM, etc. And even if it seems restricted to 2D. That’s because waves generally obey the superposition principle and are vectors. Which can thus be decomposed on 2D basis by projections. But there was still needs for similar algebraic representation of rotations in 3D (and higher dim), as the « complex numbers » were the god given perfect simple tool in 2D representation of Rotations and Dilations. Such tool Hamilton kept seeking for 10 years before he realised that they might be none in 3D, but one similar one in 4D. And they are indeed the « Quaternions », which behave much like « complex numbers », and that are « surprisingly » very useful to represent 3D rotations, in a much better way than classic « Euler angles » which are a bad and dangerous map. But that was not at all the end of this fascinating story, since two contemporary of Hamilton, Grassmann and Clifford, made even greater breakthrough. Especially Clifford, who discovered a deeper structure that unifies all of the previous ones and which works in ANY dimension. His work gave a new complementary understanding of the old « imaginary i ». Since he understood that better than a extended « number » which squares to minus one, better that a quarter turn positive rotation, better than an « Hamilton pair » (i.e. a 2D « vector »), the deepest way to understand such a universally useful badly named « imaginary » i, was to see it as an ORIENTED UNIT SURFACE. Grassmann and Clifford thus generalised the extension of planar vectorial geometry in which « complex numbers » revealed the multiplicative hidden structure that actually exist among vectors, to all dimensions. Points and Vectors were no more the exclusive stars of Geometry! Now areas, volumes, hypervolumes, were fully part of the algebraico-geometric arena. But in a unified revolutionary way. Scalars could be now add to vectors, and areas, volumes, etc, all together in a unified algebraic new calculus called « Clifford Algebra ». That was the most important discovery since Euclide. It revolutionized the « Euclidian geometry » by giving unified algebraic tools to generalized geometry in any D. Dot product and « vector product » were unified in the so called « algebraic product ». And the «3D « vector product » was generalised by the « wedge product » that works not only in 3D but in any dimension. Geometry and any of its application will never be the same. A new world has emerged, even if so few are still even aware of it, besides Grassmann and Clifford discovery 200 years ago! Human inertia is atrocious…and general ignorance, abyssal. So in a nut shell, « complex numbers », and more generally « Clifford numbers » are Rotations in any dimensions, even in non euclidian metrics, giving hyperbolic rotations, relativistic boosts, GR maps, etc. They capture in algebraic form and fulgurant calculus, the archetype of change and movement : rotation, translation, dilation, mirror, boost, spin, etc. No wonder they are unavoidable ! How can you become an engineer without knowing all of that. This is foundation! Fundamental stuff! And by the way, the are not only one type of « complex number » in 2D, but besides the « usual » known one, two others, which makes three types! The two others are not « fields », but nevertheless « algebras », which is good enough and even richer, since they are « relativistic complex numbers » in a natural way. In these two twin geometry, « null-vectors » exist (which square to zero without being themself zero, as light cone vectors in Relativistic dynamic). Good meditation!
@ How on Earth is that possible! « Complex numbers » have slowly emerged from Cardan initial discovery of an « illegal » arithmetic/algebraic « trick » useful to solve some cubics that he couldn’t solve otherwise. He noticed indeed that even if sqrt(-121), which sprung out « naturally » or « naively » in one of those, had no apparent arithmetic « legality », it was nevertheless useful as a intermediary « trick », since in the final result, such « monstrous » illegal quantity disappeared anyway by « conjugation » of the type : (a+sqrt(-121))(a-sqrt(-121)) = a^2 + 121. Such trick allowed him to crack cubics that couldn’t be solved without! That was the first hint and piece of the puzzle. But this puzzle took three century to be fully understood, by gradual steps, even though very soon after Cardan, the Italian Bombelli makes extensive use of it in algebra and equation solving, like quadratics whith negative determinant. But things remained very obscured in the « closed » arena of Algebra. The decisive step out, was to shift from algebra and equations, to (planar) geometry and transformations. Argan had the genius idea that this weird arithmetic « extraterestrial » and somehow still « imaginary » illegal object that was long written sqrt(-1), before Euler named it « i » (for « imaginary », which shows how it was seen for long), had in fact the same property as a half turn planar rotation, which square makes a central flip. That was the breakthrough since what it actually meant is that people opened their mind and start understanding that this -1 which equals to i^2, might not be a « usual number », or at least that it could be seen as something else than a « number » : namely a 180° planar ROTATION. Thus as a GEOMETRIC TRANSFORMATION, instead, or complementary to a « number » opposite to unity! That was the flash of lightening in Yeats poem : « The waste land »! The mysterious egg was cracked. The rest was the history of his actual birth in human mind with further steps of maturation. And finally Hamilton fully understood that what had been there discovered, was in fact a hidden multiplication on vectors that remained hidden from the « night of time », but that was there from the very beginning, and which is a omnipresent pattern used daily by Mother Nature : ROTATION. This for long « mysterious i » (of « Shiva »), was in some deep regards a positive quarter turn. And by such the building brick of any planar