As a second year university math student, I love this video. Not only it is constructing number sets without mentioning axioms and algebraic properties, but it also builds upon real life concepts that make sense intuitively. Well done, I am really glad this video exists, good job
And what make all of that insanely awesome is those creative new real life concepts that has never existed before especially if they make the last ones more sense it's like an old perspective had been expand
Finally, after all these years of searching, I have found it. Intuition for imaginary and complex numbers. This video is simply brillant, I really hope it blows up more because more people need to see this! Fantastic work. This makes me want to try making a quick lil 2d game where the coordinates are just complex numbers, and after watching this I can actually see that as a very feasible thing to do. Genuinely thank you for this one, it's great. +1 new subscriber!
@@imaginaryangle I'd've used c++, and you can import the complex package to make that work. But realistically I won't have the time to really work something like that out, unfortunately
I loved the video. You explained some of the most fundamental and philosophical ideas in counting and number theory so very succinctly. The first few minutes were a blast. No one usually goes into such topics but did it so easily. The ideas behind counting and what it means for our mind may not even be important to most mathematicians, but for me it is fundamental to realise what exact process is happening in my mind and the physical substrate of the universe itself - from which math is made of.
i'm going to get my friends with the walk -2 +2i steps foward.... that analogy was brilliant. i always tought the complex numbers, as a modified version of the cartesian plane, and everyone exept one person got it (the one who didnt get it was a guy that avoided math like the plague, he is also studying psychology funnily enough)
Thank you! My second option for describing them were dance moves, but for that I would have needed to use clips from movies and I'm not experienced enough with fair use yet.
I got confident understanding the complex numbers as rotational number long ago but it never clicked to me that indeed they could still be used for counting, exactly like negative numbers can be used for missing quantities. Incredible viedo
Could you do this same sort of construction for: - algebraic numbers - in particular, algebraic integers (is there a way to relate those to whole cheeses rather than partial ones?) - geometric algebra (extending to arbitrarily many orthogonal directions, and also giving a new meaning to imaginary numbers that very intuitively extends to any number of dimensions) - finite fields (I think this one is probably quite easy) - p-adic numbers - adelic and idelic rings?
I know this will sound subjective (and it is), but I find Complex Numbers more natural than those other sets, so I wouldn't pick an approach like this when talking about them. I'm also quite impressed by content done on those topics by other creators and don't feel like I have a significant contribution to make there.
The point about places you can cut cheese that you cant describe with even pieces just made me realise how incredible the discovery of irrational numbers actually is
I looked away from the video and I came back and I just heard "It looks like paradoxically there are whole numbers hiding within the cheese, but they also make up all the cheese"
I have no doubt that your way of thinking is really capable for changing the world if we just learned more about, just to taste what it means to see things differently using our independent framework that is inspired by others
Thank you! We all internalize knowledge differently, and we need more than one angle for a new concept to find its home in our minds. I know I did and I'm grateful to teachers who reminded me by their own example that it's possible.
This is an excellent video! I've been working with math for so long that I've long forgotten what it's like to seriously think about numbers -- especially whole numbers and rationals, since I normally work in the reals, a vector space, or the complex plane. The video raises some excellent questions about what they are, what they can mean, the differences in their interpretation, what counting really is, etc. This was very eye opening, and it was a wonderful feeling having my mind blown by counting oranges.
Small point, but at 14:49, it says that we know 5/4*5 is not an integer because 5 is not a multiple of 4. This doesn't technically follow, since for example 6/4*6 is an integer even though 6 is not a multiple of 4. This argument relies on the fraction being in lowest terms, which wasn't made totally clear. Great video by the way!
Thank you! Right when I introduce 5/4, I say "5/4 doesn't simplify", which was my natural language way of saying that it's in lowest terms. I agree I could have hammered that stipulation in a bit harder, but I wanted to keep especially this part as digestible as possible for people that might be unfamiliar with the jargon.
