Quaternions EXPLAINED Briefly

[@Math_oma]

I just started a Patreon if you appreciate the work done on this channel: www.patreon.com/Mathoma Thanks for viewing the channel!

[@ffggddss]

"Now oftentimes in math, we have many different ways of viewing the same thing." Bingo!! One of the ways I like to define the quaternions, is as complex numbers over the complex numbers, where the 2 imaginary units are distinct, and anticommute. If we define q = A + Bj where A = a + bi, B = c + di you then have q = a + bi + cj + dij So then you just call this new, "product" of imaginary units, a new, third imaginary unit, k, q = a + bi + cj + dk And now, all the multiplication rules for imaginary units can be worked out. From our premises that i² = j² = -1, and ij = -ji = k, we have ij = -ji = k jk = jij = -ij² = i ki = iji = -i²j = j ik = iij = -j kj = ijj = -i k² = ijij = -iijj = -1 Fred

[@segurall1]

@Mathoma when multiplying both side by a variable is it a rule that, that variable comes first. Eg.. j*j*k=j*i vs j*k*j=i*j or is this only the case because were talking about higher dimensions here? It is it dependent on the side of the variable which is being multiplied?

[@michaelcharette5124]

%

[@k6l2t]

I haven't finished watching the video yet, but I came down to thank you for using black background and white text. EVERYONE SHOULD DO LECTURES THIS WAY!!! My eyes are not burning for once holy shit dude.. THANK YOU

[@mydogbrian4814]

Me thinks NOT!

[@jbexta]

Spoken like someone watching at 3am. Also me

[@wisdomokafor9631]

Thanks so much you really saved me. I could not find anyone who could explain this topic better than this

[@peternolan814]

Hello, Many thanks indeed for uploading this set of five videos about quaternions. Most interesting. I'm 63 and now at last I know about quaternions that were discovered by Hamilton an Irish mathematician as you know. I live not too far away from the place a canal where the plaque with his formulation of quaternions is written at the spot where it all became clear to Hamilton while he was walking on his way to his university Trinity College Dublin from where he lived that was what is Dunsink Observatory today and then. All the best, Peter Nolan.(Ph.D., experimental physics). Dublin. Ireland.

[@Math_oma]

+Peter Nolan Even though people don't often learn of quaternions anymore, the concepts live on whenever you write a dot product or cross product. You might be interested in geometric algebra, too (or Clifford algebra more generally). I'm putting together videos on that topic but when I eventually start talking about the geometric algebra of 3D, the quaternions will pop up yet again, this time with a different interpretation than "arrow" or "vector".

[@peternolan814]

Hello, I am just about to watch all of your videos a second time starting with the first one above. The astronomer I was telling you about above was telling me that quaternions are used to steer satellites and as you undoubtedly know there are many other applications for them as well. The i, j and k in quaternions are not the same as the i, j and k that we used when we were being taught about vectors starting in secondary school and I found that a bit confusing to start with. All the best and many thanks, Peter Nolan.(Ph.D., experimental physics). Dublin. Ireland.

[@Math_oma]

+Peter Nolan Yeah, they mean different things but those i,j,k are nice historical artifacts from the quaternion days of physics. Of course, nowadays they are just placeholders for x,y,z with no algebraic significance.

[@peternolan814]

Many thanks for that clarification. I had not heard the word "placeholder" before. As they say in America every day is a day at school. I'm 63. All the best, Peter Nolan.(Ph.D., experimental physics). Dublin. Ireland.

[@Math_oma]

+Peter Nolan Perhaps "placeholder" wasn't a good way to describe the current function of i,j,k now that I think of it. They certainly represent the basis vector in the x,y,z directions so they do indicate something.

[@nodoesart]

I have a presentation to do in my first year of university (i'm doing a double degree in maths and physics) and we chose quaternions with my two friends. I'm the only one who speaks English so I'll be able to brag and make them think I'm a quaternion expert because I watched your video. It was really well-explained and easy to understand, thank you !