rotation. But to fully grab that the « exponential map » was useful. And from that, complex numbers, which are in general, rotations of points on the unit circle, composed with dilation/contraction, became the natural algebraic tool to model any waves, any circular motion, any cyclic phenomenon. It’s thus everywhere in Physics and engineering : electricity, optic, EM, electronic, mechanical, QM, etc. And even if it seems restricted to 2D. That’s because waves generally obey the superposition principle and are vectors. Which can thus be decomposed on 2D basis by projections. But there was still needs for similar algebraic representation of rotations in 3D (and higher dim), as the « complex numbers » were the god given perfect simple tool in 2D representation of Rotations and Dilations. Such tool Hamilton kept seeking for 10 years before he realised that they might be none in 3D, but one similar one in 4D. And they are indeed the « Quaternions », which behave much like « complex numbers », and that are « surprisingly » very useful to represent 3D rotations, in a much better way than classic « Euler angles » which are a bad and dangerous map. But that was not at all the end of this fascinating story, since two contemporary of Hamilton, Grassmann and Clifford, made even greater breakthrough. Especially Clifford, who discovered a deeper structure that unifies all of the previous ones and which works in ANY dimension. His work gave a new complementary understanding of the old « imaginary i ». Since he understood that better than a extended « number » which squares to minus one, better that a quarter turn positive rotation, better than an « Hamilton pair » (i.e. a 2D « vector »), the deepest way to understand such a universally useful badly named « imaginary » i, was to see it as an ORIENTED UNIT SURFACE. Grassmann and Clifford thus generalised the extension of planar vectorial geometry in which « complex numbers » revealed the multiplicative hidden structure that actually exist among vectors, to all dimensions. Points and Vectors were no more the exclusive stars of Geometry! Now areas, volumes, hypervolumes, were fully part of the algebraico-geometric arena. But in a unified revolutionary way. Scalars could be now add to vectors, and areas, volumes, etc, all together in a unified algebraic new calculus called « Clifford Algebra ». That was the most important discovery since Euclide. It revolutionized the « Euclidian geometry » by giving unified algebraic tools to generalized geometry in any D. Dot product and « vector product » were unified in the so called « algebraic product ». And the «3D « vector product » was generalised by the « wedge product » that works not only in 3D but in any dimension. Geometry and any of its application will never be the same. A new world has emerged, even if so few are still even aware of it, besides Grassmann and Clifford discovery 200 years ago! Human inertia is atrocious…and general ignorance, abyssal. So in a nut shell, « complex numbers », and more generally « Clifford numbers » are Rotations in any dimensions, even in non euclidian metrics, giving hyperbolic rotations, relativistic boosts, GR maps, etc. They capture in algebraic form and fulgurant calculus, the archetype of change and movement : rotation, translation, dilation, mirror, boost, spin, etc. No wonder they are unavoidable ! How can you become an engineer without knowing all of that. This is foundation! Fundamental stuff! And by the way, the are not only one type of « complex number » in 2D, but besides the « usual » known one, two others, which makes three types! The two others are not « fields », but nevertheless « algebras », which is good enough and even richer, since they are « relativistic complex numbers » in a natural way. In these two twin geometry, « null-vectors » exist (which square to zero without being themself zero, as light cone vectors in Relativistic dynamic). Good meditation!
Agreed. Mathematics for the most part isn't taught but rather read to you. Hence most maths professors could easily be replaced by AI. It's a mess and needs a solution
Haven't taken a university math course in 30 years, but I was happy to see that I understood what he was explaining.
It's trivial. School-level maths.
@@JupiterThunder no complex no.s at school level in the UK, only in Further Maths.
I haven't been in school for 12 years and I'm not that old, and I don't understand jack being said
I love the way he writes something down, turns to the audience and smiles. I can easily imagine his disappointment when he sees that only few students really understand why he's smiling - because the result is beautiful. Fundamentalism of this theorem is not in dealing with some stuff we've just invented and named "complex numbers", it's fundamental because we don't need any other stuff, we are at the end of arithmetics, everything we can build from these numbers has roots described by only these numbers. It's a very beautiful theorem and it deserves to be shown to everyone.
Wow, it must be one of the proud moments in his life. Wow
Now I can tell my friends that I have attended a lecture at Oxford University.
Albeit a very simple one, covering subject-matter normally done at school!
Shh the little details don't matter@@ruperttristanblythe7512
@@JupiterThunderAs in highschool. 😅
you know what? i was amazed to see the whiteboard that could be lifted up😮
annoying that he didn't lift it at the centre. Obviously not a physics lecturer.... :-)
Imagined him writing on his projected shirt!
I love how he does really care about making sure everybody is seeing the board and focusing, this lecture gonna help me to prepare for my high school exams
Introduction to Complex Numbers
● [0:14]. Introduction to complex numbers.
● [0:33]. Notation for real numbers (ℝ) and complex numbers (ℂ).
● [1:12]. Definition of a complex number: z ∈ ℂ means that z = a + ib, where a and b are real and i² = -1.
● [1:42]. Real part of a complex number: Re(z) = a.