This was phenomenal! Even though I finished my bachelor's in mathematics a number of years ago, I never really gained an intuitive understanding of the imaginary numbers pertaining to anything in the real world. If I should ever go back to teaching maths, I shall use this video as supplementary teaching material. Bravo!
All these squares make a circle. All these squares make a circle. All these squares make a circle. All these squares make a circle. All these squares make a circle. All these squares make a circle. All these squares make a circle. All these squares make a circle. All these squares make a circle.
At about 14:45 you use red and green to distinguish integers from non-integers. These two colors are very similar in brightness, so for someone with the most common form of colorblindness, like me (and a decent chunk of the entire male population), it's pretty hard to distinguish the two colors. In future math videos, would you mind distinguishing sets like this with brightness and/or pattern as well? One being solid and the other dotted or striped would make the distinction immediately obvious even to those more colorblind than me.
Similarly, at 23:00 the red and orange look nearly identical to a colorblind person; I didn't realize they were meant to be different until you mentioned "red and dark blue" - I saw the dark blue but was wondering where the red was, until I realized you must have been referring to the positive Y axis.
I realized this after making the video and in the videos I'm working on now this is taken into consideration. Sorry this was not as accessible to you as it should have been, I'll take more care in future videos. Thank you for suggestions on how to make it clearer!
Fun fact: In polish irrational numbers are called "Niewymiernie" meaning unmeasurable, which I think is a little better name, because it doesn't have an association with stupid ideas like the word irrational
I didn't know that. But to me it seems at least equally likely to confuse, since measures of an irrational amount definitely exist. Irrational as a word is supposed to be interpreted as "without ratio", which isn't the first association and even that has its own problems. What I find interesting about it is that the same double meaning ("without ratio" and "not sensible") also exists in Greek, both today and in Pythagora's time.
In your last example about cheese inventory and order state accounts for imaginary direction. If you could extend it for non-90 degree directions like the phase in electrical circuits that will be a great video. Thanks for your clear explanation.
The 8th root of unity comes up - you could conceive of a cycle with 8 steps instead of 4 by using it. But that's more imaginary accounting than I want to get into 😅
At least there’s a clear cut between constructables and unconstructable, and with a continued fraction, there is a clear cut definition for rational and irrational numbers, which is cool.
Severely underrated video. I was hooked and got interesting insights out of it despite having watched a bajillion videos on the interpretation of imaginary numbers already. Great job!
This question has been bothering me relentlessly for a year. I've been trying to learn as much as I can but I'm a true amateur. I'm trying to look at analytic extensions for tetration to all complex numbers and then look where there is a hole in the domain, and then use the inverse of tetration to define the new number like we did with exponentiation
I'm researching this topic because I also want to know (beyond just the obvious properties). There might be a video about this coming up at some point.
@@imaginaryangle how exciting!!! I like to call the inverses of tetration troot(x, p) and tlog(x, b). Also R4(x,p) and T4(x,p) for short, as tetration is fourth order hyperoperation. I was thinking 🤔 could you define a new type of mean (like arithmatic, geometric, harmonic, RMS) by using the troot()? Since the geometric mean is basically R3(a*b, 2), what is tetra mean uses the term R4(a^b, 2)? Since exponentiation isn't commutative, we might combine R4(a^b, 2) and R4(b^a, 2) in some meaningful way to get a mean of some kind. The real trouble with tetration is the noncommutativity. It makes everything difficult and the usual algebra I do fails at a basic level with tetration. But I don't think it's impossible to deal with. I've heard of multiple extensions of tetration from the naturals to the complex, and people have been able to do cool stuff with it. Hope you find the new number system!!! I feel like its out there
Isn't this a consequence of conventional choices we use when drawing axes, or in electromagnetism, when we choose North to be positive? Am I missing something? It would be interesting to know if there is some natural preference at play here that I am not aware of.
I think imaginary number should have been called perpendicular number, that give a much more intuitive sense of what they are, number in another direction/dimension.