[@antoinedevldn]

This is how beautiful a concept can be explained! Flawless, thanks a lot :)

[@SuperAcousticDude]

This was wonderfully explained. I cant tell you how many channels ive been thru trying to find something like this. Many try to flex their knowledge of the topic without realising that some of their viewers are there to learn the topic from the ground up. Amazing work!

[@benjaminv3748]

I still learn very (relatively speaking) math, in fact I have only briefly touched complex numbers and i, yet this video was very intresting and taught me some basic quaterion rules. Now I probably won't use this knowledge for another few semesters but videos like these always keep me motivated to learn more and get there!

[@pootisspangle6636]

What college you at? I like Harvard because math 55 teaches me about topology, where there are subsets, elements, n and k cells, intersection area of, total area of, neighborhoods, hausdorff neighorhoods, euclidean space, hausdorff space, converting topology into metric, paths, quantifications, including uniqueness quantification, existential quantification, and universal quantification, closures, interiors, boundaries, CW Complex, isotopy, homotopy, homomorphism, morphing into other shapes, paths, congruent, changed to, if and only if, functors, open balls, closed balls, fibers, and more.

[@PARISAROMAN]

No angry comments, the opposite, I am happy to see and listen your awesome exposition. Thank you very much Mathoma.

[@tombruckner2556]

That's the first time I actually understood what Quaternions are all about. Kudos to you!

[@VladTepesh409]

I find the quaternion multiplication table very informative, as well as a refresher from the other parts of your video. Thanks!

[@efraingbj]

Thanks for the video, I'm currently working on a robotics project and it's the first time I heard about quaternions, your explanation was pretty spot-on as an introduction to the topic! Also your channel seems really interesting, great job! Greetings from the other side of the wall!

[@xfry]

OMG OMG OMG!! I was stuck reading this topic in a book and you finally has given to me the final understanding about this topic! You are amazing dude

[@DWORLD-xl4pb]

I was with you Doc all the way until you got here: 11:00 When you multiplied by ijk = -1 => iijk = -I... All these calculations take graduating layers of abstract thought and my thoughts failed after the third level of complexity. I love math but I could never get beyond algebra 1 to precalculus because so many assumptions of the abstract seem incorrect. I don’t know if that makes sense, but hopefully we can help a whole new generation break the fear of beyond. Live long and prosper sir! 🖤💪🏽👌

[@davidmurphy563]

Ok, I'm replying to a two year old comment but I paused at the very same point. This is me gathering my thoughts - it helps me to write it down - so just ignore me. The issue is that normally -1 * -1 = 1. So vec3(-1, -1, -1) * vec3(-1, -1, -1) = vec3(1, 1, 1). Which is the same as vec3(1, 1, 1) * vec3(1, 1, 1) when obviously it's not the same. I mean, -1 * -1 = 1 is just a definition but it's a crappy one in this context because it's asymmetric. Assumptions of the abstract as you say. Just because a vector is pointing in the negative direction doesn't mean we want it to behave differently when we multiply it. Obviously we don't. So let's just change that rule. So now, when it's negative, let's multiply its positive and then add the negative sign back after the multiplication is done and return that result. Do this and -1 * -1 now outputs -1. I think proper maths takes the sqrt(-1) but the result is the same so who cares? I'm not sure, but I suspect all this mathematical hocus-pocus is just to make negative numbers behave like positive numbers and not magically flip on you. In computer terms you'd call this patching a bug. Clearly, -1 * -1 shouldn't =1, at least in this context.

[@oscarruiz8834]

I liked the video. It leaves many things un-answered, but it it useful. It is just natural that one should read and search more on the topic before one understands what is going on.

[@lukastheinfinite7311]

You think and explain in the way I do it. Kudos sir! And thank you for your effort!

[@nexusclarum8000]

I like how you write the letter 'q'. Also good explanation.

[@elimenendez237]

Thank You! That's the only explanation of Quaternions that I could understand properly, very well explained

[@rohan.fernando]

A brilliant explanation of quaternion algebra. William Rowan Hamilton would surely be impressed. Well done.

[@DevShah-z2d]

at 1:31 when you explain so simply the R^2, you have my like on the video

[@arongil]

Khan Academy 2.0! This is great!