● [2:01]. Imaginary part of a complex number: Im(z) = b.
Basic Operations with Complex Numbers
● [2:18]. Addition of complex numbers: (a + ib) + (c + id) = (a + c) + i(b + d).
■ The addition of complex numbers is commutative.
● [3:14]. Multiplication of complex numbers: (a + ib)(c + id) = (ac - bd) + i(ad + bc).
■ The multiplication of complex numbers is also commutative.
● [4:31]. Division of complex numbers, z = (a + ib) and w = (c + id) :
■ The conjugate of the denominator is used: (c + id)(c - id) = c² + d².
■ (a + ib) / (c + id) = [(a + ib)(c - id)] / (c² + d²).
■ The expression is simplified to obtain the real and imaginary parts of the result.
● [6:40]. Division is defined as long as the denominator is not zero (w ≠ 0).
● [7:18]. Equality of complex numbers: Two complex numbers are equal if and only if their real and imaginary parts are equal.
● [7:43]. Conjugate of a complex number: z = a + ib, then z̅ = a - ib.
■ The product of a complex number and its conjugate is a non-negative real number: zz̅ = a² + b².
● [9:29]. Real part of z: Re(z) = (z + z̅) / 2.
● [9:39]. Imaginary part of z: Im(z) = (z - z̅) / 2i.
● [10:12]. Properties of the conjugate:
■ (z + w)̅ = z̅ + w̅.
■ (zw)̅ = z̅w̅.
■ (z / w)̅ = z̅ / w̅ (if w ≠ 0).
Complex Numbers and Polynomials
● [11:49]. Motivation for complex numbers: They arise from the need to find roots of polynomials that do not have real solutions.
■ Example: The polynomial x³ + 2x² + 2x + 1 has a real root at x = -1.
■ The other two roots are complex: -1/2 ± i√3 / 2.
● [14:45]. Fundamental Theorem of Algebra: Every polynomial of degree n has n complex roots (counting multiplicities).
● [18:17]. Theorem (unnamed): If all coefficients of a polynomial are real, then:
■ The polynomial can be factored into linear terms corresponding to the real roots (z - α₁, z - α₂, ..., z - αᵣ).
■ And into quadratic factors corresponding to pairs of complex conjugate roots: (z - γ₁)(z - γ̅₁), (z - γ₂)(z - γ̅₂), ..., (z - γₛ)(z - γ̅ₛ).
● [22:15]. Consequence: If a polynomial has real coefficients and γ is a complex root, then its conjugate γ̅ is also a root.
Polar Form of Complex Numbers
● [23:26]. Graphical representation of complex numbers:
■ The horizontal axis represents the real part (x).
■ The vertical axis represents the imaginary part (y).
● [24:46]. Polar form: A complex number z = x + iy can also be represented in polar coordinates (r, θ).
■ r: modulus of z, denoted as |z|, is the distance from the origin to z: |z| = √(x² + y²).
■ θ: argument of z, denoted as arg(z), is the angle between the positive real axis and the line connecting the origin to z.
● [26:46]. The argument is undefined for z = 0.
● [27:22]. Special cases of the argument:
■ If y > 0 and x = 0, then arg(z) = π/2.
■ If y < 0 and x = 0, then arg(z) = 3π/2.
● [27:58]. The argument is periodic: θ, θ + 2π, θ - 2π, θ + 4π, etc., define the same argument.
● [28:59]. The principal value of the argument is in the interval 0 ≤ θ < 2π.
● [29:57]. Properties of the conjugate in polar form:
■ |z̅| = |z|.
■ arg(z̅) = -arg(z).
● [30:47]. Relationships between Cartesian and polar coordinates:
■ x = r cos(θ).
■ y = r sin(θ).
● [31:27]. Proposition:
■ |zw| = |z||w|.
■ |z/w| = |z| / |w| (if w ≠ 0).
■ |z|² = zz̅.
■ arg(zw) = arg(z) + arg(w) (if z ≠ 0 and w ≠ 0).
■ arg(z/w) = arg(z) - arg(w) (if z ≠ 0 and w ≠ 0).
● [35:35]. No information is provided about the modulus or the argument of z + w in this lesson.
● [35:49]. Triangle inequality: |z + w| ≤ |z| + |w|.
Proofs of Some Properties
● [36:41]. Some proofs are omitted, but can be found in the course's online notes.
● [36:49]. Example of proof: (zw)̅ = z̅w̅.
● [39:58]. Example of proof: if a polynomial has real coefficients and γ is a root, then γ̅ is also a root.
thanjs
@ you are welcome
@@iñigotehow do you make the
timing of the course ?
@@vincentvejdovsky2795 First, I provided the video to NotebookLM along with the complete transcription and timestamps. Then, I asked it to create a professional summary for a UA-cam video. Then, I provided the result to ChatGPT and asked it to correct the indentation as well as any errors in the theory. Finally, I reviewed and adjusted the timestamps better because the result isn’t perfect, as well as corrected any typographical errors.