@@imaginaryangle , @anteeko Yes .. and the trouble with perp numbers is they'd be remembered as perpetrators ;-} Mind you, there are many places in physics where they're called propagators; so maybe 'prop numbers' could do service. And because any unitary (I mean modulus 1) imaginary number is a rotator, the analogy with the propeller is not amiss. Still, you wouldn't want a pilot getting mesmerized by thinking how his props are describing massive imaginary numbers modulo 2\pi . Now if you put all this text to music, or a skit like Who's on First, it would be called a real number about imaginary numbers.. Have fun! 🤣 (smiley cis \pi/4 )
A cool number, say \omega used in some solutions of the Ramanujen conjecture for the diophantine problem (x^2+7=2^n) is 1/2(1+\sqrt(-7)) complex but similar in form to the golden ratio. A continued fraction expression for the golden ratio is 1+1/(1+(1/(1+...))). As \omega has modulus 2, a similar continued fraction expression for it is 1-2/(1-2/(1--2/(1-2/(1-...)))). Curiously enough (spot the flaw if you can .. there is one), this means you might define i formally as i = \sqrt(1/7)-1-4\sqrt(1/7)/(1-2/(1--2/(1-2/(1-...)))) using all real numbers. The trouble is, this doesn't escape its unique character of imagination, because of the fatal flaw in the formalism . Have fun finding it!
What I love about this video is that everything you've explained holds up in algebra. For instance, making four quarter turns in the circle brings you back to the positive reals. Mirroring this, i^4=1. It just furthers the whole idea that geometry and algebra are inherently inseparable and I love it.
@@imaginaryangle Yeah, the idea that e^ipi = -1 (or more accurately e^ipi + 1 = 0)is honestly mind boggling that someone actually bothered to calculate that, but even without dabbling in the subject I can imagine just how monumental a discovery like this would be
So if negative is a descriptor, I just say store ax bx cx dx, then any number is just a descriptor except 1 becausse its defines the unit descriptor, however any descriptor can be defined as you wish, but all other descriptors are infinitely dimensional, or not orthogonal. To achieve trheir grouping means to define an ordered grouping which ties them to a descriptor path plan. Unless we assume a descriptor is recursively permutatable, transmutable, or zero defined as 1. The only thing missing is a scale of zero's size, so zero's close to zero, might also be called zero's. Of course, we imply a war so there's terror, or a golden age, so there's gayety whatnot what have you then. Extremely short range hyper-dimensionalities, but a more kosher plane is further away, ok, there;s still a truck there, but I expect to see a not dog, lol!
Yo, such a nice video! Some notes but you totally don't need it. I think The example in the explanation is too detailed and makes bored. Like chess board cheese or something. People who watch this kind of video, are might bad at math, but most of them already have the basic math knowledge. Anyhow, I love your video, its really fun to watch!🎉
Excellent video. As you discussed forward and backward walking and it's interchangeability, have you thought about how the notion of "forward" and "backward" would mean when it is applied to time. Isn't it true that that time only flows in "before" to "after" direction and thus use of "forward" and "backward" as it related to time is meaningless. Thoughts?
How time flows (or how we and other things we observe flow through it) looks like we all involuntarily chug along at the same speed on a common axis. This model works for everyday intuition and non-relativistic physics, but it's not enough to answer such a fundamental question.
@@imaginaryangle Right. I guess what I am saying is that when scientists talk about reverse flow of time, it does not make sense to me. I can understand time flow is ordered set of events. But what ever that order is - the "before" event to "after" event is the definition of "forward". So even if the causality is somehow inverted i.e. effect comes before cause - it will be weird but the time would have flown in "forward" direction. So when the scientists talk about (reverse) showing a the film of broken wine glass - where the broken wine glass on the floor comes together, the film is running past the projector light in reverse direction but time still flows forward. BTW There are other similarities with imaginary numbers and light cones of Minkowski space. Just like in your example that you can go from +ve integer direction to -ve integer direction by first going through orthogonal imaginary direction of i, in the light cone - going to future (upper half of light cone) and if you want to reverse the time orientation to past (lower light cone) you have to go through spacelike volume outside the light cones i.e. so space like direction volume is like complex plane and your present moment that extends into space like volume is like the 90 degree imaginary axis. Does that make sense. Of course in case of numbers +ve direction axis is a line. In case of time the +ve direction is a light cones with the slopes of world lines of constant velocity inside that light cone differ based on your speed. So the line kind of inflated into a (actually 3D half) cone pointing up. And may be because it is physically not possible to enter the space like direction we cannot go backward in time. BTW I love the way you systematically developed the ideas one by one to give some intuition about how to think about imaginary number counting.