[@Math_oma]

+Arongil Productions Yeah, his videos are some of my main inspirations, except I work at a slightly higher level.

[@mrigank8822]

Mathoma khan academy is a little bit too basic even for beginners

[@redrevelations]

Thank you very much for explaining quaternions in a very simple way. It helped me a lot.

[@DavidFosterZen]

I do all my calculations with Octonians. Excellent Video. Keep up the hard work.

[@Math_oma]

That's interesting. What sorts of calculations are they? It's funny you mentioned the octonions because I was going to make a video on the octonions and the Cayley-Dickson construction (unless you beat me to it).

[@DavidFosterZen]

It was purely in jest. I have a hard enough time time finding a practical uses for quaternions. But if I had a use for them, I would definitely use them, even if my heart is really with Clifford Algebras.

[@Math_oma]

Oh damn, I was actually looking for a nice application of the octonions, other than bizarre string theory stuff.

[@DavidFosterZen]

Clifford Algebras... that is what you really want. Are you familiar with those? And if not, would a video on those be helpful? Much more intuitive and versatile.

[@Math_oma]

I'm not too familiar with Clifford algebras even though I have heard the name in association with hypercomplex numbers. The topic is on my long to-do list of math topics to read about. Unfortunately, the Clifford algebra videos on KZbin seem to be too high-level, but it would certainly be interesting if you (or someone else reading this) made a video on the topic.

[@pootisspangle6636]

This is the most simple genious idea ever. 👌

[@craigruchman7007]

My math only goes up to abstract algebra, so this was a nice intro for me, thanks

[@rlenclub]

I thought that *brief* was the wrong word to use here until I remembered you're explaining quaternions.

[@OomptzMusic]

Love the "blackboard" presentation. Great video 👍

[@gurunath72]

Very nice explanation, especially liked the bit about how an abstract mathematical idea became an useful one.

[@youssefgaaloul]

I dont get why ijk=-1. Is it something I just have to accept? Assuming so then I understood the general way of multiplication of quaternions as you explained it.

[@leochinchillaa]

same i thought ijk would have to be i^3 = -i =-j =-k

[@skub22do]

At 12:28, why is the 'j' placed to the left of 'i', rather than the right?

[@TheSam1902]

Thank you for this video ! I'm currently in high school and I'm developing some video games. Since I didn't know how quaternions worked I was using Euler's angles which are (magically) translated to rotation by some modern game engine (i.e. Unity3D). Now I'll be able to work directly with quaternions so thanks for quenching my thirst of curiosity !

[@Math_oma]

+Black Rainbow You're welcome; quaternions are one of my favorite topics in math so I always enjoy talking about them. This particular video probably won't tell you much on why they work in rotations, but hopefully that will make sense when you see my other quaternion videos. The more I study this topic, the more I'm convinced Euler angles and other contortionist routines using matrices are the wrong way to think of 3D rotations. Quaternions and more generally the geometric algebra of R^3 are too natural and the formulas are too concise for them to not be the best way to understand rotation.

[@darovan4398]

You are a legend ! Thanks for a wonderful lecture

[@ignitiondj4025]

I AM ANGRY THAT THIS VIDEO EXPLAINED QUATERNIONS SO WELL there you go you're welcome

[@rohanbandaru]

How is the equation at 9:10 even possible? Shouldn't i*j*k = -1^1.5? How do i = j = k = -1^0.5 multiply together to get -1?

[@MuffinsAPlenty]

i, j, and k are all different square roots of -1. Different square roots don't have to multiply together to give the number you're interested in. For example, 2 and -2 are both different square roots of 4. 2(-2) = -4, not 4. Even with the real numbers, you can't just write all of the square roots of 4 as 4^0.5. You need some way to distinguish the two square roots, and we use direction (one being positive and one being negative). For the quaternions, you get _lots_ of square roots of -1, 6 of them, actually! This requires that we need more "directions" for square roots. You can think of i, j, and k as directions that these square roots live in. One of the other important things to keep in mind is that quaternion multiplication is not commutative, so ij is not equal to ji. This often messed people up when they try to do computations with quaternions for the first time.

