Hypercomplex numbers | Math History | NJ Wildberger

[@AnimeLover-su7jh]

People who do programming... Actually I am a game programmer and I am watching this as my historical introduction to rotations Very nice and clear thanks

[@invictus327]

Norman, your lecture series and KZbin channels are exceptional for clarity, concision, insight and enduring educational value. Thankyou.

[@yoichihiruma4291]

This students are lucky they have a passionate teacher who includes history of math

[@becomepostal]

They are not so lucky, given he says stuff that is false (like "reals are decimals").

[@TheRosyCodex]

@@becomepostal lol stfu nerd

[@Ash-vi3gg]

but it's an actual whole subject lmao

[@maxwang2537]

44:21 It is amazing as always to see some beautiful results can be achieved from entirely different directions yet in equally elegant ways. I have a picture in my mind - poor humans are desperately trying to poke through the clouds, from different angles, to have glimpses of how the universe works. I came across the concept of quaternions long time ago, being led by curiosity, but it very much puzzled me. This lecture helped me a lot in this. Thank you prof Wildberger. I hope I can progress more to bridge the gap between Hamilton’s theory, vector analysis, and quantum mechanics.

[@reimannx33]

Be careful - the bigger the ambition, the higher the probability of failure and the deeper the fall. For every success after tortured years like Hamilton's, there have been countless other ideas relegated to the trash bins of hostory , and even some highly successful ones found home in mental asylums like Cantor, godel, or "lost it completely" like van Gogh, boltzmann, nietchtze.

[@markosskace514]

Which would be next video about more extensions of octonians (and simple Lie groups)?

[@mathbbn2676]

The teacher's teaching has been very good, and I will follow the teachers every day

[@TheDavidlloydjones]

Congratulations: excellent audio. Despite the fact that you're recording in front of a hard blackboard -- no echo. All other educational video makers: please note! 'Course one of the virtues of broadcast media like television and KZbin is that we don't have to sit around watching people write on the blackboard because they can prepare their grapics before they record the show... Huge saving of time for hundreds, even thousands, of people who will then watch the efficiently produced show, pay attention to the content, etc. etc. But yeah, you did get the audio right.

[@Slay_and_Play]

I liked this lecture. I'm taking Discrete Structures in CS and this lecture covered some of the stuff on the prerequisites list. Thanks so much!

[@MrJosephArthur]

Very interesting historical exposition about the hypercomplex numbers which are an extension or generalization of the complex numbers. In the same way, the extension of the mathematical analysis as real analysis, complex analysis and hypercomplex analysis. It is interesting to see the evolution of mathematical thought concerning to this topic of mathematics. Some algebraic proposition from the ancient Greek mathematician Diophantus inspired later mathematicians as Fibonacci, Fermat, Euler and others to develop new results; hence, W. R. Hamilton obtained inspiration from previous mathematical results which came from those mathematicians inspired by the Diophantus' work. In this way, Hamilton proposed his "quaternions" which were the first approach to the hypercomplex numbers. Afterwards, as usual, the sophistication was increasing in the mind of mathematicians with respect to the extension of the complex numbers, and then, other more sophisticated objects appeared in mathematics, as for example, biquaternions, octonions, sedenions and so forth. Another interesting observation on Hamilton's discovery about the quaternions is the projection of this result on other fields or contexts. At the beginning, theoretical interests of Hamilton for the extension of complex numbers became important results for both pure mathematics and applied mathematics.

[@binjm3a]

Thank you very much . I asked you for these lectures about 5 months ago . So thank you and i really appreciate your works

[@AkamiChannel]

I'm looking for your discussion of D the algebra that is sort of on the same level with H (quaternions), but different. I know it's somewhere in your playlists, but I'm not sure where! Discussions of j, which squares to 1 is also of great interest.

[@ikocheratcr]

Great lecture, I love it. Clear and simple explanations. Also, pretty good all the historical notes. Sadly the camera guy is not up his task: focus problems and pointing camera somewhere else.