@SandipChitale Thank you! I haven't really thought about light cones on a complex plane. Usually when I think of time shenanigans, it's an exploration of entropy/information/energy. This will come up on the channel, probably some time next year. So I won't spoil anything here 🙃
my personal favorite at this particular moment in time is the claim that there is a continuous Real axis... this one is fun because infinity is not Real, and neither is its inverse, but the inverse of infinity exists on the Real axis, and the Reals are supposed to be all of the things that can be limits, and infinity can be a limit. so... what's goin' on with something like f(x) = 1/x in the neighborhood of x = 0? lim x->0+ f(x) = positive infinity f(0) = undefined lim x->0- f(x) = negative infinity so the immediate neighbors of 0 are the positive and negative inverses of infinity, which aren't Real by definition, yet they're what we're evaluating the limit at, and thus definitely fall on the Real Axis, which means they exist within a gap between some Real values, but the Reals are allegedly continuous by virtue of being an Archimedean Group, which is strange because the Rationals are also Archimedean but discontinous... seems like this was all made up by idiots.
quadratics are fun, too. so the Imaginaries are supposed to be perpendicular to the Reals, by definition. but have you ever tried figuring out where the Imaginary solutions to a quadratic plot to? the result doesn't make any sense if you plot on an axis perpendicular to the Real axis. but... if you're not a moron, and you pay attention to what's going on, you'll notice that quadratics plot parabolas. and parabolas are conic sections taken from a plane slicing a mathematical cone parallel to the edge. this slicing plane is thus also the plotting plane, and so the 'Real' axis and the 'Imaginary' axis of the plotting space should be perpendicular to not only the y-axis of the plotting plane, but also the coordinate system of the cone itself. and it turns out that there's only one possible axis which satisfies this condition, which means that when it comes to quadratics the Real axis is identical to the Imaginary axis. and a nice little detail that emerges from this realization is that the mathematical cone we've been taught about is incomplete. its two lobes are actually only 1/3 of the full 3-dimensional cone, since there are two other copies of the same shape, all set perpendicular to each other, filling 3-space. and the full graph of any quadratic plots the cross section of two of these perpendicular double-lobed cones. with the Imaginary solutions merely indicating the cross section of the lobe which is taken to be less fundamental to our randomly chosen orientation within the full 6-lobed cone's coordinate system. not to mention that the nicest way to express the quadratic is never used by anyone. namely: ax^2 +bx +c = 0 r = -b/2a q = c/a x = r +/-i v(q -r^2) and if we tinker with things a bit we find that when q = r^2 it's necessarily true that r/0 = r/2 for all Real r.
if you want things to work out best, addition isn't a mathematical operation. the dogma goes that addition is an operator which operates over numbers, like the Naturals. and if that were the extent of mathematics, then that might be acceptable. however, extending things to include subtraction and Integers reveals that we've really only been working with lists of Integers all along. which means that arithmetic is about vectors, not numbers. and as such it's impossible to add or subtract Naturals. for clarification consider the following: 4 -3 = 1 -3 +4 = 1 4 -3 = -3 +4 so when we swap positions between the 3 term and the 4 term, where did the + sign come from on the 4? and if - indicates a subtraction operator, why did the - sign stick to the 3? well, if what's really going on is we're just listing off the Integers +4 and -3, and that list is unordered, then there's absolutely nothing to explain. there is no paradox here, because the + and - symbols are simply the (partial) units of the Integer vectors. and this interpretation is further supported by the fact that + itself is literally just an abbreviated form of the Latin word 'et' meaning 'and'. so from the beginning addition was literally just meant to indicate a list of vectors, but the founding fathers of Axiomatic Set Theory were too dumb to realize this, and made up some other crap that doesn't work and we all pretend to follow, but in reality we must ignore in order to handle things like basic Algebra. classic cult shit.
