Oh, no worries. Just several hundred years of mathematical discoveries explained geometrically in a calm, soothing voice in a way a ten year old could understand. Great job bro.
It's a real pity that people who are paid to teach can't even come close to making things as understandable as you, among other educational creators, manage to do. Your explanations and the visualizations are excellent, and I'm eagerly waiting for the following videos!
Damn, this is wild. And in only 30 minutes! This could've been a single class literally anywhere along the way in high school required math courses. It stuns me that as a college student with years of mathematical experience, I've never seen anything so solid and grounded about the reality of complex numbers. The fact that they're the only possible number system that satisfies our requirements, not like we just made up some crazy thing so that 16 year olds would hate math, not that we're using fundamentally strange and broken mathematics to try to model our reality, but that this is the only possible way any of those things could be. Damn
Thank you so much. Your comment made my day ;-) I've always been convinced that it should be possible to make mathematics more concrete and "grounded" as you call it. But also more "inevitable". Many definitions seem arbitrary at first glance, but they serve deeper purposes that a high school class does not always go into. I'm glad to hear that this alternative approach is landing.
I’ve discovered that a lot of mathematical knowledge is more accessible and understandable outside of the classroom. With some inventiveness, someone could probably self-study most college-level mathematics in as much time or even a little faster.
Great series so far, though I wonder if 30 minutes is a bit too much for beginners. Personally the rhythm seems great but I already had complex analysis so I can't tell. Anyways, keep up the good (and hard) work, I hope your channel explodes like it deserves
As someone new to self studying math but very interested in it, I found this video very interesting and easy to follow. Watched the whole 30 minutes of it. 10/10
I will briefly mention them in the 4th (and final) video of this series. I don't plan to go into the details, although I will argue why it can be useful to have a non-real number that squares to 0 or 1.
I heard somewhere that the name "Orthogonal numbers" was proposed instead of imaginary numbers. If you think about it "Real" numbers are also imaginary, they are just abstractions of the world that are useful to us to represent ideas (such as quantities, proportions, measures, etc ...), its not like you can actually point somewhere in the natural world and say, oh there I see a "2" or something like that (I say natural because of course we have created physical representations of these ideas) ... in that sense the "Imaginary numbers" are also just an abstraction to understand ideas of the world. It is just that it's a higher level of abstraction (do not even mention quaternions) that makes it mysterious to people ... that plus the term "Imaginary" which doesn't seem friendly when talking about numbers.
Around 26:22 , you say "only one formula does the trick...no other formula exists that has all the nice properties", which I think is a little misleading. For examples, (a,b) * (c,d) = ( ac-2bd, ad+bc+2bd ) or (a,b) * (c,d) = (ac-4bd, ad+bc) both work just fine and have all the properties you listed. What was proven is that any formula that works (and isn't just the reals) secretly encodes the complex numbers in another way. For instance, if I did my algebra right, my first (a,b) is like a+b+bi and my second is like a+2bi.
@@AllAnglesMath From your linear algebra background, the details of the formulas aren't too important. Basically, I was just using different bases for the complex plane.
I have a question. If we can invent imaginary axis and make imaginary real plane. Why can't we add third dimension j = 1/0 dots will be rotated 90⁰ around imaginary axis and if we look at imaginary-real plane it will be collapsed into dots which have zero gap between them? Is 3D number system possible?
Look up Frobenius Theorem. As Hamilton wrote to Graves the day following his well-known walk to the Royal Irish Academy on Oct. 16, 1843: "And here there dawned on me the notion that we must admit, in some sense, a fourth dimension of space for the purpose of calculating with triples ... An electric circuit seemed to close, and a spark flashed forth." The above of course refers to his discovery of quaternions.
I think you turned down (ac + bd, ad + bc) a bit too quickly. Sure, a number (a, b) doesn't have an inverse when a² = b², but you still could've taken a look at the geometry that results from this product! It's really interesting and might not be what you'd expect. Another interesting product is (ac, ad + bc). This also has several non-invertable elements (technically called "zero-divisors") (a, b) when a = 0, but it also gives an interesting and very useful geometry. It's _also_ surprisingly closely related to calculus, and in a similar vein, the previous product is very useful in physics.
There's a great story about a teacher who was explaining this to his students. He said: there are languages in the world in which a double negation becomes an even stronger negation (as in "I didn't see nobody"), but then other languages in which a double negation cancels into a positive, just like (-1)*(-1)=1. But, he claimed, there are no languages anywhere on earth in which a double *positive* cancels to become a negation. At that point, one of his students shouted: Yeah, right!
It's really weird, when you think about it. multiplication = rotation. The complex numbers are hiding in the space between -1 and 1. Thank you for illustrating this so clearly for us.
