The Multiplication Multiverse | Infinite Series

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What happens if you multiply things that aren’t numbers? And what happens if that multiplication is not associative?

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Multiplication of numbers is an associative property and we can make sense of “multiplication” between things that aren’t numbers but that’s not considered as associativity. And since we’re talking about associativity, you might wonder about that other property of real numbers: You know: when multiplying two numbers, swapping their order doesn’t change the answer. This property is called commutativity. But keep in mind: it’s a very special property to have! Not everything in life is commutative. For example, getting dressed in the morning... putting on your socks and then your shoes is NOT the same as first putting on your shoes and then your socks.

Link to Resources:

The Fundamental Group: ‘Loop concatenation’

Written and Hosted by Tai-Danae Bradley
Produced by Rusty Ward
Graphics by Ray Lux
Assistant Editing and Sound Design by Mike Petrow

Thanks to Matthew O'Connor and Yana Chernobilsky who are supporting us on Patreon at the Identity level!

And thanks to Nicholas Rose and Mauricio Pacheco who are supporting us at the Lemma level!

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Комментарии

Hi everyone, Tai-Danae here. It's nice to e-meet you all! Feel free to drop me your questions below! You can also find me on Twitter at @math3ma.

pbsinfiniteseries

Hey y’all! Many of you have asked a great question: Why are we calling loop concatenation a “multiplication”?? Technically, loop concatenation is a “binary operation, ” i.e. a way to combine two inputs and get one output. Addition and multiplication of numbers are probably the most familiar examples.

In practice, mathematicians often use the word “addition” to describe binary operations that are commutative. And although I don’t mention it in the video, loop concatenation is not commutative! (Do you see why?) So we just call it “multiplication” instead.

And thanks for all the feedback! I’ve got another idea to help us get used to this notion of multiplying things that *aren’t* numbers, and it won't involve fancy algebra language. I’ll share it on my blog (www.math3ma.com) hopefully next week. Stay tuned!

- Tai-Danae

pbsinfiniteseries

"Now, to do this each car must travel at twice their original speed... but that's fine." I did not expect to crack up like that, quality delivery.

placidesulfurik

my favorite commutative non-associative operation is winning in rock-paper-scizzors. B)

(✊ vs ✋) vs ✌ ≠ ✊ vs (✋ vs ✌)

sofia.eris.bauhaus

Shout out to the green screen guy for somehow getting her hair keyed correctly. I know that wasn't easy! You can see the insane amount of motion blur they used every time her head even slightly moves!

BBand

Infinite series is still is good hands!

shubhamshinde

Hey Tai-Danae, great first episode! Im a swedish 27 yo that has got a masters in engineering and your presentation pushed the boundry of what I know - thanks! Since I've taken quite a lot of math classes in uni, following your presentation wasnt that hard - but for somebody that still is in uni or in high school, the speed of which you introduce and jump between concepts and terms might be hard to understand. I've followed the PBS series on youtube since the first episodes, and all presenters did this in the beginning - so im confident you'll do a great job!

damirradoncic

"We'll discover that all of the different ways of multiplying 100 different loops in a topological space can be encoded in a 98-dimensional polyhedron called an associahedron." That sounds absolutely mad! I love it. I wonder what you'd consider the identity in loop multiplication.

nom

Do not slow down.
This is a video. We people can go back and listen, pause. Cant wait for the next one!

magne

"The cool thing is that this forms what we call a Lie ok, moving on."

unvergebeneid

I miss this series. And I think Tai-Danae Bradley was one of my favorite hosts for it.

Omnifarious

2:55, 5th line, I assume that's a (-1) rather than a (=1)?

Kram

Happy to see we’re going faster! Go go go!

Swiftclaw

I don't know why people are complaining about a fast pace. I had no trouble keeping up, even though the idea of multiplying loops is completely new to me. You're doing great, and welcome to Infinite Series! I loved your first episode and I'm hyped for the next one :D

kamoroso

I love where you're taking the channel. And I like how you're showing the notation while you're saying the statement to low-key introduce people to the language of mathematics.

Also, dropping the definition of a Lie algebra in for funsies - dig it. I'd love a short series on the importance of Lie algebras in modern math and physics, with some explanation of their properties (it might be tough to hit the right level for this channel, granted).

bentoomey