dude that was so informative and interesting! it's perhaps the first time complex numbers makes sense to me, thank you! and make more videos like this exploring other areas of math and making it so much fun!
Thanks for adding the captions. I found that setting the speed of the video to 0.75 and reading the captions made it very easy to follow. It is indeed a fantastic basic explanation of imaginary and complex numbers, how they work, and what they can be used for. I wish I had this video when I was STRUGGLING in math classes back in high school and college. Thanks much!
In electrical engineering "j" is used for imaginary numbers, because "i" is used for current (amperage, the flow of electricity). Complex numbers and their algebraic use only comes round when AC circuits are discussed, while "i" for current is used in learning the fundamental laws in DC circuits.
Thank you so much, IceDave, for sharing your knowledge. With complex or abstract topics, many can understand but few can explain or teach in easily understandable terms to others. You have given us an awesome video and explanation!! Much appreciated!
Thank you very much for this. I've been trying to wrap my mind around imaginary and complex numbers for a while now, and your video really helped to solidify my understanding. Keep up the great work!
After decades, imaginary numbers and complex numbers make sense, thanks to this video. Great Job IceDave33! The video is a genius insight and provides a fresh perspective to the world of complex numbers which I know have always been very real to physicists (eg in the realms of electronics (where they are used to represent phase) and in quantum mechanics). I no longer see them as a trick to help solve problems and can now appreciate them for what they are.
One more thing: I recommend keeping the graphic of the person fixed at the origin (though still allowed to rotate), and instead represent the velocity vector by a bold arrow (with varying length) extending out from the person at the origin. I have several reasons for this suggestion (re common misunderstandings I've had to dispel from my students), but I don't have the space to get into that here. Again, thanks for sharing these. You did a great job, and I look forward to your future videos.
Hi Matt, Yeah, certainly there are other ways to model rotations (eg with vectors and matrices etc). And as you say, the video illustrates that the imaginary part of a number can be considered as an imaginary dimension. The main point I wanted to get across was that the symmetry of complex numbers allows it to represent a co-ordinate AND a corresponding transformation (rotation and enlargement) in the 2D plane at the same time.
I like the instructor's methodology. We must remember that there are different learning styles. After forty years of instruction, I have learned to use different approaches to help the student understand. Sir Walter Besant “If a child can’t learn the way we teach, maybe we should teach the way they learn.”
An interesting approach to complex numbers - will show to my Further maths students tomorrow as we're currently studying them! Very clearly explained. Thank you :o)
Completely agree. In some ways it's bizarre that complex numbers are actually an integral part of our world, but then I realise actually they're so amazing and versatile no wonder they're everywhere!
Yeah, I completely agree - it is rather confusing - the representation you suggested is much clearer. If I ever get an opportunity to do something similar again, I'll do as you suggest! :) Also, thank you for the recommendations on Geometric Algebra - I've looked at the paper and it's very beautiful and just so neat! Will link to it in the description of the video. (And will try to see if I can take some courses on it next year!) Thanks again for all your detailed feedback and suggestions! David
... We are perfectly happy with measuring (say length) along a 1d scale - and we are happy to measure angles... But actually lengths and angles are clearly interwoven, and are described most naturally as complex numbers) And finally I should add that by expanding real numbers to complex numbers you don't lose anything (eg we DO still have commutativity etc - [this is lost in QM from operators etc]), you just gain a more symmetric system AND (rather importantly) algebraic closure. All best, Dave
I really like the fact that you introduce imaginary numbers and the complex plane all in one step, and the walking velocity is an excellent vehicle (no pun intended). I wonder if you have tried this on high school students and what problems, misconceptions, etc you encountered. I am thinking specifically of using it with students who have never heard of imaginary numbers before.
The standard i,j,k basis of 3d space deals only with addition and subtraction. It's the additional multiplicative structure of complex numbers which give them their power. For example, if I represent a point in the plane as a complex number then I can do things with angles much more naturally (because, as the video tries to explain, angle addition is part of complex multiplication) :). However, this is only a minor use of complex numbers, and they're found in so many different areas of maths!
Not sure what's "novel" here, as this looks like the standard way of introducing Argand diagrams and complex multiplication. I just checked my Asmar text (Applied Complex Analysis), and the ideas in this video are found in sections 1.2 & 1.3. Now that I think about it, it's probably true that most high school teachers do not present this. So I'm sure people will benefit from being exposed to this early on. With that in mind, I do want to thank you for making/sharing this excellent tutorial.
not sure if someone has said this already, but Ive never learned it as sqrt(1/2)+sqrt(1/2)i You can think of it this way: In trig, a 45 degree right triangle has an a and b value of 1, and the hypotenuse is root(2). Therefore You can apply sine to both 45 degree angles, to achieve 1/root(2) + 1/root(2)i , which is rationalized to root(2)/2+root(2)/2i (0.7071+0.7071i approx) You will already have to know trig and polar form stuff though. I figured out the square rrot of i without looking it up, purely based on polar form and trig
It is still true that -1 x -1 = 1, but when we start thinking about complex numbers off the real axis (eg i, or 2+i etc etc), we can no longer think of them as positive or negative numbers - that classification just doesn't make sense - so whilst the rule doesn't break down as such, it just doesn't make sense in the majority 2d complex plane. We can get some intuition back however if we consider complex numbers as rotations, eg -1 represents "turn around 180 degrees".
Things get a little more interesting and a little less intuitive - see the wiki page on "Quaternions". Basically, in 3D, the order of applying rotations matters, so there's a more complicated structure. If you think about it, there are 3 perpendicular planes to rotate in (the flat 2d plane, and then two perpendicular directions sticking up) - this means we need 3 symbols for the square root of -1, which we call i, j and k. But -1 is also different, so we actually need a '4d' number: a+bi+cj+dk.
... Practical uses of i are pretty widespread, because this symmetry extends even further (eg exponentiation of imaginary numbers yields trig functions [ie rotations!]) and this means i can be used for solving differential equations involving oscillations etc. In some sense, complex numbers are like 2 dimensional vectors with other stuff thrown in, if that makes sense? All the best, Dave PS: Sorry for the three posts! haha
Maybe word "number" isn't the best term for negative and complex numbers. Because number is something we can see, something that is original. In some way, we choose that negative numbers represent debt, but there is no such thing as - 1 apple. In that way number 1 is same number as - 1 but in different content (they are same in language but different (opposite) in meaning). In the exact same way, i is not a number, it is a way of represent walking, rotations, etc. So, of course I can't have 2 + 3i apples, because we choose that original quantities, numbers, are namely positive real numbers. All else is just same as they but in different way of looking. That is all philosophy. Only "confusing" is that simply same obvious and daily-life numbers represent different things when we need. Anyway, great video!
That sounds like an interesting project! Best of luck with it! To try and answer your question; anything you do with i could in theory be reformulated into using other methods (such as with matrices or with any number of quite arbitrary representations or formulations) *BUT* these are nearly always much messier and *fundamentally* would be equivalent to using i. Mathematicians (and physicists!) long ago realised that i was a very important and fundamental part of maths...
Very Very Very Good. And I agree too fast for someone trying to learn this from scratch... also... I'd suggest slowing down on on the complex multiplication as well... I hope you have videos explaining how this relates to for example the mandelbrot sets, and complexity, etc.
I think what is amazing is how multiplication becomes rotation around the complex plain and how exponentiation becomes multiplication of rotation around the complex plain. The real puzzle to me is how the real universe seems to also adhere to complex arithmetic since the results of equations that use complex number mathematics are observed to be true under experiment.
We take i to represent a counter-clockwise rotation is just a historical thing - everything would work if we chose it to represent a clockwise rotation instead :). As for the order of operations - the order doesn't matter - you get the same answer if you have positive first and negative second (this is known as commuting in maths). To clarify, 90 degrees corresponds to i itself. The key idea is that multiplication of complex numbers is the same as *adding* their angles (not multiplying) :)
...(and of course mathematical modelling). So, in the example of oscillations, the basic second order constant co-eff differential equation d2y/dx2 + y = 0 could be solved either with the real solution [y=Acos+Bsin], or with the complex solution [y=e^(ix)] - but note that the general pattern and symmetry (with say d2y/dx2 - y = 0) is much clearer in the complex solution. ... (On a personal level, I believe that complex numbers are fundamental to the universe....