Lovely video, so many concepts with intuitive questions and answers! One nitpick: At about 14:25 you showed a nice proof and used colors to represent numbers we know are integers or are not integers. I would advise against using green and red as colors there as red-green vision deficiency is quite common around the world (tho mostly in men i think) and while I for example can still see and somewhat distinguish between the ones you used I know a handful people who would not be able to get any information out of your coloring at all. I only say this because I think it is an easy thing to change while bettering the experience drastically for people affected, for us red and green are just very similar colors. On that note tho, the colors used at about 17:30 are a perfect example of good colors to use as there is no such thing as a red-blue color blindness. Again, I greatly enjoyed your video nontheless and will look forward to future ones :D
Hey, thank you very much for the tip and sorry this was not as accessible to you as it could have been. I will take more care in future videos, especially when an explanation relies on color perception.
@@imaginaryangle Great to hear! No problem. This happens, its an easy thing to miss especially when one does not have much contact to it beforehand but as i said i also think its very fixable when being mindful. Keep up the great work!
I'm wondering what would happen if at 21:35 instead of drawing an orthogonal line up we decided to draw a line at a 60° angle and another at 120° ; could we create 2 imaginary units ?? Great video !!
If you take the cube root of -1 instead of the square root, you will indeed get two imaginary units pointing 60 and 120 degrees to the left and two more to the right. These are 4 of the 6 you would get if you took the 6th root of 1, the remaining two are -1 and 1 itself. You can express those units in terms of i, or you can go crazy, give them names and try doing a bit of complex arithmetic with them 😉
@mehdimabed4125 There's a link in the description to a wiki page from Brilliant that has a section on roots of unity. You need to copy and paste the link, RU-vid still doesn't make my links clickable because my channel is new. Let me know if that addresses what you are interested in
you could, but it wouldn't be particularly interesting. assuming your three units are 1, the 60° point, and the 120° point, you can actually do with just two of these. the 120° point can be represented as a sum of scaled version of 1 and the 60° point, and actually, in general, if you pick any two of these units, you can always express the third as a sum of scaled copies of the two you chose. basically, you'd just end up with the complex numbers in a more convoluted form. if you're interested in a better explanation of why this is the relevant search terms would be "linear algebra" in general, and "span of vectors," "linear independence," and probably "basis vectors" in particular. if you want to explore there is something that may be fruitful thats related: invent two new 'imaginary units' that live in 3-dimensional space, so now you have 1, i, and j, each orthogonal to the other, and i^2=-1 and j^2=-1 (if you want; it's a lot of fun to explore as well, like saying j^2=1 BUT j≠1 and see where that leads). to do this you'd also have to define what i*j is, and also do not assume commutativity (if you explore and get a weird result, it might not mean you made a bad choice for what i*j equals, just maybe you took something for granted. in a lot of higher-dimensional imaginary-like spaces commutativity is NOT generally true, ie x*y=y*x is not true in general)
In terms of i I use with descriptions like undefined, Quasi-defined, semi-defined, and fully defined, (i^2 = -1 falls under fully) like i^2 = X, you can further define later like i^2 = +1, the i is usually redefined as j (for obvious reasons). I count ijk = +1 as Quasi, so is ijk = 0 but those are usually defined by epsilon and ijk = -1 but not the same ijk = -1, so like (ijk)1 =/= (ijk)2, (ijk)2 as Semi. and only i is reserved for undefined like und(i)^2 = i, und(i) = "A Imaginary number" Not to be confused with a Idempotent i, as in i^2 just = itself There's Mixings you can do cause of the Forgot Functor, so you really need notes at this point so for now I'm not going any further @@imaginaryangle
I guess that means you watched the Golden Ratio video too 😄 Welcome to the channel! A guy showed up in an old British military uniform waving a pointer stick and said it was getting too silly.