[@rohanbandaru]

@@MuffinsAPlenty Ohh thanks this really helped me understand.

[@DrDanielHoward]

Thank you for the explanation and understanding of the Hamilton graffiti.

[@eliasbrassitos1]

What a nice and clear explanation, thank you!

[@FrankClautier]

Hello,Thank you for this interesting series of videos! I just have a question: around the 11:00 mark, when multiplying both sides of the equation ijk = -1 by i , do we assume associativity applies? Just wondering this, since i(ijk) is assumed to be equal to (i^2)jk . Thanks : )

[@Math_oma]

+Frank Clautier Right, we're assuming the associative property when we do this. In a different formalism, it's provably true that quaternion multiplication is associative but I prefer setting up the quaternions in this way.

[@FrankClautier]

Ah ok, thank you for the clarification :)

[@ntt3597]

Sir your content is amazing. Thank you so much.

[@maxwellsequation4887]

I LOVE THIS CHANNEL!!!! This was soooooooooo awesome!

[@TheRealAfroRick]

This is the first video that broke this down in a way that actually made some sense because it didn't try to dumb it down and hide the maths :)

[@cupajoesir]

very well explained. you have just gained a subscriber.

[@unternerdsbyalexandraelisa7550]

Thanks for this introduction to quaternions!

[@jedmaegraith417]

Please correct me if I am wrong, but wouldn't the answer to the (14:32) substituted chjk x-i (4 across, 3 down)be *Minus* chi when multiplied? Thanks,

[@addisonballif2988]

if the three imaginary numbers are not commutative, then how do you make sure that you get the right order when your are distributing? How does distributing work with non-commutative numbers?

[@MuffinsAPlenty]

You keep things on the left if they started on the left and on the right if they started on the right. So a(b+c) = ab+ac and (b+c)a = ba+ca.

[@GiraffotTV]

This is great. Thanks. One remark: It's quite confusing when you point things out on your sketch and say "here, and here..." but I can't see where you are pointing:-))

[@alyssasynakowski5748]

I'm curious how you might go about calculating the difference between two 9 axis IMU sensors using their corresponding quaternion coordinates? I'm working on a project where I have a sensor attached to the side of the chest and another attached to the arm. My objective is to calculate the position of the arm relative to the body using the quaternion coordinates. Unfortunately, I"m afraid I don't understand them enough in order to come up with an equation on my own. Any help would be very much appreciated. Thanks!

[@gewoonjulian5917]

You're great, thanks for this great explanation

[@LanceGomez]

Hamilton is the bo rai cho of mathematics. It's crazy but it works! I juggled around proving those and damned but it worked.

[@PrashantAthavale]

Thank you for your nice explanation. Which book(s) did you use in this series on quaternions? Or which book(s) would you recommend?

[@amisus1]

I'd like to see more applications of quaternions; in gyroscopes for example. Nice and clear lecture, thanks!

[@Tr0lliPop]

I am confused. why is ijk not equal to -i, and instead equal to -1? if two numbers are both the square roots of one, would'nt that make them all the same number? that would mean that ijk = -1 x (i or j or k). but it seems to not be the case. edit: wait. j times k is i? but they are all roots of one? why is that one of them multiplied with themselves equal to -1, but 2 of them multiplied with each other equal to the 3rd one? edit 2: wait why is ab = -ba? edit 3: ok so apparently I am trying to learn undergrad math in 9th grade just because I was curious what quaternion means

[@clementesantiago1763]

ijk=-1 disturb too much. What happens if I square both sides of the ecuation?

[@MuffinsAPlenty]

One thing to keep in mind here is that quaternion multiplication is _not_ commutative. For example, ij is _not_ equal to ji. One of the rules you're used to about exponentiation distributing over multiplication, i.e., (ab)^2 = a^2*b^2, actually _relies_ on commutativity of multiplication. Why? Technically, (ab)^2 = (ab)(ab) = abab. If multiplication is commutative, then abab = aabb = a^2*b^2. But if multiplication isn't commutative, it's not necessarily true that abab = aabb. So, let's look at what happens when you square ijk. You don't get i^2*j^2*k^2, rather, you get ijkijk. This ends up being 1, by definition.