It's sad that if you think about it, school and university lectures deprive you of the independent discovery of mathematical concepts. On the same day they introduce a problem, they reveal the solution immediately. I suppose that in the long run people learn faster when they're taught, and that the real skill isn't in finding the solutions but rather asking the questions, but still, I'd like it if the standard was so that people could get the satisfaction of realising it without being told.
I agree. This is not real math, it is just passively watching somebody else doing something. Moreover, everybody in the class, including the professor, thinks this is the way to learn. University is such a scam, paying thousands for this crap??!! And people still do it in 2024-25! Better off learning on your own.
Edit: In addition, observe that profs never actually even teach you HOW to think in maths. They just mumble theory at the board and stare at their notes. I thought they were good at maths, why the need for notes?. AND once again, we foolish people pay thousands for this??!! It makes no sense!
And that's the difference between a modern student and an ancient polymath.
Awesome, are you guys going to upload whole series of lectures?
Thank you prof. İt gets so clear🎉
Upload the whole course please ... 💛🙋😊
Hope i could study there 😢
you’re getting %80 of the gold for free my friend. Buy a notebook, the recommended book and study!
😢😢😢
Thank You for uploading this i need this for my preparation.
I am also the student of first year
Z minus 2 bar over 2 i
So they start teaching you from Further Maths right?
This is just A-level maths.
@@JupiterThundercomplex numbers is in Further maths (at least for edexcel)
@@xelohi4686 kinda crazy most people will have never been taught complex numbers
Thank you for this. I love these. Sorry if I'm missing something but I wasn't sure what the notation at 38:43 on the RHS represented?
it is not a mathematical symbol, it is a way to represent a difference operation.
Was this a “free lecture” by any chance? It looked like Tom and Harry felt glad to have attended it..
How does this level of mathematics help you in the real world? Like when would you need to know this?
I would need to know this during my semester exams for sure
Thank you :)
Remember when the great Vicky Neale used to teach this course, RIP
She died?!
❤❤❤
1 and 1 is ?
Can someone explain why in the complex plane the imaginary axis unit i=sqrt(-1) is depicted equal to the real axis unit 1?
The Imaginary axis is in units of i, i.e. 1i, 2i, 3i, etc. A distance i in the Im direction is drawn the same length as a distance 1 along the Re axis.
@JupiterThunder So, i's length is an arbitrary assumption. So, all that theory is an arbitrary invention. Right?
@@pelasgeuspelasgeus4634in the name of keeping things simple, I'm going to be a little loose with definitions and say that
i = sqrt(-1)
Typically it can be said that
|sqrt(x)| = sqrt(|x|)
In this case
|-1| = 1 so
|sqrt(-1)| = sqrt(|-1|) = sqrt(1) = 1
So it's not really arbitrary, just a result of its definition.
In india we learn this as....
IN ASSAM we are studying this chapter in class 10 (adv. Maths)
You're not studying it. You're merely memorizing what is i^2, geometric representation and basic equations. This lecture is not a study either. First lecture on a given topic serves as a reminder or introduction to establish common nomenclature.
It is 10 advance maths in india io_noise
I Blood BOILs to see him explain arguement and triangle inequality without explaining the geometrical significance of complex nos.! GO and learn that first!
Já estou assistindo desde o primeiro ano do ensino médio
In Bangladesh,we study this in class 12
Whay to spend so much paper?
its not paper
❤
A16yrz 12/3
We study this in class 10 in India
Do people find watching this easier than reading it?
Polite correction "principle" should be "principal"
Is that necessarily a plane ?
Obviously , it is an assertorical object that you are symbolizing with ‘ i ‘ .
There are ‘ imaginary sets ‘ that that logically generate topologies that don’t necessarily respect the metric topology of the plane .
two vectors z and w define a plane if neither are equal to 0 and kz != w where k is a real number
That ‘ s one theme .
There ‘ s also the consideration that it ‘ s not a vector space in particular : it ‘ s a field of scalars ..
Self similarity is not a theme that ought to be butchered .
Do any two nonexistent abstract objects ( by property ) define a plane ? : here , that is to say a 2 dimensional system.
As an audio engineer , when I see that space of numbers , I ‘ m thinking how are we going to model sound amplitudes …
It could be an ‘ invention ‘ to design a plane model .
There is a question of apodicity .
It ‘ s a canonical representation that imaginary numbers are required for Quantum mechanics …
That ‘ s not a priori the case . What exactly is the implication with a plane ?
Lines are certainly interesting now : I am not always attentive to my having represented them beneath awareness .
Learned this during my 11 standard 8 years and now i remember nothing lol
What's the basis for the assumption Z = a + ib?
it's not an assumption, it's the definition of a complex number.
@solidpixel Entire mathematics is standing on axioms and assumptions. Ok, for a moment let's say it's not an assumption. What's the basis for the formulation of this number? What compulsions were there?
@@RchandraMS There is no let's say. I'm telling you it's the definition. There are plenty of videos you can watch on the history of complex numbers.
@@solidpixel Forget about me watching. Define the definition of complex numbers. What led you to define that number?
I think this is an excellent question that doesn't get explicitly discussed enough; this is by no means obvious.