Cool so if you take two 1D values and then square them, you can get a 2D plain. That means you can always use math to reach a dimension higher than the one you're on.
Thanks for giving it a try! And you gave me an idea about where to link to my Golden Ratio video 😉 Maybe that's something you'd enjoy more? ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-dDgCg-zJq-E.html
Hi, we can use more and more complex number systems to compute more and more complex stuff. But, if we go back to a more and more basic level, we go back to using the basic Natural Numbers which every more complex number system is made up of. And even more basic, we go back to the number 1 (or the one God who is the source of creation, whether you think of creation as mathematical or physical). Thanks for the educational videos.
Math: Abridged I'm showing all the kids in the family this video. Especially for punishments. This everything with no cut. I call them complex because I think of them as valueless numbers. Positive reals being simple numbers & negative reals being imaginary.
I like your channel, but I really dislike this approach to imaginary numbers, saying that they are just a lateral movement in space. If that's true, we could just as well use 2D vectors. It's the algebra embedded within complex numbers that makes them special, and the deeper intuition is that complex numbers completely classify all translations, rotations, reflections, and scalings of 2D space, just like all real numbers classify translations and scalings of 1D space
They are not only a lateral movement in space, that's one aspect of their properties that comes up in the context of counting and measure. I chose to focus on that context here. I guess your dislike is directed at my use of the word "just" throughout, but that's "just" :) an issue of style to make the topic less intimidating for people that didn't really get a good handle on them in school. Thanks for your feedback!
Should not be so bad, maybe you were on low bandwidth? Or maybe your display has much better contrast in black than mine? 😅 There's faint stripes appearing on some stills, I realized my video editor was doing that too late to correct it.
I still dislike the name imaginary, because all numbers are imaginary. 5 isn't some physical object, neither is 3/4, or pi, they are more directly applicable to our physical world than negatives or imaginary numbers but they are still just an abstraction in our minds. The real numbers include the negatives, but it's not like we encounter antimatter on an everyday basis. Numbers are numbers, thinking that one group is less 'real' than another is a product of the history of numbers.
I get it. And of course it's all more than a bit subjective :) But I think the same is true for other concepts in natural language. If I tell you a hypothetical story about an elephant, it's different than telling you about some specific living elephant I could take you to go visit. One is imaginary, the other not, and the concept the word elephant communicates is also kind of immaterial - it's just an implied set of qualities something needs to have to be considered an elephant. Still, we make a difference between using an immaterial concept to refer to something imagined vs something more directly present.
I don't like that this video conflates counting with measurement. Imaginary numbers are not countable (in neither the set theory sense nor the ordered field sense) but they do have a measure (in the usual notion of distance). I know you're trying to build intuition, but you could have titled this whole video "Measuring with Imaginary Numbers" and used correct vocabulary throughout the explanations. That would probably be less confusing for students who then go to a class where they learn that complex numbers are uncountable ("but I just watched an entire video about how to count with imaginary numbers....") and would help them extend the ideas of countability and measurement to other mathematical objects without confusing the two.
I see what you mean, but I don't think it's likely that will happen. The uncountable property you are referring to is related to the cardinality of Imaginary numbers, which is the same as that of Real numbers. It's not necessary to be able to count all numbers of a certain type to be able to take one of those numbers and count in multiples of that number. There's an uncountable infinity of things you could be counting, but that doesn't stop you from counting rabbits. I don't know if you've seen the whole video, it does go into detail on the distinction between measuring and counting.
This jumps into a bunch of background without any summary that explains how this background relates to the subject of the video. I feel like I'm waiting for you to get to the point. The background subjects don't feel like steps toward the goal, but just stuff you wanted to talk about. Really, there's no purpose of talking about irrationals, since imaginary numbers don't need them.
