Imaginary Numbers are Not 'Imaginary'! In 5 Levels of Complexity

Imaginary numbers are not "Imaginary"! Despite their name, they are completely solid mathematically, and they are critical for many real-world applications. Let's explore imaginary numbers in 5 levels, ranging from the idea behind calling them "imaginary" to exploring their rectangular and polar forms, complex numbers as matrices, and real-world applications.

This video was sponsored by Brilliant.

00:00 Introduction
00:21 Level 1: "Imaginary"
02:17 Level 2: Rectangular and Polar
05:16 Level 3: Matrices
08:10 Sponsor Message
09:26 Level 4: Applications
10:15 Level 5: Abstract Algebra

[libyanmapping]

I've been watching you for weeks now under the impression that you had at *least* 50k subscribers, I was surprised that it's not even at 15k! Definitely an underrated channel!

[NickKravitz]

As a mathematician, computer scientist, and guitarist, it is interesting to see how the symbol i became overloaded. i is used to indicate imaginary number, integer variable, and index finger. Cool explanations!

[j-rey-]

This is my favorite, quick visualization for why i * i = -1:

On a number line, performing -1 * -1 is like starting at 1, then making two 180 degree turns, first to -1, then back to 1, for a full 360 degree turn. That shows how two negatives multiplied make a positive.

Similarly, performing i * i is like making two 90 degree turns, starting at 1 then first turning 90 degrees to 'i' on an axis perpendicular to the real number line (the imaginary number axis), then another 90 degree turn to -1, making a 180 degree turn in total. That shows how two imaginary numbers multiplied make a real number.

[eliteteamkiller]

_"Get real."_

_"Get rational."_

[naytte]

The claim that "imagniary numbers are made up" is right, but for the wrong reason. Most students who exclaim this do so in the belief that the real numbers are, well, more "real" than imagniary numbers. But the thing about that is that all numbers are made up in essence. There is no such thing as a "number" in the universe, they are constructed objects which only exist in our heads. That goes for real numbers as well. So if we are being pedantic, imagniary numbers are "imaginary", but so is every other kind of number in existance as well.

I write this without having watched the video yet (it's in my watch later).

[DJ-ovit]

I dont think you should leave out the "5 levels" out of your youtube title. Subscribers might recognize the color thing on the thumbnail but Id imagine, new viewers dont. Its an inviting concept to know beforehand that the video in question explains something in increasing levels of complexity.

[AndresFirte]

I really like that this channel focuses on trying to speak at everyone’s level in the same respectful way. Good work as always Sean!

[bayleev]

ah, i was kinda hoping the math major would bring up isomorphism classes of algebras! the thing is, a mathematician doesn't really think any of these definitions of the complex numbers are any more correct than any other. a set of formal symbols, a subalgebra of End(R²), the polynomial ring R[x]/(x²+1), the algebraic closure of R, a real 2D vector space with an automorphism squaring to -id: all of these constructions are "basically the same."

the precise definition of this uses abstract algebra. in this context, we define an algebra (over R) to be any vector space over the real numbers with a bilinear product of vectors. examples of this structure are the real numbers themselves, the complex numbers, and the quaternions, but also strange things like R³ with the cross product. we also say that two algebras are isomorphic if there's a linear map between them which doesn't change products; that is, A and A' are isomorphic if there is some bijective linear f: A -> A' so that f(uv) = f(u)f(v). intuitively, this is saying that the only real difference between A and A' is the labels, since you can use f to essentially change the labels without changing the structure of addition or multiplication.

now, if you look at the multiple definitions of the complex numbers, they're all different objects. however, they're also all vector spaces over R with a product between elements, so they're all algebras over R! and it turns out that they are all isomorphic: given the formal symbol i, you map it to the matrix ((0, -1), (1, 0)), the polynomial x, a solution to x²+1 = 0, or the automorphism squaring to -id. (also send 1 to 1 or id in each case; because each formal symbol is a+bi, this determines a map on all formal symbols.) this determines an isomorphism in each case, so every representation is isomorphic!

in that sense, the thought of picking one representation is a bit silly. the complex numbers are kind of just what they are, and these explicit realisations are just concrete versions of a more general phenomenon. because we know they are all isomorphic, we can switch between them whenever we want and whatever we do with them remains valid.

[NicholasAngelidis]

another great video! probably the best summary one can get in 10 minutes.

thanks dr Sean!

[randomz]

This content is incredibly high quality. Your channel will definitely blow up soon.

[Axacqk]

Complex numbers are the real numbers. "Real" numbers are a feature limited demo.

[LeTtRrZ]

I’ve been learning quantum mechanics recently, and that has given me an entirely new appreciation for the complex plane, especially when it comes to phase. The Euler equation is a way to split a number up into complex components without altering its absolute value. This is a very important property when we want to adhere to conservation principles while still allowing interference patterns to emerge under certain circumstances.

[andrearesnati]

Most clear explanation of the definition of C = R[X]/(X^2+1) I have ever heard, congratulations! An anedocte: my calculus 2 teacher used to say: I know tho types of numbers, integers and complex numbers. As integers are too complicated, I will talk about complex numbers :)

[JayTemple]

10:13 When I was an adjunct instructor, I would tell my students that complex numbers were used in electrical engineering. Finally I made a promise to myself and future classes that the next time I covered the topic, I would actually learn how it was used. I was essentially able to show the real-world interpretation of addition, subtraction and multiplication by a real number or a pure imaginary number. The only thing I didn't understand is what it meant when you multiplied by a nontrivial complex number, i.e., one where neither a nor b was 0.

[captnmaico]

Its ineteresting that for some cases the more "difficult" explanation is far easier to understand and for others its the "simple" one. Just shows people understand tasks completly different and may ace certain problems and struggle heavily on others.

[QuillPGall]

taking a summer class on complex variables rn, so the abstract alg review is appreciated. banger vid as always

[s.leeodegard]

I feel that the term "imaginary" is just a misnomer that stuck. I would prefer the term "lateral", and I believe that for some languages, that is the case.

[alexmantzoros]

great video, I knew a lot about complex numbers but I still learned!

[robfielding]

the most profound thing is to add directions in space as primitives, INSTEAD of using complex numbers. "i" does not specify the plane of rotation. There are an infinite number of objects that multiply to -1. So, even though i^2 = -1, it does not mean that sqrt[-1] is definitely i. Directions in space as an example:

right*right=1
up*up=1
right=-left
up=-down

right*up = up*left = left*down = down*right

you can derive from this that:

right*up = -up*right

it anti-commutes. this means that multiplication does not commute in general. but note that these objects square to -1:

right * up * right * up
=
(right*up)*right*up
=
-(up*right)*right*up
=
-up*(right*right)*up
=
-up*1*up
=
-up*up
=
-1

"i" is a 90 degree rotation in an unspecified plane. (right*up)^2 is a 90 degree rotation in the plane specified by (right*up).

You need to be really really careful with 3 directions in space; Because there are 3 separate planes of rotation!

[Peterotica]

Thank you for this video! you have no idea how much I’ve needed this! I always say the same to people who try to tell me my best friend is “imaginary”. That is just so hurtful and toxic. Through math they’re now proven wrong