Why do we need at least a+bi?
- Because we need the Real numbers to be part of the new set ( using b=0 we have this)
- and of course we need the imaginary part
- But what why we need to define i^2 = -1?
Historically, complex numbers were introduced by Cardan as far back as the 16th century. While people did not really understand them at the time , introducing this famous notation a+b√−1
served in intermediate computations for finding explicit solutions to cubic equations (with real solutions!).
Also, it turns out that they have this wonderful property that every polynomial (real or complex) can be factored into a product of complex polynomials of degree one. This means that complex numbers are enough to study any polynomial with real coefficients.
In Pakistan we learn it in 9th
I have never seen a joke like this even real numbers are taught in grade 9 and you are talking about complex number I bet you don't even know the value of i
The way in which he addressed that students question was very arrogant
I am ethiopia student so the concept of complex nember i was learnet in grade 11 unit seven
Nobody could replace Vicky Neale 💔
In my time that was taught in high school.
It still is, but usually in the first year of university they focus on making sure everyone is on the same level of knowledge before introducing more complex ideas.
@@nodeathuif those indians can understand the same thing. Lol they think they learnt this in high school and Cambridge just teaches this so they are superior in some sense, even though they are just bollocks down to the drain.
In India 11 class syllabus
Use a blackboard instead! It’s so unnerving!
İvr ch1? Cat
Sir I think it's better not write the exact neglecting un imaginary nos.
Am I right...
Can someone explain where is the location of the imaginary axis with respect to the 3 real axes that define our world? I mean the world has 3 real axes, right? Latitude, longitude and height.
Z axis you mean? Okay u/down, left/right, side to side ..what axis are you covering when "twisting"
@agentsmidt3209 don't you understand English?
@@pelasgeuspelasgeus4634 Latitude and Longitude, and height are not axes.
@agentsmidt3209 Really? You just destroyed physics, architecture, etc. How would you describe our 3d world?
Imaginary axis IS the y axis!! That’s the beauty of complex no.! It takes us from 1D (numbers like 1234...) to 2D world (x+iy)
Maths is a language. so we can say that... vectors and complex numbers and geometry and matrix are like synonyms you know? To represent the same coordinates in diff ways
1×1-0×0, 2×2-1×1, 3×3-2×2, 4×4-3×3, 5×5-4×4, 6×6-5×5, 7×7-6×6, 8×8-7×7, 9×9-8×8...odd numbers
n^2 - (n-1)^2 = n^2 - (n^2 - 2n +1) = n^2 - n^2 + 2n -1 = 2n-1 where n is an element of the set of integers or alternatively 2n+1 if we start with (n+1)^2 - n^2
I have struggled to understand the why (why do we need these numbers). They say it is because certain equations can only be solved "in a different dimension". And they "lift" a 2D curve into a third dimension. But that is just a 3rd dimension.
We have that there exist any root for the complex numbers and it belongs to the complex numbers, but we don't have the same thing in any other subset:
2 is an integer and a rational, but the root is just real, so you need to extend to the reals in order to find any root of 2.
Then -1 is an integer, a rational number, a real number, but the root is not real, so you just suppose to have i=sqrt(-1), meaning that i^2=-1 and that's called imaginary.
You just introduce the operations with other real numbers and you get z=a+ib complex.
(Complex: Consisting of interconnected or interwoven parts; composite of real and imaginary)
Then it depends in which field you want to study these numbers, but in algebra you have the amazing property that allows you to write any polynomial equation as a product of complex factors of degree 1 or 0:
x^2+x+1=(x+1/2-i*sqrt(3)/2)(x+1/2+i*sqrt(3)/2)
This can be useful in some cases.
Then there are some integrals that you can solve thanks to the complex numbers, such as the integral of sin(x)/x.
You can't graduate in mathematics, physics or engineering without studying these numbers.
@@vortanoise.2625 thanks, I'm ashamed to admit I have a PhD (in immunology). However, I got obsessed with quantum mechanics, and there you ABSOLUTELY need to understand these numbers. It's not easy: "Then -1 is an integer, a rational number, a real number, but the root is not real, so you just suppose to have i=sqrt(-1), meaning that i^2=-1 and that's called imaginary." The square root of -1 simply shouldn't exist
@@jaimetorres950it is hard to understand, but it's powerful abstraction and expansion of expression. Remember, even the notion of negative numbers was strange at some point. I know qbit is described with a complex function. Also mandelbrot set, fractal, where a dimension becomes a real number, like 1.14 😂. This is what the notion of a complex number enables.
Zkr
Fancy paying all that money to study at Oxford and be lectured to by this bloke who's obviously just a nervous wreck who can't even get sign rules right!
Why does everything get written down on the board. Very inefficient. Just use a book and read from it.
r u dum?
Because the beain can sometimes grasp things and be more active visually than verbally.
It works much better to show students this way, especially when doing derivations etc.
I have a geometry professor just reading from a book and it's so boring that after a while I almost fall asleep.
I can read a book by myself.