Well done. Many people mistrust numbers, having the sense that, sure, you can do this or that and get the right answer, but it doesn't have anything to do with reality. (In that respect, the fact that these particular numbers are called "imaginary" is very unfortunate.) So I'm always looking for real-world examples to show students that i (and e, pi, logarithms, etc) came out of looking at the world, not out of mathematicians sitting around thinking up difficult problems to vex math students.
There seems to be a mistake in the end - rotating by 45 degrees is equivalent to multiplication by i^1/2 (square root of i) - not by i, as it follows from this video
Fine job man a good illustration. I wish I was thought the concept that way in my control engineering and advenced Calculus classes, Learnt it the hard way though...
Quaternions are fun, but I strongly recommend looking up "Geometric Algebra." It very naturally subsumes complex numbers, quaternions, and much more, into one powerful and elegant system. Truly beautiful mathematics there. For a nice intro to the subject, do a google search for the paper "Imaginary Numbers are not Real." You can also find a video of a 70min lecture on Geometric Algebra, split into 5 parts here on youtube. I recommend the paper first, as the lecture is more advanced.
When I was first introduced to imaginary numbers I would have big problems finding any applications for it, and it seemed to me like they were just introduced so that you could write down an answer for negative square roots. =P But now when I am starting to study higher level mathematics (first years at university) I am starting to notice that they are in fact useful in many ways. For examaple, it is used to calculate interferences of waves more easily, and also when adding various values of circuits.
complex numbers could be used for trig functions. I'm not sure if they are better or worse though. They weren't invented for that reason I guess and in fact were probably just given that purpose. Is the complex plain just the x, y plain arrived at via negative square roots? Or maybe they are polar coordinates.
Hi DJDaTonio! I completely agree!! I should have left much longer gaps and spoken much more clearly, apologies for that! (Lots of English people can't understand it without pausing and rewinding too even!) Thanks for the advice :), and I'm glad the explanation was useful. Best of luck with your self-study! :)
Lovely explanation. Please allow me to suggest something that might get your video's used even more, because you do explain it well. Take a breath after every sentence. People who don't know all this stuff, like me, might have to watch it twice, or pause the video, in order to absorb the information. In other words, after saying something, give that information some time to be understood. I am not English, you see. Not criticism btw, just a hint. Thank you for helping me with self-study Peace!
Can someone please PLEASE explain? 2:05 why is it something x something (x multiplying) not something x 2 when you need double the size? Because for example 9 x9 = 81 but 9 x 2 = 18. I never understood this..
+TheiLame Hi there! Yeah, it's a tough thing to get your head around. The point I was trying to get across is that complex numbers (such as i) behave as angles when you multiply by them. Adding them just adds the separate parts together. Eg i+i = 2i, which if you plot it on the Argand diagram, it's at the same angle as i, but twice as far away.... Multiplying by two makes a number's size ("magnitude") larger, but doesn't affect its direction. To affect its direction, we have to multiply by a number with a non-zero imaginary part. (Eg, i). And to repeat this rotation twice, we have to multiply twice. This is the same as multiplying by (i*i) = i^2. In other words, instead of doing i x 2, we do i^2, to get the behaviour we want. If it still doesn't make much sense, feel free to ask some more and I'll try to clarify further!
+IceDave33 Hi!! :) "Multiplying by two makes a number's size ("magnitude") larger, but doesn't affect its direction." [..] - Does that mean that when it comes to imaginary numbers, we dont care about the magnitude anymore? The only important thing about imaginary numbers is the direction/rotation? so all the magnitudes we dont care about become/(sum up?) in the "imaginary" on imaginary line /magnitude loses its importance? (So in the end only real numbers have magnitude and for imaginary its just all about rotation and magnitude is ignored?) i hope i make sense.. ?
+TheiLame mmm not exactly. Imaginary numbers have a magnitude too. Eg i has a magnitude of 1, and 2i (which is equal to 2 x i) has a magnitude of 2. If you plot the points on the argand diagram, then the "magnitude" of the number is simply its distance measured away from 0. Eg, 3+4i has a magnitide of 5 (try drawing it on square paper and measuring it's distance directly from 0). When we multiply complex numbers (complex numbers being a sum of real and imaginary numbers, so including real numbers and Imaginary numbers as subsets), we multiply their magnitudes, but add their angles. Eg, if we multiply 3+4i by 4+3i (both of which have a magnitude of 5), we get 25i which has a magnitude of 25 :). Butttt if you plot them, and measure their angles from the real number line, you'll note their angles sum to 90 degrees, which is the angle that 25i is at.
If I find time over the summer I'll have a go! The intuition is more complicated to grasp than with imaginary numbers, so I might have to introduce it as part of a video on geometric algebra, will see what I come up with! But cheers for the feedback!
I'd hate to sound cocky or anything, but I found that speed was perfect, of course english is my first language. Even when watching lectures I play them at 150% speed, I thnk my short term memory is awful, because my lecturer speaks so slowly I forget what she was on about 4 words in!
At last, a video which begins to make some sense of "i". If I understand this correctly, it's not so much the "value" of i which is the important thing to understand, but rather, that we wanted something which would produce this rotational effect in practise, and "Sqr root of -1" was found to be a value which, although meaningless by itself, produced the desired effect when plotted on the y-axis, like your video shows. Also, it immediately strikes me that as a circle is produced, then i, (or rather, seeing as i by itself isn't a value, then some function of i) must have some relationship to pi?
Great observation at the end, indeed it does! It turns out that the equation I present at the end: Complex number representing angle θ in radians = cosθ + i sin(θ) Actually has an even neater form, because (due to some clever maths), we have: cosθ + i sinθ = e^(iθ) In other words, if we let our angle be 180 degrees, then in radians, this is π radians, and a turn of 180 degrees obviously is represented by multiplication by -1, as per the video. Substituting into the above, we get: -1 = e^(iπ), or rearranging: e^(iπ) + 1 = 0 This is the fabled "Euler's identity", often quoted as one (if not the) most beautiful formula in mathematics. Personally, I sort of consider it a simple result of thinking about complex numbers in the right way, but it's still pretty cool! :)
IceDave33 Thanks for your reply. I had to look up "e" first of all, because (although I think I had heard of it before) I didn't know what it was, and I was shocked to see such a simple looking equation into which pi fitted! (I say shocked, because I do know that you cannot "square the circle", so therefore it seems illogical that pi can be fitted into such a simple equation. After all, you could transform that equation so that it reads "pi = " and it seemed too easy to be able to define pi in such a way). I was actually reassured to see that e is also an irrational number. You see, when I first learned that you can't square a circle, the PHYSICAL implications of that astounded me: Given that there is some physical limit to how small a unit of substance can be (say an atom, or smaller still, an electron... and we think that there is such a limit), then, it follows that, even if the entire universe was a "perfect sphere", consisting of a finite amount of atoms, or even electrons, you still could not "draw" a perfectly straight line one electron thick through it, using a finite integer of electrons! So, either there is no such thing as a perfect sphere, or else no such thing as a perfect straight line! The two cannot coexist in perfect relation to each other in any totality. Therefore, these "circles" and "straight lines through their centres" which we draw on paper, must be only abstractions, incomplete representations of the real physical universe. But then, I considered Einstein's relativity, and it seems to me that therein lies the answer: There IS no "perfect circle" such as we draw on paper. Gravity bends space and time, so that even the "circle" we draw on a piece of paper isn't really a circle but only an approximation - time itself is different at one end than at the other end, and the gravity of even the charcoal of the pencil line affects the space-time of the piece of paper, bending it slightly. Sorry to ramble on, but I just have this feeling that this is the key: Once we factor in gravity and time, we will somehow arrive at an equation which rationalises pi and "squares the circle", so to speak. I'm not even a mathematician, this is just a sort of half-understood intuitive feeling that I have. Does it make any sense?