[@Kurdoman]

You are excellent in explaining it. Thanks for the effort

[@xorxpert]

OMG. Your handwriting is 😍

[@baqtronic7704]

Great explanation, congrats!

[@mohameds3354]

How is it used in the computer graphics? It always is good to listen to somebody who knows what he is talking about.

[@lyrimetacurl0]

Here's an angry comment: "you didn't cover how it actually applies to 3d rotations / transformations." I am still left imagining what exactly i, j and k actually represent. I will find out from a different video, I guess. Currently my understanding of 3D is "six degrees of freedom" (covering position / sliding and rotation) but I have the feeling quaternions also cover skewing / stretching (which are generally not required in 3D games). Also not heard if Octonions are required for 4D, etc (and if so, why would the number of dimensions for transformations in d spacial dimensions be 2^(d-1)?). But that would be for a different topic.

[@DanielArnett]

Check out this webapp, and specifically hit the switch at the middle top so it shows the sines and cosines. The video is great too, but in the sine/cosine form it clearly shows how the quaternion is defining an axis and then rotating by an angle around that axis. eater.net/quaternions/video/intro

[@monatsend]

at 12:27 you multiply both sides by j. why do you get jjk=ji and not jjk=ij? How do i know where I should write that j down? Is there a specific rule?

[@Math_oma]

+monatsend Just keep left-multiplication distinct from right-multiplication.

[@bin9294]

why?

[@harshitagupta4430]

Explained so well💖

[@seanocansey2956]

Thanks man, these are really helpful!

[@davidfuentes7806]

Thank you so much! This video was really usefull!! Thankss!!

[@ViliamF.]

0:38 Well, that's a very unusual way of drawing N

[@garrytalaroc]

What's the difference between 4d vector and Quaternion?

[@tjzx3432]

I find a good way of thinking about quaternions is to imagine to objects approaching to impact in order to cause the sum vector. This is of course a non elastic reaction, however the complex solution would be elastic and would have a loss of energy equal to the elastic energy expended. Following this train of thought this would mean that 5th dimensional dynamics would include internal variables that affect the overall motion, this would be analogous to planets colliding, or in qed with atomic emissions and interactions. But this concept seems very useful, so long as you are able to intuitively integrate the imformation.

[@lvlotives]

Many thanks for sharing your knowledge 😊

[@aikensource]

I'm here because I'm trying to use Quaternions for rotation in 2D Unity game dev. It's hurting my brain, but I did learn that the w component (x,y,z,w) of a unity quaternion may be a scalar. lol.

[@Math_oma]

+Sparkplug1034 Yeah, it's possible that the "w" is the scalar there and they write the scalar as the fourth component (whereas I write it first). Let me know if you have any questions (comment or email me). I also have further videos on this rotation topic.

[@aikensource]

I'm trying to do a Quaternion Slerp in Unity so an enemy game object gradually rotates to turn towards the player before firing a weapon. I spent 2 days working on the trig before finding out that unity quaternions have looking at something built in :(

[@aikensource]

DUtOO Thank you. I did start using Quaternion.Euler(Vector3) , but because of what I need my objects to do, I am using other methods. Thank you though!

[@420praiseit2]

Are you rotating the y-axis (euler) of the enemy or z-axis (euler) of the enemy weapon? In either case, you're overcomplicating it.

[@aikensource]

crestfallenllama I'm not over complicating it in the slightest, thank you though. I'm rotating the Z axis. And it does rotate correctly. It's not supposed to just rotate though. I'm coding it's AI, using some quaternions and it's just new to me.