@ Had thermodynamics lectures where the lecturer would just put up page after page of derivations. What I found when I lectured is you can put up the first couple of lines of a derivation and then you might have an integral, so you can talk about what you could do to solve that integral,then proceed todo it.
In India we study this in class 11
If in india, you were given the introduction to complex numbers in this type, then you would be lost to see the fundamental theorem of algebra in a first introductory lecture... And c'mon, this is Oxford , and the professor knows what and how to teach!
You must be very slow, because here they are studying it in class 1.
@@jfndfiunskj5299 😂😂😂😂
In Italy we learn this in kindergarten
Ji
This is probably the worst lecture I have ever witnessed. Back to the audience...reading from something that should hae been passed out ahead of time...insufficient explanation of processes...Oxford should be better than this.
In China we learn this in 3rd grade
That's nothing. My culture is very advanced. I learned this in the womb.
It would be nice if manners and respect were also taught in China.
People criticizing a Introduction, yes, a introduction. Straight bs comments he cant even have notes. I have meet extraordinary mathematicians that used lecture notes for their classes specially in the introductory ones since they don’t want to miss anything.
they might be extraordinary mathematicians but that does not imply that they are extraordinary lecturers/teachers
miserable lecture when you have to read everything from your concept sheet
isn't that just being well prepared to make sure you don't miss anything or make any dumb mistake that might affect students?
@@Mikael26BE No
Most courses (though not this one) comprise 16 one-hour lectures. That's a lot to deliver when it might not be the subject of your research and has to be 100% spot on, hence the notes which are meticulously prepared. And it needs to tie in with the problem sheets that the students are given after the lecture. This way of teaching may not be to everyone's liking, but it does take the students through each step of the learning. The students then go on to tutorials in pairs with their tutor. Thanks for the feedback.
@@OxfordMathematics ooops ... not his research😅 But seriously, I believe it is important to present lectures without any notes in absolute clarity (or as close as you can get). It shows the students that it is not a regurgitation but a logical flow of ideas and concepts. In my own experience, looking at lecture notes means that you have turned off your brain and you are now the puppet of your notes. Anyway, I think people are just surprised to see this from a good university
You'd think that he wouldn't have to read this lecture from notes given that it's all such basic stuff......I'm pissing off to Cambridge to do maths.....to hell with Oxford!!!
I'm assuming every student is focussed on their electronic device because this is rudimentary not to mention bats**t boring. Time to retire, sir!
Lost me after a minute. Like complex or simple numbers where your not actually using a number but letters confuses me. I’m sure he’s right but whats the actual practical use for this in the real world? Other than to justify a lecturers salary?
It's practically essential for many fields of engineering (largely because of their use in the study of aerodynamics, and fluid dynamics in general). Electrical theory uses them. Generally, physics relies heavily on them.
Plus, they're useful in statistics and analytic finance.
I feel very grateful to have this lecture come with more videos sir 😅😅😅🎉
even for a first lecture this is very weak. and they think they are an elite university.
It's a recap in the first week. We have students from high-schools all over the world. This is to make sure they are all up to speed. There are over 100 more lectures on our UA-cam Channel. Thanks for the feedback.
then what are you doing on youtube? Go there and get full marks bruh
@@OxfordMathematics if they don't know those basics before entering Oxford then what's the point of Oxford being Oxford then?
@@PauloConstantino167 Not every topic is tested to its full extent in the final examinations for many examination boards. By getting everyone up to speed, everyone is starting on the same level for this topic.
@@Hasnain. no.
This is embarrassing.
Are sir paper lekar kyu padate ho
Bad « introduction » because all the historic, pragmatic, geometric and theoric justification of « i^2=-1 » is totally skipped. That’s the core, the heart, the spirit. The rest, more or less presented here is « trivial » consequences, that should not be exposed before the existence and pragmatic feeling of « the core thing » is made clear. It’s a perfect example of upside down « pedagogy », i.e. anti pedagogy
i kind of agree, although that should have been taught in highschool.
nonetheless, i am currently 2nd year in mecatronics engineering, almost learning delayed differential equations, and to be honest, i have no clue of the usefulness of complex numbers...
maybe you can share some videos/documentaries? or even written documents, i haven't found anything related to the "origin" (i.e. historic pragmatic etc justification...)
thanks :)
@@rAdiAntOn How on Earth is that possible! « Complex numbers » have slowly emerged from Cardan initial discovery of an « illegal » arithmetic/algebraic « trick » useful to solve some cubics that he couldn’t solve otherwise. He noticed indeed that even if sqrt(-121), which sprung out « naturally » or « naively » in one of those, had no apparent arithmetic « legality », it was nevertheless useful as a intermediary « trick », since in the final result, such « monstrous » illegal quantity disappeared anyway by « conjugation » of the type : (a+sqrt(-121))(a-sqrt(-121)) = a^2 + 121. Such trick allowed him to crack cubics that couldn’t be solved without! That was the first hint and piece of the puzzle.
But this puzzle took three century to be fully understood, by gradual steps, even though very soon after Cardan, the Italian Bombelli makes extensive use of it in algebra and equation solving, like quadratics whith negative determinant. But things remained very obscured in the « closed » arena of Algebra.