579enact Mmm most of what you say is certainly true in some respect, but as a mathematician, some of your intuitions/arguments don't really hold from my perspective.. For starters, just to clear up a minor point - the inability to square the circle specifically relates to the fact that with a compass and ruler, starting with a circle, you cannot create a square with exactly the same area. This problem can be reduced to showing that pi cannot solve any polynomial equation (ie any equation of the form Ax^n + Bx^(n-1) + ... + Yx + Z) - this was only shown a couple of hundred of years ago. It is much easier to show that pi is irrational - ie that the circumference of a circle / its diameter is not an integer fraction, which I think is what you mostly refer to above! On more philosophical grounds, as a mathematician, my personal acceptance of a concept/ the existence of a perfect circle follows simply from my ability to describe it (eg - the set of points equidistant from a given point, or equally the (x, y) points such that x^2 + y^2 = d^2 for some d). Maths/logic doesn't require anything we describe to be physically realisable, just that we can envisage a set of rules, and create things from those rules. Sure, we choose rules that we can use to model the universe, and no, there is no way of putting a set of electrons in a circle (well for starters, quantum mechanics forbids them from staying still so it's completely fruitless), but that doesn't matter... Even if you can't "create" an actual sphere of physical objects, any process which looks at things an equal "distance" away (where "distance" can be any sort of concept of measurable something), then by definition, we have a sphere. This inevitably means that virtual spheres exist everywhere as the universe fluctuates and decides what to do. So actually, having the concept of a sphere turns out to be *very* useful, even if we can't physically create one. (if that makes sense). Basically, tldr: maths and physics transcends what you can do with physical particles. While we're on the topic of physics, then I just wanted to add that the universe is almost certainly not a sphere... and even if we were to know the "shape" of the universe, in a topological sense, this doesn't give us any distance information for measuring the width of the universe. (And actually in some sense, due to the combination of special relativity and issues of expansion, the width of even the observable universe is very hard to define consistently. And the entire universe is possibly infinite in size, we just can't see the rest of it due to the combination of a finite speed of light and expansion!). Finally, another quick clarification - the value of pi at a point can't change as the spacetime of the universe bends... For example, you can think about it like this: Imagine you live on a sphere that's absolutely massive (eg the Earth). Locally, the Earth seems flat, right? Well if you imagine the Earth shrinks to 1000th the size, then suddenly to the giant you, you can see over the horizon a fair way, and the Earth doesn't seem very flat any more to you... But if you shrink to the size of a microbe, then the Earth would seem pretty darn flat again. The same is true in a similar sense in special relativity. No matter how curved space time is, (even if you're inside a blackhole!), you can always look in a small enough region around you for spacetime to appear basically flat. And by looking closer and closer, and measuring the diameter of a circle you draw, you could measure more and more accurate values for pi. I'm really sorry if this comes across at all derisive, it's certainly not meant to be (quite the opposite, it's supposed to be encouraging!), I just thought it made sense to give my thoughts in case they make things clearer, or just to present an alternative viewpoint :). The philosophy of maths / physics is very interesting and worth learning about - and it's not an area I'm amazingly knowledgable about - certainly my viewpoints above on the philosophy side of things are just my point of view! (Though I should add the mathsy bits, including the bits about special relativity are true from a mathematical/logical standpoint!) Anyway, very interesting questions, cheers enact :).
I like the explaination, but how can your velocity even be -1. Even though you are turning 180° which is a complete switch of direction, saying your velocity is changing -1 is like saying you turned -180 degrees. The problem gets more complicated if you look at the unit circle. So if you were to turn 45° then and you called your velocity 1. Then you turned -45° then your velocity would not be -1. Becuase 45° or pi/4 does not equal -45 which is -pi/4 which is 315° which is also 5pi/4. And even if you assume that you turn 225° which is the true oposite of 45° then the velocity would not be the same. If you cover 225° in the same time as you cover 45° then your speeds would be completly different making your velocity different as well. In real life we dont refer to things negatively. For instance we dont travel 55mph negative north west. Or we dont say i turned -315° which is 45°. Or we dont walk -4ft.
Okay, I think my explanation wasn't fully clear, and may have caused a bit of confusion between relative and absolute velocity. In my explanation, interpret "velocity 1" as "velocity to the right". That way, "-1 to the right" is clearly "1 to the left". The confusion arises if you think of -1 as relative to you... In that sense, you consider -1 to mean "turn around". But this is what I describe as "multiplying by -1", not having a velocity of -1. Multiplying by a number that's not 1 indicates a change of velocity, but just having a velocity of -1 does not denote any change. (if that vaguely makes sense). In your explanation, neither 45 degrees or -45 degrees are represented as multiplying by (1+i)/sqrt(2) or (1 - i)/sqrt(2). (They are not plus or minus 1.) Hope that makes a little more sense?
i always thought of i as x, it is real, but it can't be simplified in any way, just as the quadratic formula can never be simplified to result in y=c for ax^2+bx+c, unless a=b=0, at which time you are simply saying 'i'm not looking at that number anymore, so i'll multiply by 0', which is why i is simple to me, it's just there. nothing else. (also, clockwise 90 degrees i always interpereted as velocity/i and since -1=i^2 and 1=-1^2, 1=i^4, and i^4/i by the law of exponents =i^3=i^2 * i=-1 * i=-i) (may be less intuitive, but people still need to learn how to divide by i) (you should totally do a video on everything needed in dealing with i, a/(1-i) x^i, all that.)
Imaginary numbers weren't created to deal with 2d rotations - it's just the kind of symmetry that they have and thinking about them as 2d rotations can help you get your head around them :). There are neat extensions of this symmetry to algebras in higher dimensions - but to be honest, the i,j,k etc are easier to grasp. Though the i,j,k are vectors which don't come equipped natively with a way to deal with rotations (you need matrices/tensors/similar).
Hi! Interesting Presentation :) You may also make imaginary numbers even more real by showing the link with phases (which was historically from were they came if I'm not mistaking) ;-) Thanks for Sharing :)
...I created this at the end of school, before starting university, and my target audience was most certainly high-school students... In my school, complex numbers were introduced simply as an algebraic quantity, and their elegance and simplicity were completely obscured. So, by 'novel', I simply meant that the introduction of i as a rotation instead of an algebraic quantity was a bit different (and in my opinion, gives a much better intuition about the symmetry of complex numbers)
+RAVI POHANI multiply by j. for the rule set i^2=j^2=k^2=ijk=-1, for rotations by j would be acw going straight towards you from 1, and i is up and k is going towards you from i, there is a theorem stating you need i j and k to plot 3d vectors and orientations, and as you see by rotating i you get to i from 1, then by k you get to straight towards you, and j then rotates it clockwise from the view looking down from i, arriving at -1, but all of these when doubled arrive at -1 from 1, thus it all fits, and to orient to a direction, set your current orientation to 1 and record the position of your destination, such is the multiplier for your current orientation, and no matter where the origin direction points you will arrive at that destination. yay for math!
Firstly, thanks for all these comments! I fully agree the fundamental idea isn't the slightest bit new, and if I made it now, I would title it alternative instead of novel (actually, just realised I can changed this so have done so!)...
@FullWaveElectronics - I'm glad everyone likes it so far! I'm rather busy at the moment, but I might try and make another one over the Christmas period. Any topics in particular you'd like me to have a look at? Maybe more complex numbers, or some other part of maths completely?
5:29 what does it even mean to square that whole thing...? How does it even verify anything (without any explanation it's like you're just doing some random operation)?
+nomealow Initially we concluded that to rotate 180 degrees, we multiply by -1. To rotate 90 degrees, we multiply by root(-1), or just i. So that must mean to rotate 45 degrees, we multiply by root(i). We got an answer of root(1/2) + root(1/2)i, and to show that this is indeed equal to root(i), we square the whole thing and see if we get i. After squaring, we did get i so root(1/2) + root(1/2)i is equal to root(i).
Thank you for flagging this, I'm seeing it too! I'm not sure what's happening - I've contacted UA-cam support for help. Also, I'm very happy to hear you've found this to be a good introduction for your Year 12 students - I originally made this at the end of Year 13, reflecting on how I wish the curriculum had started with the argand diagram, so it's great to hear that it meets that bar. Hopefully UA-cam support will be able to help. I'll let you know if I hear any updates.
Looks like video has corrupted on UA-cam's servers. I didn't know this was possible. I'll see if UA-cam support can help re-upload it over this video. If not I'll need to re-upload it to a separate video.