[@raymondrogers3797]

I am enjoying these lectures :) Is there one or one planned covering Spinors; as per Penrose and Dirac? In theory I know them; in practice not so much.

[@njwildberger]

Hi Raymond: sorry, no plans for such a lecture in the immediate future.

[@OnlyPenguian]

Raymond Rogers Of course there is a connection between hypercomplex numbers and spinors via the representation theory of Clifford algebras. The history of hypercomplex numbers doesn't end with Graves and Cayley. Look up Grassmann, Clifford, E. Cartan, Wedderburn, etc. etc. There are connections with quadratic, forms, sums of squares, orthogonal designs, etc. Look up Seberry and Gastineau-Hills.

[@elitemathematicsclassbyabw8069]

Can't we write 65 as 8² + 1² too? As the sum of squares of two sides be the length of hypotenuse which is 65. Or is necessary to represent 65 as 7² + 4²

[@ffggddss]

Another engaging lecture. Thanks once again! So as we go from reals, 1 dimension; to 2-D, complex numbers; to 3-D, which were found not to yield a coherent arithmetic; to 4-D, quaternions; to 8-D, octonions; the question naturally arises, how were 5, 6, and 7-D systems eliminated? And what about 9 or more dimensions? Well, I recall from an abstract algebra course way back, that there is a truly remarkable theorem, due as I recall, to Fisher (?): If the product of two sums of n squares can be written as a sum of n squares in a way that is bilinear in the components of the factor sums, then n = 1, 2, 4, or 8. (!!) This pretty well limits the club to those four systems. Have you heard of this theorem? And is my memory accurate? Secondly, upon fiddling with quaternions and their arithmetic years ago, I hit on that vector form of their product; but I also realized that there's what is perhaps an easier way of remembering the rules: namely, that you can picture a quaternion as a complex number over the complex numbers. That is, q = A + Bj , where A = a + bi, and B = c + di So it becomes automatic that i² = j² = -1; and if you add just one more rule: ij = -ji then from this reduced rule-set, it now follows by calling k = ij, that k² = (ij)² = ijij = -iijj = -(-1)(-1) = -1 jk = jij = -ijj = i ; kj = ijj = -i ki = iji = -iij = j ; ik = iij = -j so that encapsulates all the arithmetic rules for ℍ. I'm confident that this isn't original; it falls right out of one of the matrix representations of ℍ; but it doesn't seem to get much "press."

[@njwildberger]

You are quite right. There are essentially algebraic reasons such as the sum of squares result you mention that limit the possibilities for higher dimensional division algebras. And it is well known that quaternions can be viewed as complex numbers over complex numbers: but it is also the case that this breaks the symmetry somewhat that underlies this remarkable algebra.

[@ffggddss]

Good point about breaking the symmetry. But I guess all I was saying was that this is a conceptual bridge to the ultimate set of multiplication rules for ℍ, as well as a convenient mnemonic for them. And I think it has some beauty on its own. And now something else occurs to me. Namely, I suppose it helps show why 3-D complex numbers won't work; as soon as you introduce the second hyperimaginary unit, a third one becomes necessary; and once that third one is introduced, the system closes. This isn't fully compelling, but maybe it suggests a way that is? And OBTW, that theorem about products of sums of squares, is one I've tried to track down without success; I find it to be one of the more striking results in mathematics. Like, "How do you even *begin* to show something like that?" Do you know, off the top, whose it is, for sure? I'm doubtful of my own memory on this.

[@peterclarkson8215]

Hurwitz perhaps did job for quarter circle as one eighth sphere. pdfs.semanticscholar.org/5ea5/379d7480fd9fd22a19f420c1912e7cb0a2c7.pdf

[@feandil666]

I discovered quaternions with video games programming, though I had never heard of them before, and I did advanced maths. it's a shame they're not taught, they're fun, and very useful. (the reason games graphics engines use quaternions is that they're faster, computationally, than matrices, and they're an equivalent representation of 3D space rotations. and video games do a LOT of 3D rotations)

[@AkamiChannel]

By advanced math do u mean that you majored in math? Just curious bc I didn't go to college.