The decisive step out, was to shift from algebra and equations, to (planar) geometry and transformations. Argan had the genius idea that this weird arithmetic « extraterestrial » and somehow still « imaginary » illegal object that was long written sqrt(-1), before Euler named it « i » (for « imaginary », which shows how it was seen for long), had in fact the same property as a half turn planar rotation, which square makes a central flip.
That was the breakthrough since what it actually meant is that people opened their mind and start understanding that this -1 which equals to i^2, might not be a « usual number », or at least that it could be seen as something else than a « number » : namely a 180° planar ROTATION. Thus as a GEOMETRIC TRANSFORMATION, instead, or complementary to a « number » opposite to unity!
That was the flash of lightening in Yeats poem : « The waste land »! The mysterious egg was cracked. The rest was the history of his actual birth in human mind with further steps of maturation.
And finally Hamilton fully understood that what had been there discovered, was in fact a hidden multiplication on vectors that remained hidden from the « night of time », but that was there from the very beginning, and which is a omnipresent pattern used daily by Mother Nature : ROTATION. This for long « mysterious i » (of « Shiva »), was in some deep regards a positive quarter turn. And by such the building brick of any planar rotation.
But to fully grab that the « exponential map » was useful. And from that, complex numbers, which are in general, rotations of points on the unit circle, composed with dilation/contraction, became the natural algebraic tool to model any waves, any circular motion, any cyclic phenomenon. It’s thus everywhere in Physics and engineering : electricity, optic, EM, electronic, mechanical, QM, etc.
And even if it seems restricted to 2D. That’s because waves generally obey the superposition principle and are vectors. Which can thus be decomposed on 2D basis by projections.
But there was still needs for similar algebraic representation of rotations in 3D (and higher dim), as the « complex numbers » were the god given perfect simple tool in 2D representation of Rotations and Dilations.
Such tool Hamilton kept seeking for 10 years before he realised that they might be none in 3D, but one similar one in 4D. And they are indeed the « Quaternions », which behave much like « complex numbers », and that are « surprisingly » very useful to represent 3D rotations, in a much better way than classic « Euler angles » which are a bad and dangerous map.
But that was not at all the end of this fascinating story, since two contemporary of Hamilton, Grassmann and Clifford, made even greater breakthrough. Especially Clifford, who discovered a deeper structure that unifies all of the previous ones and which works in ANY dimension. His work gave a new complementary understanding of the old « imaginary i ». Since he understood that better than a extended « number » which squares to minus one, better that a quarter turn positive rotation, better than an « Hamilton pair » (i.e. a 2D « vector »), the deepest way to understand such a universally useful badly named « imaginary » i, was to see it as an ORIENTED UNIT SURFACE.
Grassmann and Clifford thus generalised the extension of planar vectorial geometry in which « complex numbers » revealed the multiplicative hidden structure that actually exist among vectors, to all dimensions. Points and Vectors were no more the exclusive stars of Geometry! Now areas, volumes, hypervolumes, were fully part of the algebraico-geometric arena. But in a unified revolutionary way. Scalars could be now add to vectors, and areas, volumes, etc, all together in a unified algebraic new calculus called « Clifford Algebra ».
That was the most important discovery since Euclide. It revolutionized the « Euclidian geometry » by giving unified algebraic tools to generalized geometry in any D. Dot product and « vector product » were unified in the so called « algebraic product ». And the «3D « vector product » was generalised by the « wedge product » that works not only in 3D but in any dimension.
Geometry and any of its application will never be the same. A new world has emerged, even if so few are still even aware of it, besides Grassmann and Clifford discovery 200 years ago! Human inertia is atrocious…and general ignorance, abyssal.
So in a nut shell, « complex numbers », and more generally « Clifford numbers » are Rotations in any dimensions, even in non euclidian metrics, giving hyperbolic rotations, relativistic boosts, GR maps, etc. They capture in algebraic form and fulgurant calculus, the archetype of change and movement : rotation, translation, dilation, mirror, boost, spin, etc.
No wonder they are unavoidable ! How can you become an engineer without knowing all of that. This is foundation! Fundamental stuff! And by the way, the are not only one type of « complex number » in 2D, but besides the « usual » known one, two others, which makes three types! The two others are not « fields », but nevertheless « algebras », which is good enough and even richer, since they are « relativistic complex numbers » in a natural way. In these two twin geometry, « null-vectors » exist (which square to zero without being themself zero, as light cone vectors in Relativistic dynamic).
Good meditation!