Just to update - I've re-uploaded it here: ua-cam.com/video/2h3mrS7sX0A/v-deo.html - unfortunately, the quality isn't so great, but it wasn't too great to start with. Hopefully it's still sufficient for your needs. Who knows, maybe I'll find time to create it again using 3Blue1Brown visuals and a much clearer narration at some point.
Nice. But I still can't get how it coexists w/ the 2 negs multiplied produces a pos rule. How could the rule be better stated? The linguistic contradiction is too much. It must be oversimplified? Pertinent only in a single dimension?
i 'm not that good on maths but each time i see imaginary numbers.i see two dimensions like an (x,y) plane.were x is the real part and y is the factor of the imaginary part. my question is it really a good idea to define i as i x i = -1. can't we use a function? : squareroot(x) if x ≥ 0 then squareroot(x)= y as y*y = x. ,if x ≤ 0 squareroot(x) = - absolute value of (y) as y*y= x. what is really useful in the i number.if we already can define a plane of two axis and define position of points with two coordinates.?
So, actually I think mathematicians make misconceptions about complex numbers and its relation with rotation. You can use a + bi (a complex number) to represent rotations, in the same way as r cos(β) i + r sin(β) j, but use complex numbers for rotations only really make sense when you are handling problems with negative square roots. For standard rotation problems, is better forgetting about the imaginary number and focus in Linear Álgebra notations. I think 3D rotation with quaternions can be understand as a 4 dimensional vector in geometry problems, and as a 4 dimensional complex number in this category of problems. I really don't understand why people handle rotation with complex numbers in problems which doesn't involve imaginary numbers.
I'd hazard a guess that it's because calculating the values of trigonometric functions is computationally expensive, while operations complex numbers require are extremely fast to perform.
Why can't you start from the right instead of the left? Why does the 180 degree turn have to be negative first, and positive second? Also, if 90 degrees is the square root of i, then how is 180 degrees i? If you square 90, you get 2700 degrees. Shouldn't i be 2700 degrees?
Is it fair to say that i rotates a vector through 90 degrees in 2 dimensions? I love the explanation but is this real space and what if you need to rotate it in the 3rd plane (z or whatever)? If it is real space in 2 dimensions, would you not simply use x and y coordinates to begin with to describe the line segment (vector) that is being transformed? I think there is more to i than meets the eye (art always has to save humanity in the end). Is it not called an Argand diagram because I think (i could be wrong) its not real space, and if it is not real space then where the hell is this rotation taking place and so are we not getting circular (pun intended) in our reasoning, just a bit? Like other forms of art, math is a fiction that works (just like mythology).🙃
if you want to rotate 90 degree then why don't you you add an extra speed in the Y direction? All you show me is basic 2D trigonometry equations, so why not use Y. I am still searching for why i is necessary
You're right, you don't need i for this. I wanted to demonstrate its symmetry of multiplication is equivalent to 360 degree rotations... And give a feel for what complex numbers are actually about. They really become used / useful when the answer to a problem is intrinsically somehow periodic. Eg quantum mechanics, electronics,...
It's just that I am looking for wether the imaginary number is really necessary, and if it's really a thing of nature, or more a human cultural thing. I feel like we took a wrong turn in mathematics and am searching for the real thing. math should conform more with nature. we took the wrong turn by accepting a negative number as 1 piece of information, but it is 2 pieces of information: the sign and the number. For god there are no negatives. After that sticking of the sign to a number, our society needed a trick to work with more difficult mathematical problems: imaginary numbers. Ah well I can go on rambling forever about this. But I am searching for an alternative to this imaginary stuff, and want to know why math people say that complex numbers are really necessary.
Hey, +IceDave33 Have you seen +3Blue1Brown 's video? He introduces numbers as points, adders and multipliers. Anyway, its really cool! But you did it 4 years before! I wonder how long ago this idea was originally thought up, and how most students aren't explained this is math class..
+PT Yamin That's really neat! An extension of the intuition in my video. I'll add a link to it (and I've subscribed to 3Blue1Brown, the channel has some fantastic videos!). It is common intuition / interpretation in maths circles that the exponential function transforms addition to multiplication in a natural way, but I love the way it is presented in the video, with multiplication seen as a stretching action! Thanks for the link PT! (and love your avatar). Re the question about how long it was originally thought up - I know the "intuition" I present in my video has been around / known for hundreds of years, but I made the video when back at school when it was still a bit new to me, and I'd pieced together a lot of the understanding myself at that point. I think it's unfortunate that maths classes often focus on solving equations instead of explaining things, because at the end, you don't come out with nearly the same level of enjoyment or intuition. Anyway, rant over ^_^.
+IceDave33 According to +njwildberger 's video on hyper complex numbers, the relationship between complex number multiplication and rotations and dilatations of a plane was understood by the 1830s.
Your presentation was nice (thanks!) but the very concept itself is still a bit eerie for me. Basically, what's happening is, math folks seem to be developing the theory (of complex number) from the applications (geometric rotation etc), instead of the other way round. Shouldn't a theory be able to stand by itself, without any reference to its applications?
.... But I should say the main reason I made the video wasn't to study rotations, but was to help people get over the "complex numbers don't make sense! the root of -1 doesn't exist" feeling. Because, if viewed in the right way, "i" is a 90 degree rotation - and then this also explains other symmetries (such as |a|*|b|=|ab|, arg(a)+arg(b)=arg(ab) [+-360 degrees], and formulas with the complex conjugate) in a more satisfactory way :) ...
Hey can you explain how two completely imaginary numbers multiply into a real number using this complex plane...Algebraic calculation makes sense..but what does it really mean?..can u explain it using the complex plane..that would be really helpful
Chamath Samarawickrama Indeed! Two complex numbers multiply to give a real number if their angles (measured clockwise from the real axis) sum to a multiple of 180 degrees. In particular, completely imaginary numbers (with positive co-efficient, so eg 2i, 3/5i etc etc) have an angle of 90 degrees, so when multiplied, we sum their angle representations... Thus the answer is represented as a point at 180 degrees, in other words, a negative real number. As the video tries to explain, multiplying complex numbers is equivalent to adding their angle representations (and multiplying their lengths).
IceDave33 oh now I see..Thats why a complex number and its conjugate always multiply into a real number ryt..I always understood the algebraic calculation but not whats actually happening in the complex plane.. thank you so much for the explanation..keep up the good work
This is probably the best (intuitive) explanation of imaginary numbers I have ever seen.
dude that was so informative and interesting! it's perhaps the first time complex numbers makes sense to me, thank you! and make more videos like this exploring other areas of math and making it so much fun!
Thanks for adding the captions. I found that setting the speed of the video to 0.75 and reading the captions made it very easy to follow.
It is indeed a fantastic basic explanation of imaginary and complex numbers, how they work, and what they can be used for.
I wish I had this video when I was STRUGGLING in math classes back in high school and college.
Thanks much!
This is the best explanation for imaginary numbers I've ever heard! I will definitely share with my students! Thanks so much!
In electrical engineering "j" is used for imaginary numbers, because "i" is used for current (amperage, the flow of electricity). Complex numbers and their algebraic use only comes round when AC circuits are discussed, while "i" for current is used in learning the fundamental laws in DC circuits.
This is the coolest explanation I have seen on complex and imaginary numbers
Thank you so much, IceDave, for sharing your knowledge.
With complex or abstract topics, many can understand but few can explain or teach in easily understandable terms to others. You have given us an awesome video and explanation!!
Much appreciated!
Thank you very much for this. I've been trying to wrap my mind around imaginary and complex numbers for a while now, and your video really helped to solidify my understanding. Keep up the great work!
Excellent video. This is so simple and clear. Should get many many more views and mandatory viewing for preliminary complex number courses
After decades, imaginary numbers and complex numbers make sense, thanks to this video. Great Job IceDave33! The video is a genius insight and provides a fresh perspective to the world of complex numbers which I know have always been very real to physicists (eg in the realms of electronics (where they are used to represent phase) and in quantum mechanics). I no longer see them as a trick to help solve problems and can now appreciate them for what they are.
One more thing: I recommend keeping the graphic of the person fixed at the origin (though still allowed to rotate), and instead represent the velocity vector by a bold arrow (with varying length) extending out from the person at the origin. I have several reasons for this suggestion (re common misunderstandings I've had to dispel from my students), but I don't have the space to get into that here.
