Geometry of addition and multiplication | Complex numbers episode 2

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#complexnumbers #algebra
Are complex numbers just a trick, or is there something more fundamental about them? We answer that question by uncovering their geometry and their algebra. You will discover that complex numbers are inevitable, because no other number system has exactly the nice properties we expect.
The other videos in this series are already available on Patreon: www.patreon.com/user?u=86649007
Follow these links to study further:
[3B1B 1] • Euler's formula with i...
This is where Grant Sanderson explains how to visualize operations on a number line and on a complex plane.
[3B1B 2] • Complex number fundame...
Live lesson: Lots of great insights about complex numbers. With quiz questions that make you think on your feet. Good visuals for operations on complex numbers. Several trig identities magically pop out!
[BRIGHT 1] • Start Learning Complex...
The bright side of mathematics: a short but good introduction to complex numbers. Offers another compelling argument for why we need to go into the 2nd dimension.
[KHAN 1] • Introduction to i and ...
Khan Academy: A good series about complex numbers, with many examples.
0:00 Introduction
1:39 The geometry of real addition
3:46 The geometry of complex addition
7:10 The geometry of real multiplication
10:27 The geometry of complex multiplication
14:55 Polar coordinates
17:38 'i' is a 90 degree rotation
18:25 Geometry wrap-up
19:28 Discovering complex multiplication via algebra
26:02 Conclusion
This video is published under a CC Attribution license ( creativecommons.org/licenses/... ).

Опубликовано:

1 июл 2024

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Комментарии : 48

@tobiasgertz7800 Год назад

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.

@whannabi Год назад

yessir

@jcloewe8692 Год назад

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?

@georgelaing2578 Год назад

This is a much richer treatment of operations than is usually presented. Bravo!!

@atreidesson Год назад

All Angels already rushes towards the 3b1b popularity

@pmmeurcatpics Год назад

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!

@bigbluebuttonman1137 Год назад

There is just something wrong with education, even higher education. I could go into detail, but it’s a lot of problems.

@Abstract3030 Год назад

Excellent.

@eqwerewrqwerqre Год назад

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

@AllAnglesMath Год назад

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.

@bigbluebuttonman1137 Год назад

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.

@AllAnglesMath Год назад

@@bigbluebuttonman1137 The only thing that you can't easily do online (yet), are exercises. And those are a major part of learning math.

@bigbluebuttonman1137 Год назад

@@AllAnglesMath Agreed.

@thezipcreator 3 месяца назад

@@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

@antonioaraujo8257 Год назад

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

@jcloewe8692 Год назад

This is beautiful, thank you. I've always been curious about maths but I work in a different field, you're expanding my mind.

@billcipher3737 Год назад

Very high quality! I hope you continue making these videos :)

@moralboundaries1 9 дней назад

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.

@samylahlou Год назад

Incredible job !

@ivanperkovic274 Год назад

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

@VanDerHaegenTheStampede Год назад

Incredible job. Will you approach other 2-dimensional number systems such as split-complex (hyperbolic) numbers and dual numbers?

@AllAnglesMath Год назад

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.

@eNicMate Год назад

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.

@AllAnglesMath Год назад

The "negative numbers" also used to have a bad reputation for no reason.

@AtoEactor 4 месяца назад

Amazing. Thank you.

@Nashvillain10SE Год назад

Can you believe he gave us homework?! 😂

@corbinwilson660 Год назад

Great channel. I am loving the videos. I’m curious, what do you use to make your videos?

@AllAnglesMath Год назад

I use a custom-made python library with OpenCV. I get this question a lot, so maybe I should make a video about it ;-)

@corbinwilson660 Год назад

@@AllAnglesMath You definitely should. It is some of the most relaxing and satisfying visuals I’ve ever seen. Keep up the good work man 👍

@poopiecon1489 Год назад

@angeldude101 11 месяцев назад

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.

@elgunsadiqli6912 Год назад

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?

@pietergeerkens6324 Год назад

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.

@elgunsadiqli6912 Год назад

@@pietergeerkens6324 Thank you for answer. I will read about it

@AllAnglesMath Год назад

The final video in the series will talk about higher-dimensional number systems, but it won't go into detail.

@diribigal Год назад

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 Год назад

Very cool. I should try to work out these formulas in detail to see what is going on.

@diribigal Год назад

@@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.

@tombouie Год назад

I would suggest you take a look at geometric algebra

@bigbluebuttonman1137 Год назад

Geometric Algebra is probably something I’d enjoy.

@will14fredo 2 месяца назад

Never thought it would take me so long in my life to understand why -1*-1=1😂

@AllAnglesMath 2 месяца назад

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!

@penguinjuice7543 9 месяцев назад

Good explanation but i didnt like how you explained complex addition in terms of vector addition when vectors came after complex numbers.

@AllAnglesMath 9 месяцев назад

When you say "came after", do you mean historically, or do you mean that we explain vectors in a future video?

@tintincruz8660 7 месяцев назад

But what about division? 😭

@AllAnglesMath 7 месяцев назад

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.

@05degrees 9 дней назад

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.