@ How on Earth is that possible! « Complex numbers » have slowly emerged from Cardan initial discovery of an « illegal » arithmetic/algebraic « trick » useful to solve some cubics that he couldn’t solve otherwise. He noticed indeed that even if sqrt(-121), which sprung out « naturally » or « naively » in one of those, had no apparent arithmetic « legality », it was nevertheless useful as a intermediary « trick », since in the final result, such « monstrous » illegal quantity disappeared anyway by « conjugation » of the type : (a+sqrt(-121))(a-sqrt(-121)) = a^2 + 121. Such trick allowed him to crack cubics that couldn’t be solved without! That was the first hint and piece of the puzzle. But this puzzle took three century to be fully understood, by gradual steps, even though very soon after Cardan, the Italian Bombelli makes extensive use of it in algebra and equation solving, like quadratics whith negative determinant. But things remained very obscured in the « closed » arena of Algebra. The decisive step out, was to shift from algebra and equations, to (planar) geometry and transformations. Argan had the genius idea that this weird arithmetic « extraterestrial » and somehow still « imaginary » illegal object that was long written sqrt(-1), before Euler named it « i » (for « imaginary », which shows how it was seen for long), had in fact the same property as a half turn planar rotation, which square makes a central flip. That was the breakthrough since what it actually meant is that people opened their mind and start understanding that this -1 which equals to i^2, might not be a « usual number », or at least that it could be seen as something else than a « number » : namely a 180° planar ROTATION. Thus as a GEOMETRIC TRANSFORMATION, instead, or complementary to a « number » opposite to unity! That was the flash of lightening in Yeats poem : « The waste land »! The mysterious egg was cracked. The rest was the history of his actual birth in human mind with further steps of maturation. And finally Hamilton fully understood that what had been there discovered, was in fact a hidden multiplication on vectors that remained hidden from the « night of time », but that was there from the very beginning, and which is a omnipresent pattern used daily by Mother Nature : ROTATION. This for long « mysterious i » (of « Shiva »), was in some deep regards a positive quarter turn. And by such the building brick of any planar rotation. But to fully grab that the « exponential map » was useful. And from that, complex numbers, which are in general, rotations of points on the unit circle, composed with dilation/contraction, became the natural algebraic tool to model any waves, any circular motion, any cyclic phenomenon. It’s thus everywhere in Physics and engineering : electricity, optic, EM, electronic, mechanical, QM, etc. And even if it seems restricted to 2D. That’s because waves generally obey the superposition principle and are vectors. Which can thus be decomposed on 2D basis by projections. But there was still needs for similar algebraic representation of rotations in 3D (and higher dim), as the « complex numbers » were the god given perfect simple tool in 2D representation of Rotations and Dilations. Such tool Hamilton kept seeking for 10 years before he realised that they might be none in 3D, but one similar one in 4D. And they are indeed the « Quaternions », which behave much like « complex numbers », and that are « surprisingly » very useful to represent 3D rotations, in a much better way than classic « Euler angles » which are a bad and dangerous map. But that was not at all the end of this fascinating story, since two contemporary of Hamilton, Grassmann and Clifford, made even greater breakthrough. Especially Clifford, who discovered a deeper structure that unifies all of the previous ones and which works in ANY dimension. His work gave a new complementary understanding of the old « imaginary i ». Since he understood that better than a extended « number » which squares to minus one, better that a quarter turn positive rotation, better than an « Hamilton pair » (i.e. a 2D « vector »), the deepest way to understand such a universally useful badly named « imaginary » i, was to see it as an ORIENTED UNIT SURFACE. Grassmann and Clifford thus generalised the extension of planar vectorial geometry in which « complex numbers » revealed the multiplicative hidden structure that actually exist among vectors, to all dimensions. Points and Vectors were no more the exclusive stars of Geometry! Now areas, volumes, hypervolumes, were fully part of the algebraico-geometric arena. But in a unified revolutionary way. Scalars could be now add to vectors, and areas, volumes, etc, all together in a unified algebraic new calculus called « Clifford Algebra ». That was the most important discovery since Euclide. It revolutionized the « Euclidian geometry » by giving unified algebraic tools to generalized geometry in any D. Dot product and « vector product » were unified in the so called « algebraic product ». And the «3D « vector product » was generalised by the « wedge product » that works not only in 3D but in any dimension. Geometry and any of its application will never be the same. A new world has emerged, even if so few are still even aware of it, besides Grassmann and Clifford discovery 200 years ago! Human inertia is atrocious…and general ignorance, abyssal. So in a nut shell, « complex numbers », and more generally « Clifford numbers » are Rotations in any dimensions, even in non euclidian metrics, giving hyperbolic rotations, relativistic boosts, GR maps, etc. They capture in algebraic form and fulgurant calculus, the archetype of change and movement : rotation, translation, dilation, mirror, boost, spin, etc. No wonder they are unavoidable ! How can you become an engineer without knowing all of that. This is foundation! Fundamental stuff! And by the way, the are not only one type of « complex number » in 2D, but besides the « usual » known one, two others, which makes three types! The two others are not « fields », but nevertheless « algebras », which is good enough and even richer, since they are « relativistic complex numbers » in a natural way. In these two twin geometry, « null-vectors » exist (which square to zero without being themself zero, as light cone vectors in Relativistic dynamic). Good meditation!
Agreed. Mathematics for the most part isn't taught but rather read to you. Hence most maths professors could easily be replaced by AI. It's a mess and needs a solution
@@mpcformation9646which book should I read to learn all these stuffs?