Again, thanks for sharing these. You did a great job, and I look forward to your future videos.
Very nicely done! The connection between trig and the complex plane is a tasty bite.
Hi Matt,
Yeah, certainly there are other ways to model rotations (eg with vectors and matrices etc).
And as you say, the video illustrates that the imaginary part of a number can be considered as an imaginary dimension. The main point I wanted to get across was that the symmetry of complex numbers allows it to represent a co-ordinate AND a corresponding transformation (rotation and enlargement) in the 2D plane at the same time.
I like the instructor's methodology. We must remember that there are different learning styles. After forty years of instruction, I have learned to use different approaches to help the student understand.
Sir Walter Besant
“If a child can’t learn the way we teach, maybe we should teach the way they learn.”
An interesting approach to complex numbers - will show to my Further maths students tomorrow as we're currently studying them!
Very clearly explained.
Thank you :o)
Completely agree. In some ways it's bizarre that complex numbers are actually an integral part of our world, but then I realise actually they're so amazing and versatile no wonder they're everywhere!
You're more than welcome, sorry that I went a bit too in-depth there!
Best of luck with it,
Dave
Actually.... this has been very helpeful I'd been trying to figure that out for so long now
Yeah, I completely agree - it is rather confusing - the representation you suggested is much clearer. If I ever get an opportunity to do something similar again, I'll do as you suggest! :)
Also, thank you for the recommendations on Geometric Algebra - I've looked at the paper and it's very beautiful and just so neat! Will link to it in the description of the video. (And will try to see if I can take some courses on it next year!)
Thanks again for all your detailed feedback and suggestions!
David
Exceptional video btw, loved it! Its opened up so much andd things are making some sense!
... We are perfectly happy with measuring (say length) along a 1d scale - and we are happy to measure angles... But actually lengths and angles are clearly interwoven, and are described most naturally as complex numbers)
And finally I should add that by expanding real numbers to complex numbers you don't lose anything (eg we DO still have commutativity etc - [this is lost in QM from operators etc]), you just gain a more symmetric system AND (rather importantly) algebraic closure.
All best, Dave
Thanks for the subtitles. This guy is more difficult to understand than the waiter at my fav Indian restaurant...
amazing video ...the best and simple explanation about complex number ever seen
I really like the fact that you introduce imaginary numbers and the complex plane all in one step, and the walking velocity is an excellent vehicle (no pun intended). I wonder if you have tried this on high school students and what problems, misconceptions, etc you encountered. I am thinking specifically of using it with students who have never heard of imaginary numbers before.
they've started setting this video for homework at Ags! haha good video, cheers man!
The standard i,j,k basis of 3d space deals only with addition and subtraction. It's the additional multiplicative structure of complex numbers which give them their power. For example, if I represent a point in the plane as a complex number then I can do things with angles much more naturally (because, as the video tries to explain, angle addition is part of complex multiplication) :).
However, this is only a minor use of complex numbers, and they're found in so many different areas of maths!
Not sure what's "novel" here, as this looks like the standard way of introducing Argand diagrams and complex multiplication. I just checked my Asmar text (Applied Complex Analysis), and the ideas in this video are found in sections 1.2 & 1.3.
Now that I think about it, it's probably true that most high school teachers do not present this. So I'm sure people will benefit from being exposed to this early on. With that in mind, I do want to thank you for making/sharing this excellent tutorial.
Beautifully done, Clear and concise.
Now I just have to confirm if a Magnetic Spin 1/2 and is just in the opposite direction of a spin of -1/2.
Good approach and easy to understand but can you slow your speech down a bit ?
Very nice way to explain "geometrical side" of complex numbers. Thanks
not sure if someone has said this already, but Ive never learned it as sqrt(1/2)+sqrt(1/2)i
You can think of it this way: In trig, a 45 degree right triangle has an a and b value of 1, and the hypotenuse is root(2). Therefore You can apply sine to both 45 degree angles, to achieve 1/root(2) + 1/root(2)i , which is rationalized to root(2)/2+root(2)/2i (0.7071+0.7071i approx)
You will already have to know trig and polar form stuff though. I figured out the square rrot of i without looking it up, purely based on polar form and trig
It is still true that -1 x -1 = 1, but when we start thinking about complex numbers off the real axis (eg i, or 2+i etc etc), we can no longer think of them as positive or negative numbers - that classification just doesn't make sense - so whilst the rule doesn't break down as such, it just doesn't make sense in the majority 2d complex plane.
We can get some intuition back however if we consider complex numbers as rotations, eg -1 represents "turn around 180 degrees".
Things get a little more interesting and a little less intuitive - see the wiki page on "Quaternions".
Basically, in 3D, the order of applying rotations matters, so there's a more complicated structure. If you think about it, there are 3 perpendicular planes to rotate in (the flat 2d plane, and then two perpendicular directions sticking up) - this means we need 3 symbols for the square root of -1, which we call i, j and k. But -1 is also different, so we actually need a '4d' number: a+bi+cj+dk.
You should make more videos, You have a very intuitive way of explaining things
this made me finally truly understand quaternions, thanks.
this made me finally understand quaternions
...
Practical uses of i are pretty widespread, because this symmetry extends even further (eg exponentiation of imaginary numbers yields trig functions [ie rotations!]) and this means i can be used for solving differential equations involving oscillations etc.
In some sense, complex numbers are like 2 dimensional vectors with other stuff thrown in, if that makes sense?
All the best, Dave
PS: Sorry for the three posts! haha
This is an excellent explanation of this interesting concept. Thanks, great job.
Maybe word "number" isn't the best term for negative and complex numbers. Because number is something we can see, something that is original. In some way, we choose that negative numbers represent debt, but there is no such thing as - 1 apple. In that way number 1 is same number as - 1 but in different content (they are same in language but different (opposite) in meaning). In the exact same way, i is not a number, it is a way of represent walking, rotations, etc. So, of course I can't have 2 + 3i apples, because we choose that original quantities, numbers, are namely positive real numbers. All else is just same as they but in different way of looking. That is all philosophy. Only "confusing" is that simply same obvious and daily-life numbers represent different things when we need. Anyway, great video!
That sounds like an interesting project! Best of luck with it!
To try and answer your question; anything you do with i could in theory be reformulated into using other methods (such as with matrices or with any number of quite arbitrary representations or formulations) *BUT* these are nearly always much messier and *fundamentally* would be equivalent to using i.
Mathematicians (and physicists!) long ago realised that i was a very important and fundamental part of maths...
Very Very Very Good. And I agree too fast for someone trying to learn this from scratch... also... I'd suggest slowing down on on the complex multiplication as well... I hope you have videos explaining how this relates to for example the mandelbrot sets, and complexity, etc.
I think what is amazing is how multiplication becomes rotation around the complex plain and how exponentiation becomes multiplication of rotation around the complex plain. The real puzzle to me is how the real universe seems to also adhere to complex arithmetic since the results of equations that use complex number mathematics are observed to be true under experiment.
We take i to represent a counter-clockwise rotation is just a historical thing - everything would work if we chose it to represent a clockwise rotation instead :). As for the order of operations - the order doesn't matter - you get the same answer if you have positive first and negative second (this is known as commuting in maths).
To clarify, 90 degrees corresponds to i itself. The key idea is that multiplication of complex numbers is the same as *adding* their angles (not multiplying) :)
Nice introduction to visualising complex numbers. Bravo.
Just amazing explanation
...(and of course mathematical modelling). So, in the example of oscillations, the basic second order constant co-eff differential equation d2y/dx2 + y = 0 could be solved either with the real solution [y=Acos+Bsin], or with the complex solution [y=e^(ix)] - but note that the general pattern and symmetry (with say d2y/dx2 - y = 0) is much clearer in the complex solution. ...
(On a personal level, I believe that complex numbers are fundamental to the universe....
this is nice, good work.
Well done.
Many people mistrust numbers, having the sense that, sure, you can do this or that and get the right answer, but it doesn't have anything to do with reality. (In that respect, the fact that these particular numbers are called "imaginary" is very unfortunate.) So I'm always looking for real-world examples to show students that i (and e, pi, logarithms, etc) came out of looking at the world, not out of mathematicians sitting around thinking up difficult problems to vex math students.
