The Rise and Fall of Quaternions: Why We Use i, j, and k in Vector Calculus | Deep Dive Maths

[@hohinng8644]

We love you Jeff! Make some more videos/ lectures on various mathematics topic! You are the best online Math lecturer I have ever seen!

[@ProfJeffreyChasnov]

Thank you!

[@lingarajpatnaik3870]

Professor! This is perhaps the best exposition of quaternions I ever came across. May many more people revel in mathematics and learn from teachers like you.

[@maharajjinkb7824]

Vector Calculus emerged out of the theory of hypercomplex numbers, discovered by Hamilton. 'i' was already used to denote the imaginary root of minus one in the traditional complex number system. Hamilton pondered the possibility of adding an additional, uniquely different imaginary root of minus one, j , to fashion a three-dimensional complex algebra. The geometric interpretation of such an algebra might be very useful in the study of the physical sciences. But the algebra failed, unless a third square root of minus one was introduced, such that ij=k, etc. In their purest form, vectors are just geometric representations of numbers. But this vector space was four-dimensional. Hamilton decided to demote the real axis of the space to a number, a 'scalar' quantity, without geometric interpretation. Hence, Hamilton's 'quaternion' was the sum of a three-dimensional vector and a number. This interpretation led to a great deal of confusion, to the quaternion wars, and to the eventual separation of the vector algebra into scalar and vector parts, as it is in modern vector analysis. Notice also that modern vector analysis is completely divorced from the concept of hypercomplex numbers and the original meaning of i, j and k.

[@guidopaoluzicusani2185]

I had a superficial knowledge of quaternions but didn't know about the vectors war... nice video! Now cross product and vector product make more sense, too! 😀

[@TV-ll3kh]

I am a physics major and I was completely unaware of this history. That you for teaching me!

[@ProfJeffreyChasnov]

Most people are unaware. Quaternions have really been buried in physics and math.

[@gabrielcazaubieilh4342]

This was so instructive and interesting! Thank you Jeff!!!

[@qhwho7410]

Respect history, put knowledge into proper perspective and context, a great deeds!

[@gh0stable]

Thanks Prof. Chasnov! As a physics student who is also into computer graphics, I find this video to be very informative!

[@BalrogIsCool]

Amazing video I learned a lot! Really liked the historical parts and I didn't know how strongly connected Quaternions were to vector calculus before watching

[@shayan3354]

Thankyou sir for this insight on this topic .... Really learnt something new today 🙏

[@johnzander9990]

Very interesting. Please, more math history.

[@musawwirahmad6396]

I was hoping if you could explain what Hamilton meant by the square point and how the allure thereof led him to make his subsequent assumptions.

[@suleimanbas]

brilliant explanation. thank you.

[@ahmedmuhammed6905]

Thank a lot for such beautiful video

[@mustafaalneami5193]

Thank you professor for this amazing video

[@ElboxD]

Great video. Thanks!

[@falvarezarzate1]

Amazing Professor, I also thought, when you talk about the cyclical and anticyclical multiplication, in the Levi Civita symbole that you taught us in your vector calculus course. Thanks for this material. See you soon with much more content too learn from you.

[@adeirman7368]

Professor, i think you should add statistics part on Mathematics for Engineers Specialization. Because statistics is super important part of engineering. Thank you Prof.

[@EtherealFlowerette]

I love your videos, I wanna a teacher like you. Could you tell your journey of maths?

[@segads]

Hey there! Thanks for the video!

[@loading...2876]

Legend!

[@Nickle314]

Would be interesting to do a video on why the cross product doesn't work in more than 3 dimensions. Which leaves onto Geometric Algebra. Interesting that there were attacks on Grassman. I didn't know that.

[@Mik1604]

I have never onderstood why a complex number is written as a sum, when in fact it’s just a pair of values and they are not added at all.

[@NaseerAhmad-dd6pp]

Thank U sir for mix up maths with real life example by animation interesting vision

[@snehangshuroy]

Thanks!

[@watching4410]

22:30 I^2=j^2=k^2=-1 I•i=j•j=k•k=+1 I•i=0 when perpendicular

[@amrabdellatif9956]

I love math history!

[@geertdejonge4194]

Thanks, great

[@marioabrantes4932]

Very good!

[@ramazanmert7686]

Perfect

[@wadesammohamed]

Okay this make sense now

[@amrabdellatif9956]

Hey Jeff! Cool video! I found out about it through your email. Keep sending me emails if they're gonna include videos like these!

[@douglaswolfen7820]

Great video! I found the title a little misleading though. I associate vector calculus with the gradient operator, and div and curl and so on. I was looking forward to seeing how quaternions work when using those operations. But the explanation seems to stop before it reaches the "calculus" part of vector calculus

[@angeldude101]

From what I can tell, the problem seems to be that physicists were trying to use quaternions as points, when they're most definitely not points. The computer graphics crowd actually had the right idea in using them for rotations, because they _are_ rotations. If you're not trying to rotate things, then quaternions would just get in the way. That said, a lot of modern physics, especially quantum mechanics _is_ just rotating things, which is why the physicists have since independently rediscovered quaternions and called them "spinors," which is a much more descriptive name for them, though the notation they use is even _more_ obfuscated than the already obfuscated quaternions.

[@joshuazeidner8419]

oy vey

[@godfreypigott]

"Maths" .... Wow, the correct word! I take it you're not American.

[@ProfJeffreyChasnov]

Haha! I am but I am fond of the s

[@godfreypigott]

@@ProfJeffreyChasnov Good to know. But, sorry, I feel the need to check other Americanisms: (1) Do you pronounce the c in Arctic? (2) Does "buoy" have one syllable or two? (3) [my biggest bugbear] Do you say "I could care less" or "I couldn't care less"?

[@Nickle314]

@@godfreypigottWrite me. One of the Americanisms I hate. The other is burglarized. Take a verb, turn into into a noun, then add an extension to make it back into a verb. Why not just use the verb?

[@godfreypigott]

​@@Nickle314 I can only think of "coronate" at the moment. What other examples do you have? I also hate extro, "gas" for petrol, takeout, "period" instead of full stop, cookies, "in the hospital" instead of "in hospital", "herb" without pronouncing the 'h', pronouncing "semi-" as "sem-eye", drapes, faucet, normalcy, sidewalk, "meet" instead of "meeting", and having no clue what a fortnight is. And one for any fellow mathematicians - saying "quantity squared" instead of "all squared".

[@Nickle314]

@@godfreypigott Out of your list, normalcy I agree. I also hate the rocket launches with 'nominal' instead of normal. On the write me, I suspect that was English when people left England for the US. Then in the 19th century the academics got to work and decided there has to be a preposition. Gas, short for gasoline, I don't object too. herb with no h is odd too. Another perhaps is Cilantro.

[@samholt9177]

Does this guy actually know what’s he’s talking about or is he just reading a script😂😂😂😂😂

[@ProfJeffreyChasnov]

I wrote the script. What's your point?

[@igormarkov6254]

Thanks!