[@sebastiandocktor8190]

thank you for the video. - good explained

[@silverlining6824]

All lectures on quarternions are given by mathematicians, via complex numbers. Maybe the following practical application will motivate the need. Rotations in 3D can be expressed in terms of two angles, theta and phi. One of them lies in the plane formed by two of the orthogonal axes, say X and Y; and the other in the plane involving the third axis, say X and Z. Any 3D rotation can be expressed as sine and cosine of theta and phi. So, what is the problem? Why are the quarternions useful? Trigonometric functions such as sine are computed as infinite series. (Look up Taylor Expansion for the Sine and Cosine functions). Exact solution involves infinitely many terms. Bit real time gaming demands fast computation. Yet, truncation of a series as approximation necessarily involves errors. So what is one to do? This is where quarternions come in; they involve only dot and cross products of real numbers - very fast and at the same time exact and precise. Now, are you motivated to follow this or any other presentation on quarternions?

[@dralbertomarquez]

Fantastic explanation. I'd like to see one with functions. Are there applications you know using vector functions and not just vectors

[@Math_oma]

+Alberto Marquez Oh sure, a lot of what we call vector calculus was originally done using quaternions. I'm pretty sure you can get the standard divergence and curl operators out of multiplying by a quaternion with the del operator in the vector part.

[@jonvance69]

Great video! Big shout to the Irish mathematician ☘️.

[@marcofe82]

Your explanation is really simple and perfect! Another question, which kind of graphics tablet You used? May I have the model? Thank you so much!

[@EngineerNick]

Really nice explanation great job :)

[@elvisnobregadealcantara642]

Thank you so much for the video, it is very informative. If I may have the chance to try to improve your work, I would say to you use different colors in the writings, that add information to the image

[@brendawilliams8062]

Thankyou for the generosity in sharing. Thankyou.

[@jadensmiley6297]

Thank you! I finally understand!

[@DavidFosterZen]

As I think about it, I recall using imaginary numbers with Mobius transforms (conformal transformations) and of course, fractals like the Mandelbrot set. Now, I have to wonder, if we can put Quaternions in place of the complex numbers in these things, do we get anything interesting?

[@Math_oma]

I think I've seen some quaternionic Mandelbrot sets either on KZbin or elsewhere online. Here's one link I found: kzbin.info/www/bejne/d6qvp6yoip6Gg7c I couldn't tell you anything about the Mobius transform; I haven't worked with it.

[@Eilbheis]

When you come to learn about math but end up converting to Catholicism

[@Math_oma]

Pretty much.

[@dmitrykolesnikovich]

but why ijk equals to -1 please?

[@APaleDot]

Because ij = k, so ijk = kk = k^2 = -1

[@joer7151]

Do you have a follow-up quaternion video? I'm, for one, interested

[@Math_oma]

+joe ren Yes, check out my quaternions playlist.

[@Aidiakapi]

i was trying to make a jk, but it turned out it was already equal Also, I'm angry!

[@Math_oma]

+Aiddiakapi I guess I'm not sure what you mean in your first sentence. Good, get angry!

[@Aidiakapi]

+mdphdguy1 jk = joke, i = jk

[@Math_oma]

+Aidiakapi Oh, I see. I didn't know "jk" was an abbreviation for "joke". Doesn't seem like it's a word that needs to be abbreviated.

[@Aidiakapi]

mdphdguy1 It is definitely not needed, only for bad puns occasionally :).

[@alexanderradoslavov1459]

its abbreviation of "just kidding" too

[@marksebastian6689]

Now, can we reduce this to a "simple" script to actually do such multiplications in a timely manner?

[@libertyhopeful18]

do you mind if i ask what your particular field of study is? is it more mathematics or physics. i was reading a book on quantum physics, and found that when the pauli matrices are considered along with the 2x2 identity matrix, it forms a quaternion. now i have only really heard about quaternions from this video. but I'm curious if you have any knowledge of this as it applies to quantum mechanics. book is called quantum mechanics written by leonard susskind

[@Math_oma]

+libertyhopeful18 It's actually neither - I'm a medical student with a research focus in neuroscience. I'm merely a wannabe mathematician. I'm extremely rusty on quantum mechanical stuff but I do know the Pauli matrices are basically a rediscovery of quaternions and (when multiplied by i) are isomorphic to the quaternions.

[@cortsmith1981]

What program did you use to draw with? I have seen this program used here and there lately with game development videos and such.