There seems to be a mistake in the end - rotating by 45 degrees is equivalent to multiplication by i^1/2 (square root of i) - not by i, as it follows from this video
Thanks a lot for sharing your knowledge mate!
Will be really helpful in the future : )
A very helpful metaphor for any teacher of Maths
At 6:30 he should have said please leave me any feedback, positive, negative, or imaginary below~
um if it is not positive or negative feedback would it be neutral feedback?
Fine job man a good illustration. I wish I was thought the concept that way in my control engineering and advenced Calculus classes, Learnt it the hard way though...
Fantastic explanation!
Quaternions are fun, but I strongly recommend looking up "Geometric Algebra." It very naturally subsumes complex numbers, quaternions, and much more, into one powerful and elegant system. Truly beautiful mathematics there.
For a nice intro to the subject, do a google search for the paper "Imaginary Numbers are not Real." You can also find a video of a 70min lecture on Geometric Algebra, split into 5 parts here on youtube. I recommend the paper first, as the lecture is more advanced.
When I was first introduced to imaginary numbers I would have big problems finding any applications for it, and it seemed to me like they were just introduced so that you could write down an answer for negative square roots. =P
But now when I am starting to study higher level mathematics (first years at university) I am starting to notice that they are in fact useful in many ways.
For examaple, it is used to calculate interferences of waves more easily, and also when adding various values of circuits.
complex numbers could be used for trig functions. I'm not sure if they are better or worse though. They weren't invented for that reason I guess and in fact were probably just given that purpose. Is the complex plain just the x, y plain arrived at via negative square roots? Or maybe they are polar coordinates.
thank you for the explanation, what would happens if we ad a 3er dimension??
Hi DJDaTonio! I completely agree!!
I should have left much longer gaps and spoken much more clearly, apologies for that! (Lots of English people can't understand it without pausing and rewinding too even!)
Thanks for the advice :), and I'm glad the explanation was useful. Best of luck with your self-study! :)
It's really a very simple explanation. Thank you.
Nicely explained man :)....great job....
Lovely explanation. Please allow me to suggest something that might get your video's used even more, because you do explain it well.
Take a breath after every sentence. People who don't know all this stuff, like me, might have to watch it twice, or pause the video, in order to absorb the information. In other words, after saying something, give that information some time to be understood.
I am not English, you see.
Not criticism btw, just a hint. Thank you for helping me with self-study
Peace!
...I'll watch this again tomorrow. Tomorrow I'll understand. ...tomorrow.
Or will you?
Great explanation!
Can someone please PLEASE explain?
2:05 why is it something x something (x multiplying) not something x 2 when you need double the size? Because for example 9 x9 = 81 but 9 x 2 = 18.
I never understood this..
+TheiLame is it because 2x2 is same as 2x2 because it never gets bigger than 2 or something? :( but it says i x i not i x 2!!
+TheiLame Hi there!
Yeah, it's a tough thing to get your head around. The point I was trying to get across is that complex numbers (such as i) behave as angles when you multiply by them.
Adding them just adds the separate parts together.
Eg i+i = 2i, which if you plot it on the Argand diagram, it's at the same angle as i, but twice as far away.... Multiplying by two makes a number's size ("magnitude") larger, but doesn't affect its direction.
To affect its direction, we have to multiply by a number with a non-zero imaginary part. (Eg, i). And to repeat this rotation twice, we have to multiply twice. This is the same as multiplying by (i*i) = i^2. In other words, instead of doing i x 2, we do i^2, to get the behaviour we want.
If it still doesn't make much sense, feel free to ask some more and I'll try to clarify further!
+IceDave33 Hi!! :) "Multiplying by two makes a number's size ("magnitude") larger, but doesn't affect its direction." [..] - Does that mean that when it comes to imaginary numbers, we dont care about the magnitude anymore? The only important thing about imaginary numbers is the direction/rotation? so all the magnitudes we dont care about become/(sum up?) in the "imaginary" on imaginary line /magnitude loses its importance?
(So in the end only real numbers have magnitude and for imaginary its just all about rotation and magnitude is ignored?)
i hope i make sense.. ?
+TheiLame mmm not exactly. Imaginary numbers have a magnitude too. Eg i has a magnitude of 1, and 2i (which is equal to 2 x i) has a magnitude of 2.
If you plot the points on the argand diagram, then the "magnitude" of the number is simply its distance measured away from 0. Eg, 3+4i has a magnitide of 5 (try drawing it on square paper and measuring it's distance directly from 0).
When we multiply complex numbers (complex numbers being a sum of real and imaginary numbers, so including real numbers and Imaginary numbers as subsets), we multiply their magnitudes, but add their angles. Eg, if we multiply 3+4i by 4+3i (both of which have a magnitude of 5), we get 25i which has a magnitude of 25 :). Butttt if you plot them, and measure their angles from the real number line, you'll note their angles sum to 90 degrees, which is the angle that 25i is at.
Hope that makes a bit more sense, though feel free to ask for more clarification! :p
Great vid! can you explain how this works in 3 dimensions for quaternions? Nice work man!
If I find time over the summer I'll have a go! The intuition is more complicated to grasp than with imaginary numbers, so I might have to introduce it as part of a video on geometric algebra, will see what I come up with! But cheers for the feedback!
excellent tutorial. much thanks! Would like to see more of the like
why not use the second axis with normal y?
I'd hate to sound cocky or anything, but I found that speed was perfect, of course english is my first language. Even when watching lectures I play them at 150% speed, I thnk my short term memory is awful, because my lecturer speaks so slowly I forget what she was on about 4 words in!
Great video!
At last, a video which begins to make some sense of "i". If I understand this correctly, it's not so much the "value" of i which is the important thing to understand, but rather, that we wanted something which would produce this rotational effect in practise, and "Sqr root of -1" was found to be a value which, although meaningless by itself, produced the desired effect when plotted on the y-axis, like your video shows.
Also, it immediately strikes me that as a circle is produced, then i, (or rather, seeing as i by itself isn't a value, then some function of i) must have some relationship to pi?
Great observation at the end, indeed it does! It turns out that the equation I present at the end:
Complex number representing angle θ in radians = cosθ + i sin(θ)
Actually has an even neater form, because (due to some clever maths), we have: cosθ + i sinθ = e^(iθ)
In other words, if we let our angle be 180 degrees, then in radians, this is π radians, and a turn of 180 degrees obviously is represented by multiplication by -1, as per the video. Substituting into the above, we get: -1 = e^(iπ), or rearranging:
e^(iπ) + 1 = 0
This is the fabled "Euler's identity", often quoted as one (if not the) most beautiful formula in mathematics. Personally, I sort of consider it a simple result of thinking about complex numbers in the right way, but it's still pretty cool! :)
IceDave33 Thanks for your reply. I had to look up "e" first of all, because (although I think I had heard of it before) I didn't know what it was, and I was shocked to see such a simple looking equation into which pi fitted! (I say shocked, because I do know that you cannot "square the circle", so therefore it seems illogical that pi can be fitted into such a simple equation. After all, you could transform that equation so that it reads "pi = " and it seemed too easy to be able to define pi in such a way). I was actually reassured to see that e is also an irrational number.
You see, when I first learned that you can't square a circle, the PHYSICAL implications of that astounded me: Given that there is some physical limit to how small a unit of substance can be (say an atom, or smaller still, an electron... and we think that there is such a limit), then, it follows that, even if the entire universe was a "perfect sphere", consisting of a finite amount of atoms, or even electrons, you still could not "draw" a perfectly straight line one electron thick through it, using a finite integer of electrons! So, either there is no such thing as a perfect sphere, or else no such thing as a perfect straight line! The two cannot coexist in perfect relation to each other in any totality. Therefore, these "circles" and "straight lines through their centres" which we draw on paper, must be only abstractions, incomplete representations of the real physical universe.