We focus on multiplication in the video, but division is not very difficult either. To simplify a division like (a+bi)/(c+di), just multiply the numerator & denominator both by (c-di). That gets rid of the imaginary stuff at the bottom.
A bit late to the party but: you can also try to solve for x, y in (a + bi)(x + yi) = (c + di). This looks like a single equation with two unknowns but in fact it’s two when we separate real and imaginary parts after expansion. Though it might be a bit tedious without geometric insight, and with it you can end up inventing what All Angles said first-but this blunt algebraic path is there. BTW this trick of multiplying by the conjugate is used more widely: you can rationalize a denominator of (a + √b) by multiplying the whole fraction by (a − √b), these two are sorta conjugate to each other as well. In general if you have a conjugate for your objects which allows to make a real number, you have a way to invert: if A A* = r ∈ ℝ and r is nonzero, then A (A* / r) = 1 and so A⁻¹ = A* / r. And when you can invert things, you have divide by them. Generally though it might get complicated with noncommutative multiplication (left division and right division may exist separately and disagree if both are defined) or when there are no inverses but there is still division. Personally I try to forget about the latter case. But it happens in some areas of math. I’m so lucky I haven’t needed this for my applications.
BEAUTIFUL ! Love your work ! THANKS !
Oh, no worries. Just several hundred years of mathematical discoveries explained geometrically in a calm, soothing voice in a way a ten year old could understand. Great job bro.
yessir
Right? I thought math was supposed to be extremely difficult and mysterious, that's what I was always taught in school. What's all the fuss about?
This is a much richer treatment of
operations than is usually presented.
Bravo!!
It's a real pity that people who are paid to teach can't even come close to making things as understandable as you, among other educational creators, manage to do. Your explanations and the visualizations are excellent, and I'm eagerly waiting for the following videos!
There is just something wrong with education, even higher education. I could go into detail, but it’s a lot of problems.
Damn, this is wild. And in only 30 minutes! This could've been a single class literally anywhere along the way in high school required math courses. It stuns me that as a college student with years of mathematical experience, I've never seen anything so solid and grounded about the reality of complex numbers. The fact that they're the only possible number system that satisfies our requirements, not like we just made up some crazy thing so that 16 year olds would hate math, not that we're using fundamentally strange and broken mathematics to try to model our reality, but that this is the only possible way any of those things could be. Damn
Thank you so much. Your comment made my day ;-)
I've always been convinced that it should be possible to make mathematics more concrete and "grounded" as you call it. But also more "inevitable". Many definitions seem arbitrary at first glance, but they serve deeper purposes that a high school class does not always go into. I'm glad to hear that this alternative approach is landing.
I’ve discovered that a lot of mathematical knowledge is more accessible and understandable outside of the classroom.
With some inventiveness, someone could probably self-study most college-level mathematics in as much time or even a little faster.
@@bigbluebuttonman1137 The only thing that you can't easily do online (yet), are exercises. And those are a major part of learning math.
@@AllAnglesMath Agreed.
@@AllAnglesMath I don't know if that's really true-libretext has a lot of free math textbooks that also include exercises at the end of chapters
All Angels already rushes towards the 3b1b popularity
Great series so far, though I wonder if 30 minutes is a bit too much for beginners. Personally the rhythm seems great but I already had complex analysis so I can't tell. Anyways, keep up the good (and hard) work, I hope your channel explodes like it deserves
Very high quality! I hope you continue making these videos :)
Excellent.
Incredible job !
This is beautiful, thank you. I've always been curious about maths but I work in a different field, you're expanding my mind.
As someone new to self studying math but very interested in it, I found this video very interesting and easy to follow. Watched the whole 30 minutes of it. 10/10
Incredible job. Will you approach other 2-dimensional number systems such as split-complex (hyperbolic) numbers and dual numbers?
I will briefly mention them in the 4th (and final) video of this series. I don't plan to go into the details, although I will argue why it can be useful to have a non-real number that squares to 0 or 1.
Great channel. I am loving the videos. I’m curious, what do you use to make your videos?
I use a custom-made python library with OpenCV. I get this question a lot, so maybe I should make a video about it ;-)
@@AllAnglesMath You definitely should. It is some of the most relaxing and satisfying visuals I’ve ever seen. Keep up the good work man 👍
I heard somewhere that the name "Orthogonal numbers" was proposed instead of imaginary numbers. If you think about it "Real" numbers are also imaginary, they are just abstractions of the world that are useful to us to represent ideas (such as quantities, proportions, measures, etc ...), its not like you can actually point somewhere in the natural world and say, oh there I see a "2" or something like that (I say natural because of course we have created physical representations of these ideas) ... in that sense the "Imaginary numbers" are also just an abstraction to understand ideas of the world. It is just that it's a higher level of abstraction (do not even mention quaternions) that makes it mysterious to people ... that plus the term "Imaginary" which doesn't seem friendly when talking about numbers.