[@Math_oma]

+Cort Smith I use SmoothDraw4 with a Wacom tablet.

[@wawmsey]

thanks, really well explained!

[@georgekyriazopoulos6985]

Excellent presentation

[@Kteezer]

If you imagine a 3D plane where the x axis is i, the y axis is j, and the z axis is k, then you can use the right-hand rule to cross two of the axis in order to easily recreate the multiplication table you made :)

[@Math_oma]

+Kteezer Right. This multiplication is the historical precursor of the cross product.

[@pavelp80]

I have one question about complex numbers. Solution of quadratic equation has: - two or one real solutions representing intersection of parabola with X axis - two complex solutions Is there any special meaning of those two complex numbers?

[@carultch]

Yes. Take the original parabola in the real numbers, let's call it z=x^2 (you'll see why I'm not using y, soon enough). Call the original parabola P1. Put the real number inputs on the x-axis, and the imaginary number inputs on the y-axis. Make an identical copy of P1, and call it P2. Rotate P2 by 90 degrees around the z-direction vertical axis through the vertex. Now mirror P2 about the horizontal plane through the vertex. You now have the extension of the original parabola to the domain of real and imaginary numbers. The roots of the quadratic equation that are complex, will correspond to where parabola P2 intersects the x-y plane. The x-coordinate of these intercepts is the real part of the complex root, and the y-coordinate is the imaginary part. If we had a 4th dimension to work with, we could form the full continuous parabola of all the complex inputs in all its glory, and include the complex outputs. But, because our range is restricted to real numbers in the z-direction, it gets difficult to visualize. The parabolas P1 and P2 as I defined, are the intercept where the complex part of the solution to z=x^2 equals zero, and z is exclusively real. Many times, color shading is used for depicting the imaginary part of z, so that it ends up looking like a heat map on a 3-D surface. You may also see color shading to indicate the angle of the imaginary number, and Z-position to indicate the magnitude, where the x-y values correspond to the input to the function.

[@BrettEskrigge]

Nice video as always. I've known of quaternions since Numberphile first covered them, and I have been curious since then as to how i^2=j^2=k^2=ijk=-1 was originally come up with. That is, what is the justifiable reason that this equation true? I understand that mathematics is about pushing our understanding, and I can understand the idea about having 3 different complex numbers in quaternions. But I don't understand the ijk=-1 part Is there any other reason as to why this is true other than, 'let this be true'?

[@Math_oma]

The equation ijk=-1 seems to me to be more of an insight as to how to set up the definition of quaternion multiplication than a true statement. Once you experiment around with that equation i^2=j^2=k^2=ijk=-1 and arrive at the definition, you could just throw away the ladder and just say that quaternion multiplication is more fundamental and then calculate: (0,1,0,0)*(0,0,1,0)*(0,0,0,1)=(-1,0,0,0), which is another way of writing ijk=-1. This is roughly analogous to how one might treat complex numbers; instead of philosophizing over what i^2=-1 means, what we do is just follow the definition and calculate (0,1)*(0,1)=(-1,0). There may be a more satisfying answer regarding ijk=-1 from the point of view of "geometric algebra", in which some mathematical statements are grounded in the geometry of 3-dimensional space. But that requires talking about things like the inner product and wedge product and I've only just started reading about that area of math. I might make a little video series on it in the future, seeing as how it doesn't seem very difficult to do.

[@BrettEskrigge]

+mdphdguy1 Alright, I can understand that. I have a brief understanding of inner products, so I'll just treat ijk=-1as being an inner product of quaternions. Thank you

[@Math_oma]

If I remember correctly, in geometric algebra, there are things called bivectors which are similar to the quaternions and multiply in a way to generate ijk=-1. In 3 dimensions, there are three bivectors called, e_12, e_23, and e_13, which all square to -1 and its also true that e_12*e_23*e_13=-1, where * is something called the geometric product. As I said, I've only just started reading about it so I would look up some other reference to read about it.