But then, I considered Einstein's relativity, and it seems to me that therein lies the answer: There IS no "perfect circle" such as we draw on paper. Gravity bends space and time, so that even the "circle" we draw on a piece of paper isn't really a circle but only an approximation - time itself is different at one end than at the other end, and the gravity of even the charcoal of the pencil line affects the space-time of the piece of paper, bending it slightly. Sorry to ramble on, but I just have this feeling that this is the key: Once we factor in gravity and time, we will somehow arrive at an equation which rationalises pi and "squares the circle", so to speak. I'm not even a mathematician, this is just a sort of half-understood intuitive feeling that I have. Does it make any sense?
579enact Mmm most of what you say is certainly true in some respect, but as a mathematician, some of your intuitions/arguments don't really hold from my perspective..
For starters, just to clear up a minor point - the inability to square the circle specifically relates to the fact that with a compass and ruler, starting with a circle, you cannot create a square with exactly the same area. This problem can be reduced to showing that pi cannot solve any polynomial equation (ie any equation of the form Ax^n + Bx^(n-1) + ... + Yx + Z) - this was only shown a couple of hundred of years ago. It is much easier to show that pi is irrational - ie that the circumference of a circle / its diameter is not an integer fraction, which I think is what you mostly refer to above!
On more philosophical grounds, as a mathematician, my personal acceptance of a concept/ the existence of a perfect circle follows simply from my ability to describe it (eg - the set of points equidistant from a given point, or equally the (x, y) points such that x^2 + y^2 = d^2 for some d). Maths/logic doesn't require anything we describe to be physically realisable, just that we can envisage a set of rules, and create things from those rules. Sure, we choose rules that we can use to model the universe, and no, there is no way of putting a set of electrons in a circle (well for starters, quantum mechanics forbids them from staying still so it's completely fruitless), but that doesn't matter... Even if you can't "create" an actual sphere of physical objects, any process which looks at things an equal "distance" away (where "distance" can be any sort of concept of measurable something), then by definition, we have a sphere. This inevitably means that virtual spheres exist everywhere as the universe fluctuates and decides what to do. So actually, having the concept of a sphere turns out to be *very* useful, even if we can't physically create one. (if that makes sense). Basically, tldr: maths and physics transcends what you can do with physical particles.
While we're on the topic of physics, then I just wanted to add that the universe is almost certainly not a sphere... and even if we were to know the "shape" of the universe, in a topological sense, this doesn't give us any distance information for measuring the width of the universe. (And actually in some sense, due to the combination of special relativity and issues of expansion, the width of even the observable universe is very hard to define consistently. And the entire universe is possibly infinite in size, we just can't see the rest of it due to the combination of a finite speed of light and expansion!).
Finally, another quick clarification - the value of pi at a point can't change as the spacetime of the universe bends... For example, you can think about it like this:
Imagine you live on a sphere that's absolutely massive (eg the Earth). Locally, the Earth seems flat, right? Well if you imagine the Earth shrinks to 1000th the size, then suddenly to the giant you, you can see over the horizon a fair way, and the Earth doesn't seem very flat any more to you... But if you shrink to the size of a microbe, then the Earth would seem pretty darn flat again. The same is true in a similar sense in special relativity. No matter how curved space time is, (even if you're inside a blackhole!), you can always look in a small enough region around you for spacetime to appear basically flat. And by looking closer and closer, and measuring the diameter of a circle you draw, you could measure more and more accurate values for pi.
I'm really sorry if this comes across at all derisive, it's certainly not meant to be (quite the opposite, it's supposed to be encouraging!), I just thought it made sense to give my thoughts in case they make things clearer, or just to present an alternative viewpoint :). The philosophy of maths / physics is very interesting and worth learning about - and it's not an area I'm amazingly knowledgable about - certainly my viewpoints above on the philosophy side of things are just my point of view! (Though I should add the mathsy bits, including the bits about special relativity are true from a mathematical/logical standpoint!)
Anyway, very interesting questions, cheers enact :).
I like the explaination, but how can your velocity even be -1. Even though you are turning 180° which is a complete switch of direction, saying your velocity is changing -1 is like saying you turned -180 degrees. The problem gets more complicated if you look at the unit circle.
So if you were to turn 45° then and you called your velocity 1. Then you turned -45° then your velocity would not be -1. Becuase 45° or pi/4 does not equal -45 which is -pi/4 which is 315° which is also 5pi/4.
And even if you assume that you turn 225° which is the true oposite of 45° then the velocity would not be the same. If you cover 225° in the same time as you cover 45° then your speeds would be completly different making your velocity different as well.
In real life we dont refer to things negatively. For instance we dont travel 55mph negative north west. Or we dont say i turned -315° which is 45°. Or we dont walk -4ft.
Okay, I think my explanation wasn't fully clear, and may have caused a bit of confusion between relative and absolute velocity.
In my explanation, interpret "velocity 1" as "velocity to the right". That way, "-1 to the right" is clearly "1 to the left".
The confusion arises if you think of -1 as relative to you... In that sense, you consider -1 to mean "turn around". But this is what I describe as "multiplying by -1", not having a velocity of -1. Multiplying by a number that's not 1 indicates a change of velocity, but just having a velocity of -1 does not denote any change. (if that vaguely makes sense).
In your explanation, neither 45 degrees or -45 degrees are represented as multiplying by (1+i)/sqrt(2) or (1 - i)/sqrt(2). (They are not plus or minus 1.)
Hope that makes a little more sense?
IceDave33 okay i get it now
i always thought of i as x, it is real, but it can't be simplified in any way, just as the quadratic formula can never be simplified to result in y=c for ax^2+bx+c, unless a=b=0, at which time you are simply saying 'i'm not looking at that number anymore, so i'll multiply by 0', which is why i is simple to me, it's just there. nothing else. (also, clockwise 90 degrees i always interpereted as velocity/i and since -1=i^2 and 1=-1^2, 1=i^4, and i^4/i by the law of exponents =i^3=i^2 * i=-1 * i=-i) (may be less intuitive, but people still need to learn how to divide by i) (you should totally do a video on everything needed in dealing with i, a/(1-i) x^i, all that.)
Imaginary numbers weren't created to deal with 2d rotations - it's just the kind of symmetry that they have and thinking about them as 2d rotations can help you get your head around them :).
There are neat extensions of this symmetry to algebras in higher dimensions - but to be honest, the i,j,k etc are easier to grasp. Though the i,j,k are vectors which don't come equipped natively with a way to deal with rotations (you need matrices/tensors/similar).
Hi! Interesting Presentation :)
You may also make imaginary numbers even more real by showing the link with phases (which was historically from were they came if I'm not mistaking) ;-)
Thanks for Sharing :)
If I had learned this shit back in high school, I would have gotten a hell of lot farther in math.
...I created this at the end of school, before starting university, and my target audience was most certainly high-school students...
In my school, complex numbers were introduced simply as an algebraic quantity, and their elegance and simplicity were completely obscured. So, by 'novel', I simply meant that the introduction of i as a rotation instead of an algebraic quantity was a bit different (and in my opinion, gives a much better intuition about the symmetry of complex numbers)
hi.
what to multiply with to go out of the plane
?
+RAVI POHANI multiply by j. for the rule set i^2=j^2=k^2=ijk=-1, for rotations by j would be acw going straight towards you from 1, and i is up and k is going towards you from i, there is a theorem stating you need i j and k to plot 3d vectors and orientations, and as you see by rotating i you get to i from 1, then by k you get to straight towards you, and j then rotates it clockwise from the view looking down from i, arriving at -1, but all of these when doubled arrive at -1 from 1, thus it all fits, and to orient to a direction, set your current orientation to 1 and record the position of your destination, such is the multiplier for your current orientation, and no matter where the origin direction points you will arrive at that destination. yay for math!
The cross product of the real and imaginary basis vectors will be orthogonal to both and give you a new dimension out of the Argand plane.
Firstly, thanks for all these comments!
I fully agree the fundamental idea isn't the slightest bit new, and if I made it now, I would title it alternative instead of novel (actually, just realised I can changed this so have done so!)...
@FullWaveElectronics - I'm glad everyone likes it so far! I'm rather busy at the moment, but I might try and make another one over the Christmas period. Any topics in particular you'd like me to have a look at? Maybe more complex numbers, or some other part of maths completely?
5:29 what does it even mean to square that whole thing...? How does it even verify anything (without any explanation it's like you're just doing some random operation)?
