Quaternions

[Anonymous No.10944247]

/Med/fag here, can someone explain me quaternions? Is it possible to explain it logically in a simple way?
My math knowledge is basic, my Complex number is basic as well
How is it possible that {\displaystyle \mathbf {i} ^{2}=\mathbf {j} ^{2}=\mathbf {k} ^{2}=\mathbf {ijk} =-1.} ? Can you explain me?

[Anonymous No.10944249]

>>10944247
>{\displaystyle \mathbf {i} ^{2}=\mathbf {j} ^{2}=\mathbf {k} ^{2}=\mathbf {ijk} =-1.}

The fuck is this shit lmao
sorry
pic related

[Anonymous No.10944250]

>quarter (the) onions
go back to /ck/

[Anonymous No.10944252]

>>10944250
not funny didn't laugh

[Anonymous No.10944258]

>>10944247
The Quaternions group can be defined by looking at this multiplication table. And that's pretty much it.

[Anonymous No.10944270]

>>10944258
Okay, and how do you explain in simple terms why commutative property doesn't exist for them?

[Anonymous No.10944281]

>>10944270
Looks closely at the table.

If is say a*b
Then I look for a in the left column and look for b in the top row. Then I look at the value where the row and column intersect.

For example:
k*I = j (by looking at the table)
I*k = -j (by looking at the table)

[Anonymous No.10944285]

>>10944270
Vector multiplication is not commutative
These are i-hat j-hat and k-hat

[Anonymous No.10944287]

>>10944281
>k*I = j (by looking at the table)
>I*k = -j (by looking at the table)
you lost me here

I understand why the cross product isn't commutative (studied in Physics 1 class), but I can't get quaternions

[Anonymous No.10944291]

>>10944285
ohh, so terms in quaternions are vectors?

if yes I understand it now lol >>10944287

what do you mean by -hat ?

[Anonymous No.10944306]

>>10944287

[Anonymous No.10944363]

>>10944306
Ok...but WHY?
They just "invented a new math" for this or there is a specific reason why quaternions aren't commutative?
I understand the reason why cross product isn't commutative
Is it because of cross products?

[Anonymous No.10944393]

>>10944247
>>10944363
The original reason Hamilton came up with quaternions was by the reasoning:
>if you consider the real numbers to have "zero generators"
>then the complex numbers have a single generator, i
>so can I make a similar, nice system using two generators??
He says as much in a letter to his son (easily findable through google). He spent years or decades on this problem.

The answer, it turns out, was no. You actually need three generators, and that's what Hamilton got when he invented the quaternions (or more specifically the "real quaternions"). After finding the quaternions, Hamilton devoted the rest of his life to finding applications, and they are now used in physics, in (some styles of) 3D computer graphics, and other places I don't know much about. Obviously Hamilton was insane, but he wasn't wrong.

[Anonymous No.10944466]

>>10944363
> They just "invented a new math" for this
Yeah.....sorta. But those rules were made up in order to still allow consistency in the math of the complex numbers and the real numbers.

Just like the rules of addition and multiplication are extended to the real numbers without changing the original rules for the natural numbers.

> there is a specific reason why quaternions aren't commutative?
As soon as you introduce the new numbers, j and k, and define multiplication that way...You'll see that not all quaternions commute. Only special subsets of the quaternions are commutative. And all the the complex numbers, a subset of quaternions commute.

There is no deeper reason. I personally don't use them. There just a mathematical curiosity. If I want to define rotations, I just use matrices instead.

[Anonymous No.10944826]

>all these dumb aspies ITT who want to show off to people online what a smart boy they are.
Unironically kys faggots, you remind me of that kid who always answers the professors deliberately easy questions at the beginning of the semester

[Anonymous No.10944898]

>>10944247
>How is it possible
It's taken as an axiom of an analysis, and then you try to find out what implications you can coax out of it.

[Anonymous No.10944910]

>>10944363
>They just "invented a new math"
No such thing as invention.

[Anonymous No.10944912]

>>10944910
Platonists please go and stay go

[Anonymous No.10944923]

>>10944247
No field axiom asserts that if a^2 = b^2, then a must necessarily be unique.
>http://mathworld.wolfram.com/FieldAxioms.html
Also they're missing nontriviality axiom 1=/=0, cmon wolfram. A field is what's needed to make math work basically, since quarternions do not violate the field axioms, theorems may be derived. Of course, proving the axioms for quarternions is significantly less trivial than it is for the reals, but thats left as an exercise to the reader.

[Anonymous No.10945080]

>>10944247
A division algebra defined over the field R. The dimension (as a vector space <=> an algebra) is 2^n for natural numbers n. As you go up in dimensionality, you lose "familiar" properties of the algebra, e.g. positive definiteness (C), commutativity (H), associativity (O).

[Anonymous No.10945562]

>>10944910
Yeah, I didn't meant "invention" lol
But I'm not good in english and I was missing a word

[Anonymous No.10945579]

>>10944363
I will not give you a reason, but a demonstration.
You probably know that complex numbers can be used to represent rotation in the plane (mutliplication by a unit complex number = rotation by the argument). Similarly quaternions can be used to represent rotations in a 3-space. And rotations in a 3-space are not commutative. On the other hand, planar rotations are commutative (there's no axis) and complex numbers are commutative.

[Anonymous No.10945583]

>>10945579
I'm interested in math, I've always wanted to became a mathematician but in my country (italy) you get paid meagre and I don't want to do spreadsheet and other shits
I know Calculus 1, which other subfield should I learn by myself? I'm curious about complex numbers

[Anonymous No.10945932]

[math] \displaystyle
\boxed{ \mathbb{T} \;
\boxed{ \mathbb{S} \;
\boxed{ \mathbb{O} \;
\boxed{ \mathbb{H} \;
\boxed{ \mathbb{C} \;
\boxed{ \mathbb{R} \;
\boxed{ \mathbb{Q} \;
\boxed{ \mathbb{Z} \;
\boxed{ \mathbb{N}}}}}}}}}}
[/math]

[Anonymous No.10945942]

>>10944363
Define "new".
The reason the cross product is not commutative is the same as that for the quaternions: they are defined in a way that makes them non-commutative.
Bonus note: If you'd define them differently, then they'd not be capable of modeling generations of rotations.

[Anonymous No.10945948]

>>10944923
>The qaternions are a field
>You need fields to do math
>You need the reals to do math

[Anonymous No.10945952]

>>10945583
Go to Germany, then.
Money is a meme anyway

[Anonymous No.10945971]

>>10944363
> is a specific reason why quaternions aren't commutative?
Because rotations in 3D aren't commutative. Take a cube and rotate it first about one axis, then the other. This will bring you to a different state than if you had done these rotations in the opposite order.

[Anonymous No.10945995]

>>10945932

[Anonymous No.10946025]

>>10945932
You could add the 0,1 representation of the bools.

[Anonymous No.10946926]

>>10945971
I like this answer