[@BrettEskrigge]

Yeah alright, will do. Thank you very much

[@josephcote6120]

OK, this is how I understand it. You know that multiplying by i is the same as rotating by 90 degrees in the complex plane? To describe a three dimensional point you need three planes; j and k are the 90 degree rotations in those perpendicular planes. And it is just defined so that if you make all three rotations you end up pointing the opposite direction. For a concrete example, say the starting point is the north pole of the earth. i represents rotation in the plane that cuts through 0 degrees longitude. Moving that point by i beings it down to the equator, doing it again (i^2) beings it to the south pole. Call j the 90 degree rotation through the plane of the equator, j moves 90 around and j^2 moves 180. Make k a 90 degree rotation in the plane of 90 longitude, same thing. Here comes the prize. Start at the north pole, multiply by i, point's on the equator, now multiply by j, point has swing 90 east making an L shape, last multiply by k from that point and it takes you to the south pole (same as multiplying by -1 to begin with.) so i^2 = j^2 = k^2 = ijk = -1

[@evanroderick91]

quaternions are very interesting, yet a little confusing. Awesome vid though!

[@omegaPSI2006]

Where were you all this time? I'm so angry I just found this video out instead of years ago!

[@SandburgNounouRs]

Hello, When you say "multiply 2 quaternions together", could you also say it is a cross product of 2 vectors of dimension 4 ?

[@Math_oma]

+SandburgNounouRs The cross product is really a three-dimensional concept (and 7 if you're masochistic) so I wouldn't want to explicitly say that this operation is a crossing of vectors. Remember that the cross product in three dimensions takes in two vectors and outputs a vector orthogonal to both inputs, has a length equal to the area of the parallelogram swept out by the input vectors, and is oriented by the right-hand rule. Such a thing only exists in three dimensions because as soon as you go into four-dimensions, there is no vector that has these properties. Simply put, in 3D if you take a plane, the set of all vectors orthogonal to the plane is a 1D subspace (a line) whereas when you move to 4D the set of all vectors orthogonal to a 2D subspace (plane) is a 2D subspace. Notice how the dimensions of the subspaces always add up to the dimension of the underlying space, e.g. 2+1=3 and 2+2=4. However, the whole concept of the cross product actually arises from quaternion multiplication, not the other way around. If you look further in this playlist, you'll see that the cross product (and dot product) is a part of quaternion multiplication.

[@jennifersiagian]

I guess I am a thorn in your side.. but really Thank You for freely sharing what you know with us.. I re share everywhere.

[@Math_oma]

+jennifer siagian Not at all; I enjoy reading comments and talking to my subscribers.

[@jennifersiagian]

Mathoma really I am on a Math a thon with you.... your so beautiful Thank the Father He blessed your mind

[@jennifersiagian]

Mathoma ps I re share all your Math google + g night (but I am still here dong the playlist)

[@Math_oma]

+jennifer siagian I appreciate it! Glad to see you're enjoying and learning from the videos.

[@jennifersiagian]

Mathoma where's life there's hope hah ok can you do some teaching on Fibonacci sequence? I love patterns.. ok g night

[@hovhadovah]

11:37 isn't this a contradiction of the fact that i = sqrt(-1), j = sqrt(-1), and k = sqrt(-1)? Because then you get that sqrt(1) = 1 = sqrt(-1)... implying that i^2 = i.

[@APaleDot]

??? But sqrt(-1) does _not_ equal 1.

[@cory1111]

This is the most interesting thing ever right below gravity

[@SamuelHabekotte]

"These Quaternions very famously find application in those who create video games" *busted* Thank you for this very informative video

[@aashsyed1277]

what are you using to write?

[@AhmedIsam]

1:56 "quaternions are just 4D vectors living in R4" They actually live in C4, not R4, otherwise there's nothing special about them.

[@bernhardriemann3821]

Ahh professor Isam, Another mathematical god joined the chat.

[@BanksySan]

R4 includes the imaginary part.

[@SoumilSahu]

You're wrong. They're a 4 tuple and isomorphic to R4

[@jennifersiagian]

I love your mind..Amen Jn 17:21-22 3 = 1

[@grossersalat578]

Piece of graffity, haha. Hamilton scratched it in stone!

[@tuexss]

Now do Octonions! :D