+nomealow Initially we concluded that to rotate 180 degrees, we multiply by -1. To rotate 90 degrees, we multiply by root(-1), or just i. So that must mean to rotate 45 degrees, we multiply by root(i). We got an answer of root(1/2) + root(1/2)i, and to show that this is indeed equal to root(i), we square the whole thing and see if we get i. After squaring, we did get i so root(1/2) + root(1/2)i is equal to root(i).
+Vidal J Yea thanks i re-watched the video again and understood this time.
It's one of the simplest explaination thank you
This was really helpful thanks a lot DAVE!!!
Sadly, this video no longer plays. It was so good to introduce complex numbers to my Year 12 students. Is it possible to reactivate it?
Thank you for flagging this, I'm seeing it too! I'm not sure what's happening - I've contacted UA-cam support for help.
Also, I'm very happy to hear you've found this to be a good introduction for your Year 12 students - I originally made this at the end of Year 13, reflecting on how I wish the curriculum had started with the argand diagram, so it's great to hear that it meets that bar.
Hopefully UA-cam support will be able to help. I'll let you know if I hear any updates.
Looks like video has corrupted on UA-cam's servers. I didn't know this was possible. I'll see if UA-cam support can help re-upload it over this video. If not I'll need to re-upload it to a separate video.
Just to update - I've re-uploaded it here: ua-cam.com/video/2h3mrS7sX0A/v-deo.html - unfortunately, the quality isn't so great, but it wasn't too great to start with. Hopefully it's still sufficient for your needs. Who knows, maybe I'll find time to create it again using 3Blue1Brown visuals and a much clearer narration at some point.
Please leave any feedback positive, negative, or imaginary below.
Nice. But I still can't get how it coexists w/ the 2 negs multiplied produces a pos rule. How could the rule be better stated? The linguistic contradiction is too much. It must be oversimplified? Pertinent only in a single dimension?
i 'm not that good on maths but each time i see imaginary numbers.i see two dimensions
like an (x,y) plane.were x is the real part and y is the factor of the imaginary part.
my question is it really a good idea to define i as i x i = -1. can't we use a
function? : squareroot(x) if x ≥ 0 then squareroot(x)= y
as y*y = x.
,if x ≤ 0
squareroot(x) = - absolute value of (y)
as y*y= x.
what is really useful in the i number.if we already can define a plane of two axis and define position of points with two coordinates.?
If x < 0, then any (real) y cannot have y * y = x...
If y is positive or negative, then y*y is still positive.
So the squareroot(x) for x
Maybe slow down on the hard parts. Why use the "i" variable, what it represents, and how it can be applied to every day uses?
So, actually I think mathematicians make misconceptions about complex numbers and its relation with rotation. You can use a + bi (a complex number) to represent rotations, in the same way as
r cos(β) i + r sin(β) j, but use complex numbers for rotations only really make sense when you are handling problems with negative square roots. For standard rotation problems, is better forgetting about the imaginary number and focus in Linear Álgebra notations. I think 3D rotation with quaternions can be understand as a 4 dimensional vector in geometry problems, and as a 4 dimensional complex number in this category of problems. I really don't understand why people handle rotation with complex numbers in problems which doesn't involve imaginary numbers.
I'd hazard a guess that it's because calculating the values of trigonometric functions is computationally expensive, while operations complex numbers require are extremely fast to perform.
Why can't you start from the right instead of the left? Why does the 180 degree turn have to be negative first, and positive second?
Also, if 90 degrees is the square root of i, then how is 180 degrees i? If you square 90, you get 2700 degrees. Shouldn't i be 2700 degrees?
Great idea, but I would say clock pin is a better example, or sling.
Is it fair to say that i rotates a vector through 90 degrees in 2 dimensions? I love the explanation but is this real space and what if you need to rotate it in the 3rd plane (z or whatever)? If it is real space in 2 dimensions, would you not simply use x and y coordinates to begin with to describe the line segment (vector) that is being transformed? I think there is more to i than meets the eye (art always has to save humanity in the end).
Is it not called an Argand diagram because I think (i could be wrong) its not real space, and if it is not real space then where the hell is this rotation taking place and so are we not getting circular (pun intended) in our reasoning, just a bit?
Like other forms of art, math is a fiction that works (just like mythology).🙃
if you want to rotate 90 degree then why don't you you add an extra speed in the Y direction? All you show me is basic 2D trigonometry equations, so why not use Y. I am still searching for why i is necessary
You're right, you don't need i for this. I wanted to demonstrate its symmetry of multiplication is equivalent to 360 degree rotations... And give a feel for what complex numbers are actually about.
They really become used / useful when the answer to a problem is intrinsically somehow periodic. Eg quantum mechanics, electronics,...
It's just that I am looking for wether the imaginary number is really necessary, and if it's really a thing of nature, or more a human cultural thing. I feel like we took a wrong turn in mathematics and am searching for the real thing. math should conform more with nature. we took the wrong turn by accepting a negative number as 1 piece of information, but it is 2 pieces of information: the sign and the number. For god there are no negatives. After that sticking of the sign to a number, our society needed a trick to work with more difficult mathematical problems: imaginary numbers. Ah well I can go on rambling forever about this. But I am searching for an alternative to this imaginary stuff, and want to know why math people say that complex numbers are really necessary.
Complex numbers are like the fourth dimension. Hard to imagine, but necessary to explain things.
Hey, +IceDave33
Have you seen +3Blue1Brown 's video? He introduces numbers as points, adders and multipliers. Anyway, its really cool! But you did it 4 years before! I wonder how long ago this idea was originally thought up, and how most students aren't explained this is math class..
+PT Yamin That's really neat! An extension of the intuition in my video. I'll add a link to it (and I've subscribed to 3Blue1Brown, the channel has some fantastic videos!). It is common intuition / interpretation in maths circles that the exponential function transforms addition to multiplication in a natural way, but I love the way it is presented in the video, with multiplication seen as a stretching action! Thanks for the link PT! (and love your avatar).
Re the question about how long it was originally thought up - I know the "intuition" I present in my video has been around / known for hundreds of years, but I made the video when back at school when it was still a bit new to me, and I'd pieced together a lot of the understanding myself at that point. I think it's unfortunate that maths classes often focus on solving equations instead of explaining things, because at the end, you don't come out with nearly the same level of enjoyment or intuition. Anyway, rant over ^_^.
+IceDave33 According to +njwildberger 's video on hyper complex numbers, the relationship between complex number multiplication and rotations and dilatations of a plane was understood by the 1830s.
+PT Yamin Cheers, thanks for the sharing! :)
Your presentation was nice (thanks!) but the very concept itself is still a bit eerie for me. Basically, what's happening is, math folks seem to be developing the theory (of complex number) from the applications (geometric rotation etc), instead of the other way round. Shouldn't a theory be able to stand by itself, without any reference to its applications?
.... But I should say the main reason I made the video wasn't to study rotations, but was to help people get over the "complex numbers don't make sense! the root of -1 doesn't exist" feeling. Because, if viewed in the right way, "i" is a 90 degree rotation - and then this also explains other symmetries (such as |a|*|b|=|ab|, arg(a)+arg(b)=arg(ab) [+-360 degrees], and formulas with the complex conjugate) in a more satisfactory way :)
...
This was good! :-) Thankyou
Hey can you explain how two completely imaginary numbers multiply into a real number using this complex plane...Algebraic calculation makes sense..but what does it really mean?..can u explain it using the complex plane..that would be really helpful
Chamath Samarawickrama
Indeed! Two complex numbers multiply to give a real number if their angles (measured clockwise from the real axis) sum to a multiple of 180 degrees.
In particular, completely imaginary numbers (with positive co-efficient, so eg 2i, 3/5i etc etc) have an angle of 90 degrees, so when multiplied, we sum their angle representations... Thus the answer is represented as a point at 180 degrees, in other words, a negative real number.
As the video tries to explain, multiplying complex numbers is equivalent to adding their angle representations (and multiplying their lengths).
IceDave33 oh now I see..Thats why a complex number and its conjugate always multiply into a real number ryt..I always understood the algebraic calculation but not whats actually happening in the complex plane.. thank you so much for the explanation..keep up the good work
Indeed!
And you're more than welcome! Glad I could help :)