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A layman's quest to understand wtf this is... >In mathematics, the octonions are a normed division algebra over the real numbers . wtf is a normed division al

by fifnir 8y ago

A layman's quest to understand wtf this is... >In mathematics, the octonions are a normed division algebra over the real numbers . wtf is a normed division algebra?? >In mathematics, Hurwitz's theorem is a theorem [...] solving the Hurwitz problem for finite-dimensional unital real non-associative algebras endowed with a positive-definite quadratic form. ... right. I have the same problem when I try to understand anything statistics related, I get hit by a barrage of unknown words and my brain just melts. Is there any place that explains mathematical concepts in ... different ways?

davebryand 8y ago
Here are a few that I love: 3 Blue 1 Brown: https://www.youtube.com/channel/UCYO_jab_esuFRV4b17AJtAw https://www.youtube.com/channel/UCYO_jab_esuFRV4b17AJtAw VSauce: https://www.youtube.com/user/Vsauce https://www.youtube.com/user/Vsauce Numberphile: https://www.youtube.com/user/numberphile https://www.youtube.com/user/numberphile
- throwawaymath 8y ago
  Those are good for supplementary exposition (especially 3Blue1Brown), but they're not suitable for learning on their own. They're also a bit of a hodge podge beyond calculus and linear algebra. To understand what's going on here in any meaningful sense, the parent commenter should pick up an accessible undergrad textbook on abstract algebra. It doesn't have to be particularly advanced. Then they'll have a better foundation for understanding the algebraic features of various number systems.
empath75 8y ago
Unfortunately there’s a lot of domain knowledge wrapped up in very short phrases. What you want to learn is abstract algebra, if you want to google for YouTube videos that break it down for you.
Micoloth 8y ago
Well yeah, math books... Jokes aside, i too think we need a new framework for divulgating math that actually tells what you need to know, without just handwaving at it, BUT without the amount of technical details of a mathematics class. What do you know, i think this is Possible, too.. You can communicate a surprising amount of information if you use words properly. Of course, since this has never been done except from basic maths, it would be quite a task to embark in, and one would only do it if it made economic sense..
- fifnir 8y ago
  > only do it if it made economic sense What if writing wikipedia articles was treated like publishing on prestigious journals? (somehow)
  - sigstoat 8y ago
    wikipedia already covers abstract algebra (the subject necessary for understanding the math in this article) rather well. what more do you want?
    - Cobord 8y ago
      Thank Richard Borcherds (R.E.B.) for a lot of that.
    - AnimalMuppet 8y ago
      Something that covers abstract algebra rather well if you don't already know abstract algebra well enough to know what all the terms mean? I mean, I suppose you could try to walk the graph of definitions of terms in Wikipedia to try to understand one of the articles. But I'm pretty sure that's not an acyclic graph, and it's not clear where to start in that process.
- sigstoat 8y ago
  > Jokes aside, i too think we need a new framework for divulgating math that actually tells what you need to know, without just handwaving at it, BUT without the amount of technical details of a mathematics class. oh man, have i got bad news for you about the amount of technical detail present in math classes.
  - mikec3010 8y ago
    > the amount of technical detail present in math classes You misspelled "gatekeeping".
- amelius 8y ago
  I'm still hoping for an ELI5 version of Wikipedia (e.g. as a language option).
DougBTX 8y ago
For more brain melting, here's a roadmap for learning React in 2018, React and associated tech is what, 20 years old? Here's what that looks like: https://github.com/adam-golab/react-developer-roadmap https://github.com/adam-golab/react-developer-roadmap Then consider maths, how long and how many people have been contributing to that field?
- booleandilemma 8y ago
  What’s funny is most of those projects in the leaves probably won’t even be around in another 10 years.
throwaway37585 8y ago
Some useful advice: https://math.stackexchange.com/questions/617625/on-familiarity-or-how-to-avoid-going-down-the-math-rabbit-hole https://math.stackexchange.com/questions/617625/on-familiari...
77pt77 8y ago
unital -> there is a multiplicative identity 1 such that 1.a = a for all a. normed -> there is a norm. A way of saying how big an element is. In the reals |x| - x if x is positive and -x if x is negative. In the complex numbers |a+bi| = (a^2 + b^2)^(1/2) Division -> Division (except possibly by zero) is always possible. That means for a and b not zero there exists c such that a=cb (c is a divided by b) I could be wrong about these things since I'm quite rusty
- ducttapecrown 8y ago
  I think division is normally framed rather as the existence of multiplicative inverses for all non-zero elements of the ring. That is, R is a division ring if a) there exists a 1 in R b) for all x in R, there exists y in R such that x * y = 1. I'm pretty sure these are equivalent. Proof: (==>) Let a = 1, and b in R be arbitrary. Then there exists a c such that b * c = 1. So inverses exist! (<==) Let a and b in R be arbitrary. Then a = a * b^-1 * b, so c = a * b^-1. QED.
pdonis 8y ago
> wtf is a normed division algebra?? An "algebra" is a set of thingies that have binary operations defined on them (i.e., operations that take two thingies as input and output another thingie). The thingies are usually referred to as "elements" of the algebra. Any algebra will at least have the "add" and "multiply" operations, but might also have others. A "division algebra" is an algebra where the operations are add, subtract, multiply, and divide, and all of them work basically the same as they do with ordinary numbers (but they won't have all of the same properties--see below). A "normed division algebra" is a division algebra with one additional operation called "norm", that takes as input any element of the algebra and outputs a real number, the "norm" of that element. On the ordinary real numbers, the norm is just the absolute value. The "over the real numbers" part means that you can construct the normed division algebra by starting with real numbers; or, to put it another way, all of the elements of the algebra are "made of" real numbers. The simplest way of viewing this is as a repeated operation of pairing: complex numbers are made of pairs of real numbers, quaternions are made of pairs of complex numbers (hence sets of four real numbers, hence "quater"), and octonions are made of pairs of quaternions (hence sets of eight real numbers, hence "octo"). But each step in this series loses a key property. The reals are totally ordered; the complexes are not. The complexes are commutative under multiplication; the quaternions are not. The quaternions are associative under multiplication; the octonions are not. > Is there any place that explains mathematical concepts in ... different ways? Not really, because the only way people have found to really understand mathematical concepts is to build them up out of simpler mathematical concepts. That means you can't just encounter a complicated mathematical concept and expect to understand it if you don't understand all the simpler concepts it is built from. There are no shortcuts.