[@avihonor7217]

i literally clicked on this video because i was listening to a ryoji ikeda song. i have no idea what 80% of this meant, but hey at least i learned something from this passionate guy

[@brendanward2991]

Great lecture. Thank you.

[@revenantnox]

I saw this title and thought... "I wonder if he'll talk about Quaternions". Enjoyed this

[@tannerzuleeg1229]

21:43 Allowing a1 and a2 = cosx and cosy; then by letting b1 and b2 = sinx and siny, you can see that the real part is cos(x+y) and the imaginary part is sin(x+y), just expanded using the sum of angles formulas. Gilbert Strang said it best: "r's multiply, thetas add." Simply beautiful.

[@tricky778]

Even more beautiful, ln r add along with theta. With regular complex numbers there's no ln r, of course not because there's no theta for zero and thus no polar representation

[@Bluedune86]

love the lectures!

[@xyzct]

I highly recommend _The Nature and Growth of Modern Mathematics_ by Edna Kramer

[@Death_User666]

why havent you made more of these?

[@borissklyar1415]

I propose the following "matrix sieve", that is easy to memorize and is very simple - all we have to do in order to find primes in given interval [N1;N2] - is to find positive integers that do not appear in two pairs of 2-dimensiona arrays. (Note: All primes will be in one of two forms: S1(p)=6p+5 or S2(p)=6p+7, p=0,1,2,3,...). There are two 2-dimensional arrays: |5 10 15 20 ..| 6i^2-1+(6i-1)(j-1)= |23 34 45 56...| |53 70 87 104...| |95 118 141 164...| |149 178 207 236...| |... ... ... ... | | 5 12 19 26 ..| 6i^2-1+(6i+1)(j-1)= |23 36 49 62...| |53 72 91 110...| |95 120 145 170...| |149 180 211 242...| |... ... ... ... | Positive integers not contained in these arrays are indexes p of all prime numbers in the sequence S1(p)=6p+5, i.e. p=0, 1, 2, 3, 4, , 6, 7, 8, 9, , 11, , 13, 14, , 16, 17, 18, , , 21, 22, , 24, , , 27, 28, 29, ... and primes are: 5, 11, 17. 23, 29, , 41, 47, 53, 59, , 71, , 83, 89, , 101, 107, 113, , , 131, 137, , 149, , , 167, 173, 179, .... There are two 2-dimensional arrays: |3 8 13 18 ..| 6i^2-1-2i+(6i-1)(j-1)= |19 30 41 52...| |47 64 81 98...| |87 110 133 156...| |139 168 197 226...| |... ... ... ... | | 7 14 21 28 ..| 6i^2-1+2i+(6i+1)(j-1)= |27 40 53 66...| |59 78 97 116..| |103 128 153 178...| |159 190 221 252...| |... ... ... ... || Positive integers not contained in these arrays are indexes p of all prime numbers in the sequence S2(p)=6p+7, i.e. p=0, 1, 2, , 4, 5, 6, , , 9, 10, 11, 12, , , 15 , 16, 17, , , 20, , 22, , 24, 25 , 26, , , 29, ... and primes are: 7, 13, 19. , 31, 37, 43, , , 61, 67, 73, 79, , , 97, 103, 109, , , 127, , 139, , 151, 157, 163, , , 181 .... www.planet-source-code.com/vb/scripts/ShowCode.asp?txtCodeId=13752&lngWId=3

[@RenormalizedAdvait]

Vector is more complicated in the sense that it too loses associativity as in its cross product, plus it does not form a division ring. Many have argued, although minority academicians (almost fringe) complain that Maxwell's quaternion electrodynamics is superior to Gibb-Heaviside's vector calculus notation as it allows more operations, like division for example. Is there any truth to that claim? Also, did tensors finally answer the problem of division with vectors? Sir, please make a detailed video on this topic. On a different note please explain why did Hamilton reject the then accepted holy grail of mathematical truth i.e. "commutativity" for his quarternion which led to its unpopularity or how did non-commutativity solve the problem of multiplication of vectors in 4D. 🙏