I kind of agree with you that it was stuff I wanted to talk about and there are much more straightforward ways to get to the goal. Technically you can skip irrationals, sure. I took a bit of a smelling the roses along the way approach 😄
1:29 "Group" is a mathematical term reserved for group theory. Please do not use it in this context, unless you are relating complex analysis with group theory.
3:55 "Nobody can complain." The fascinating thing about humans is that you can be sure that they will find a way to complain regardless. Especially on the internetz.
This idea for actually having a use for imaginary numbers is neat, where i = an "order" for real numbers. Though I don't see how an order multiplied by an order should equal a shipped order. Of course, five cheese times five cheeses shouldn't equal 25 cheese^2 either. So the weird revelation is that the "5" in "x5" is not really the same as "5 cheese". It's saying "+5 cheese, 5 times" totalling +25 cheese ... so I guess the definition of "+5 orders, 5 imaginary times" means "25 shipped cheeses"? Maybe we need to rename "imaginary times" to a more sensible word and get used to it, or maybe using i for real world scenarios won't ever really make sense.
Think about it in terms of addition: we are always adding things to the shop; the very first thing we do is to just add regular cheese by procuring it. Later we add cheese that we transformed by changing its "orientation". You are right that if we multiplied cheese with other cheese, we'd get not just a number squared, but also cheese squared, which would not be cheese anymore. That's why physics teachers insist so strongly on carrying units all the way through a computation 😁
Yeah although personally with the way I think about the numbers you can think about each set of real lines as having their own unit that distinguishes them from another real line. For instance say you have 2 real lines that are orthogonal one can have a sign of e_1 and the other of e_2. Then when you add these together you have a very similar structure to imaginary numbers just without the algebraic properties. Then if we consider the values e_1e_2 and e_2e_1 these can be seen as unit areas except one thing that is interesting is you can see it as defining a direction of travel too for instance e_1e_2 is going from e_1 to e_2 and e_2e_1 is going from e_2 to e_1 and so e_1e_2 = -e_2e_1. Finally if you also add that (e_1)^2=(e_2)^2=1 you get an interesting result (ae_1+be_2)e_1e_2=a(e_1)^2e_2+be_2e_1e_2=ae_2+be_2(-e_2e_1)=ae_2-be_1 which is a rotation of the original direction of 90 degrees in the direction specified if you moved e_1 to e_2. So as of currently I more so see i being the same thing as e_1e_2 except for the order in which you multiply matters and you can show that multiplying by on the left by it will rotate it 90 degrees in the opposite direction which is also the same as multiplying by e_2e_1 on the right. Also this (e_1e_2)^2=e_1e_2e_1e_2=-(e_1)^2(e_2)^2=-1 which goes to show a deeper connection to this as well. You can extend this system to N dimensions and you get very many things that square to -1. I personally like this kind of complex number formulation because it preserves the direction you want to go in as an object.
You can come up with lots of different systems by constructing axes with different units and choosing relationships between those units, and these will behave as models if the rules are aligned with some physical process. What makes imaginary numbers special in this regard is that by establishing continuity and measurement, this turns out to be where they actually pop up, without requiring additional definitions. And on top they open up the complex plane that lends itself to analytic continuation of familiar functions. So in some sense i is the most natural choice.
@@imaginaryangle I disagree that i is the most natural choice. If you think about it this is already kind of the system that we are using since we distinguish between the axis with point notation and really the basis squaring to one just is a way of mapping these numbers to another line just like multiplying by i is a way of mapping to another line. Also going off of stuff you said in the video these unit squares I am talking about actually appear in your video at 23:24 since if you look at e_1e_2, then look at (-e_1)(-e_2) they give the same answer and (-e_1)(e_2) also gives the same answer as (e_1)(-e_2) so I do think this makes the most sense. You can even get a perfect isomorphism between the complex numbers and this system if you just decide to use numbers like (a+be_1e_2) where a and b are real numbers.
@christopheriman4921 I'm not exactly following what you mean. I'm traveling at the moment so I'm a bit limited to try out what you are suggesting to see it properly. Do you think you could set this up visually in Desmos or a similar tool so I can have a closer look when I'm back near a desk?
