Geometry of addition and multiplication | Complex numbers episode 2

#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.

Follow these links to study further:

This is where Grant Sanderson explains how to visualize operations on a number line and on a complex plane.

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!

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 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

[tobiasgertz]

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.

[culater]

BEAUTIFUL ! Love your work ! THANKS !

[pmmeurcatpics]

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!

[georgelaing]

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

[billcipher]

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

[eqwerewrqwerqre]

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

[atreidesson]

All Angels already rushes towards the 3b1b popularity

[jcloewe]

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

[ivanperkovic]

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]

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

[corbinwilson]

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

[antonioaraujo]

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

[eNicMate]

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.

[NashvillainSE]

Can you believe he gave us homework?! 😂

[diribigal]

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.

[elgunsadiqli]

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?

[angeldude]

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.

[bjorntorlarsson]

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.

[tombouie]

I would suggest you take a look at geometric algebra

[willfredo]

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