by fifnir 8y ago

davebryand 8y ago

Here are a few that I love: 3 Blue 1 Brown: https://www.youtube.com/channel/UCYO_jab_esuFRV4b17AJtAw https://www.youtube.com/channel/UCYO_jab_esuFRV4b17AJtAw VSauce: https://www.youtube.com/user/Vsauce https://www.youtube.com/user/Vsauce Numberphile: https://www.youtube.com/user/numberphile https://www.youtube.com/user/numberphile

throwawaymath 8y ago
Those are good for supplementary exposition (especially 3Blue1Brown), but they're not suitable for learning on their own. They're also a bit of a hodge podge beyond calculus and linear algebra. To understand what's going on here in any meaningful sense, the parent commenter should pick up an accessible undergrad textbook on abstract algebra. It doesn't have to be particularly advanced. Then they'll have a better foundation for understanding the algebraic features of various number systems.

empath75 8y ago

Unfortunately there’s a lot of domain knowledge wrapped up in very short phrases. What you want to learn is abstract algebra, if you want to google for YouTube videos that break it down for you.

Micoloth 8y ago

Well yeah, math books... Jokes aside, i too think we need a new framework for divulgating math that actually tells what you need to know, without just handwaving at it, BUT without the amount of technical details of a mathematics class. What do you know, i think this is Possible, too.. You can communicate a surprising amount of information if you use words properly. Of course, since this has never been done except from basic maths, it would be quite a task to embark in, and one would only do it if it made economic sense..