[@Igdrazil]

Tensors didn’t solve the problem. But Grassman, Clifford and David Hestenes GEOMETRIC ALGEBRA DOES. From elementary geometry up to Gauge Theory of Gravity IN EUCLIDIAN METRIC (without need of « curved space »). Hamilton Quaternions are in fact a particular case of MULTIVECTORS. But Hamilton approach is limited and misses the deeper and wider fundamental geometric and algebraic pattern that rules in fact the Geometric Arena. Just as « commutativity » is an extremely particular and childish case of much wider non commutative GEOMETRIC ALGEBRA, which is the fundamental bed rock to build upon. Worse, 3D elementary geometry « cross product » is a tricky artifact that holds only in 3D but not in 2D nor higher dimensions. It’s a bastard curiosity which hides to student it’s general healing which is WEDGE PRODUCT, working in all dimensions. And this correct tool with the usual dot product are the two wings of the true unifying NON COMMUTATIVE BUT INVERTIBLE GEOMETRIC PRODUCT. Inversion working in ALL DIMENSIONS. Thus the fact that no « complex or quaternion structure » exist in 3D is a misleading ILLUSION, since it EXIST, just as in EVERY DIMENSION where one can multiply, invert thus devide any multivector. So GEOMETRIC ALGEBRA exist since Hamilton Quaternions, but Grassman died to early and Clifford, though understanding the weakness and particular case of Hamilton Quaternions, lacked some crucial keys to brings the theory to its complete maturity, which was achieved by David Hestenes 50 years ago. Geometric Algebra is a huge tree embracing many branches : Euclidian geometry, projective geometry, relativistic geometry, « complex analysis » in N dimensions, etc. It generalises Tensor calculus and differential forms calculus. It’s the right stand point : in NON COMMUTATIVITY BUT INVERTIBILITY. And no big wonder the world respond to such universal framework since hardly one thing is commutative in the world. Words and langages are not…Thoughts and acts are not…Rotations in 3D are not… Not even ones with common center… So letting go this childish exceptional commutativity is obviously the right thing to do in order to set up a universal algebraic theory of « geometry ».

[@rossfriedman6570]

I actually really like this because it starts out basic. I don't have a college math education

[@uberobserver]

Great work. Thank you.

[@jaanuskiipli4647]

Connection between Euler's theorem and multiplication of quaternions was hilarious.

[@dr.rahulgupta7573]

Excellent presentation of the topics. Thanks sir. DrRahul Rohtak India

[@MartenK141095]

Really good explained and i love the storys behind those numbers, but one question: Shouldn't jm = - o cause you're going against the flow or did i miss something there?

[@Sopel997]

From what I understood you take a line which contains those 2 numbers (in this case j and m. There is always only one such line) and look up direction in which shortest vector from first number to the second number points. If it points in the same direction as the line then there is no minus. And the result number is the third number on the line.

[@rangedfighter]

such a great lecture

[@JPaulDiLucci]

Possible new book: "Real Problems with Real Numbers: What is the Continuum ... Really?"

[@DavidVonR]

Interesting subject, thank you

[@dudewhoisnotfunny]

whats with the camera focus

[@stevenwilson5556]

Camera work isn't as good as the lecture is. I'd rather have a few bad focus issues and a good lecture than a great focused camera and a bad lecture.

[@ojas3464]

👍 If the Plane consisted only of Rational coordinates, Euclid's constriction of equilateral triangle would fail, such triangles won't exist. He had to enlarge to a larger set, to admit constructible numbers such as √2

[@Slywolf1992]

This guy is great.

[@fcalin21]

Great!

[@brendawilliams8062]

Thx.

[@rsassine]

Superb.