The "negative numbers" also used to have a bad reputation for no reason.
Amazing. Thank you.
Around 26:22 , you say "only one formula does the trick...no other formula exists that has all the nice properties", which I think is a little misleading.
For examples, (a,b) * (c,d) = ( ac-2bd, ad+bc+2bd ) or (a,b) * (c,d) = (ac-4bd, ad+bc) both work just fine and have all the properties you listed.
What was proven is that any formula that works (and isn't just the reals) secretly encodes the complex numbers in another way. For instance, if I did my algebra right, my first (a,b) is like a+b+bi and my second is like a+2bi.
Very cool. I should try to work out these formulas in detail to see what is going on.
@@AllAnglesMath From your linear algebra background, the details of the formulas aren't too important. Basically, I was just using different bases for the complex plane.
I have a question. If we can invent imaginary axis and make imaginary real plane. Why can't we add third dimension j = 1/0 dots will be rotated 90⁰ around imaginary axis and if we look at imaginary-real plane it will be collapsed into dots which have zero gap between them? Is 3D number system possible?
Look up Frobenius Theorem. As Hamilton wrote to Graves the day following his well-known walk to the Royal Irish Academy on Oct. 16, 1843:
"And here there dawned on me the notion that we must admit, in some sense, a fourth dimension of space for the purpose of calculating with triples ... An electric circuit seemed to close, and a spark flashed forth."
The above of course refers to his discovery of quaternions.
@@pietergeerkens6324 Thank you for answer. I will read about it
The final video in the series will talk about higher-dimensional number systems, but it won't go into detail.
Great way to explain this stuff.
But there is confusion at 15:25 where z=r and then suddenly z is r multiplied with something else.
I agree that the formulas aren't always consistent.
I would suggest you take a look at geometric algebra
Geometric Algebra is probably something I’d enjoy.
Can you believe he gave us homework?! 😂
I think you turned down (ac + bd, ad + bc) a bit too quickly. Sure, a number (a, b) doesn't have an inverse when a² = b², but you still could've taken a look at the geometry that results from this product! It's really interesting and might not be what you'd expect.
Another interesting product is (ac, ad + bc). This also has several non-invertable elements (technically called "zero-divisors") (a, b) when a = 0, but it also gives an interesting and very useful geometry. It's _also_ surprisingly closely related to calculus, and in a similar vein, the previous product is very useful in physics.
Never thought it would take me so long in my life to understand why -1*-1=1😂
There's a great story about a teacher who was explaining this to his students. He said: there are languages in the world in which a double negation becomes an even stronger negation (as in "I didn't see nobody"), but then other languages in which a double negation cancels into a positive, just like (-1)*(-1)=1.
But, he claimed, there are no languages anywhere on earth in which a double *positive* cancels to become a negation. At that point, one of his students shouted: Yeah, right!
Good explanation but i didnt like how you explained complex addition in terms of vector addition when vectors came after complex numbers.
When you say "came after", do you mean historically, or do you mean that we explain vectors in a future video?
It's really weird, when you think about it. multiplication = rotation. The complex numbers are hiding in the space between -1 and 1. Thank you for illustrating this so clearly for us.
But what about division? 😭
We focus on multiplication in the video, but division is not very difficult either. To simplify a division like (a+bi)/(c+di), just multiply the numerator & denominator both by (c-di). That gets rid of the imaginary stuff at the bottom.
A bit late to the party but: you can also try to solve for x, y in (a + bi)(x + yi) = (c + di). This looks like a single equation with two unknowns but in fact it’s two when we separate real and imaginary parts after expansion. Though it might be a bit tedious without geometric insight, and with it you can end up inventing what All Angles said first-but this blunt algebraic path is there.
BTW this trick of multiplying by the conjugate is used more widely: you can rationalize a denominator of (a + √b) by multiplying the whole fraction by (a − √b), these two are sorta conjugate to each other as well. In general if you have a conjugate for your objects which allows to make a real number, you have a way to invert: if A A* = r ∈ ℝ and r is nonzero, then A (A* / r) = 1 and so A⁻¹ = A* / r. And when you can invert things, you have divide by them.
Generally though it might get complicated with noncommutative multiplication (left division and right division may exist separately and disagree if both are defined) or when there are no inverses but there is still division. Personally I try to forget about the latter case. But it happens in some areas of math. I’m so lucky I haven’t needed this for my applications.