fifnir 8y ago
> only do it if it made economic sense What if writing wikipedia articles was treated like publishing on prestigious journals? (somehow)
- sigstoat 8y ago
  wikipedia already covers abstract algebra (the subject necessary for understanding the math in this article) rather well. what more do you want?
  - Cobord 8y ago
    Thank Richard Borcherds (R.E.B.) for a lot of that.
  - AnimalMuppet 8y ago
    Something that covers abstract algebra rather well if you don't already know abstract algebra well enough to know what all the terms mean? I mean, I suppose you could try to walk the graph of definitions of terms in Wikipedia to try to understand one of the articles. But I'm pretty sure that's not an acyclic graph, and it's not clear where to start in that process.
sigstoat 8y ago
> Jokes aside, i too think we need a new framework for divulgating math that actually tells what you need to know, without just handwaving at it, BUT without the amount of technical details of a mathematics class. oh man, have i got bad news for you about the amount of technical detail present in math classes.
- mikec3010 8y ago
  > the amount of technical detail present in math classes You misspelled "gatekeeping".
amelius 8y ago
I'm still hoping for an ELI5 version of Wikipedia (e.g. as a language option).

DougBTX 8y ago

For more brain melting, here's a roadmap for learning React in 2018, React and associated tech is what, 20 years old? Here's what that looks like: https://github.com/adam-golab/react-developer-roadmap https://github.com/adam-golab/react-developer-roadmap Then consider maths, how long and how many people have been contributing to that field?

booleandilemma 8y ago
What’s funny is most of those projects in the leaves probably won’t even be around in another 10 years.

throwaway37585 8y ago

Some useful advice: https://math.stackexchange.com/questions/617625/on-familiarity-or-how-to-avoid-going-down-the-math-rabbit-hole https://math.stackexchange.com/questions/617625/on-familiari...

77pt77 8y ago

unital -> there is a multiplicative identity 1 such that 1.a = a for all a. normed -> there is a norm. A way of saying how big an element is. In the reals |x| - x if x is positive and -x if x is negative. In the complex numbers |a+bi| = (a^2 + b^2)^(1/2) Division -> Division (except possibly by zero) is always possible. That means for a and b not zero there exists c such that a=cb (c is a divided by b) I could be wrong about these things since I'm quite rusty

ducttapecrown 8y ago
I think division is normally framed rather as the existence of multiplicative inverses for all non-zero elements of the ring. That is, R is a division ring if a) there exists a 1 in R b) for all x in R, there exists y in R such that x * y = 1. I'm pretty sure these are equivalent. Proof: (==>) Let a = 1, and b in R be arbitrary. Then there exists a c such that b * c = 1. So inverses exist! (<==) Let a and b in R be arbitrary. Then a = a * b^-1 * b, so c = a * b^-1. QED.

pdonis 8y ago

> wtf is a normed division algebra?? An "algebra" is a set of thingies that have binary operations defined on them (i.e., operations that take two thingies as input and output another thingie). The thingies are usually referred to as "elements" of the algebra. Any algebra will at least have the "add" and "multiply" operations, but might also have others. A "division algebra" is an algebra where the operations are add, subtract, multiply, and divide, and all of them work basically the same as they do with ordinary numbers (but they won't have all of the same properties--see below). A "normed division algebra" is a division algebra with one additional operation called "norm", that takes as input any element of the algebra and outputs a real number, the "norm" of that element. On the ordinary real numbers, the norm is just the absolute value. The "over the real numbers" part means that you can construct the normed division algebra by starting with real numbers; or, to put it another way, all of the elements of the algebra are "made of" real numbers. The simplest way of viewing this is as a repeated operation of pairing: complex numbers are made of pairs of real numbers, quaternions are made of pairs of complex numbers (hence sets of four real numbers, hence "quater"), and octonions are made of pairs of quaternions (hence sets of eight real numbers, hence "octo"). But each step in this series loses a key property. The reals are totally ordered; the complexes are not. The complexes are commutative under multiplication; the quaternions are not. The quaternions are associative under multiplication; the octonions are not. > Is there any place that explains mathematical concepts in ... different ways? Not really, because the only way people have found to really understand mathematical concepts is to build them up out of simpler mathematical concepts. That means you can't just encounter a complicated mathematical concept and expect to understand it if you don't understand all the simpler concepts it is built from. There are no shortcuts.