#Quaternions are a single number
kind herald
yes i see your point
kind herald
no
harsh atlas
Like it's 4 values added together
Tell me is 2 + 3 one number or two numbers
(2+3)
kind herald
a quaternion is a 4x4 matrix
kind herald
it is a matrix
harsh atlas
Aren't they literally just a+bi+cj+dk
kind herald
it's a bit more complex than that
harsh atlas
It can't be 16 numbers whuh
kind herald
huh
harsh atlas
Ok I admit I know barely anything about quaternions
harsh atlas
A 4x4 matrix consists of 16 values no
kind herald
add it up
harsh atlas
Ye it's a single number
kind herald
how much is 2x + 2y?
kind herald
2 + 2 is a single number, because it's a 4
harsh atlas
Also x and y are variables
kind herald
but you have two components in that number
harsh atlas
i j and k are not variables
They're constants (?)
Or are you talking about a b c and d
In that case I get ur point
kind herald
yes
kind herald
a single quaternion cannot be 4, or 8, or 20
harsh atlas
Much like x + y + z is one number with 3 variables
harsh atlas
a single quaternion cannot be 4, or 8, or 20
Those are real numbers
kind herald
harsh atlas
a single quaternion cannot be 4, or 8, or 20
Actually yeah, a quaternion can be all these numbers
harsh atlas
is a vector a single number
Nope, cuz they're vectors
kind herald
do you know how do you write down a vector?
harsh atlas
Quaternions are not vectors
harsh atlas
do you know how do you write down a vector?
(x, y, z) etc
You write down a quaternion as a+bi+cj+dk as I said
I think
Like I said, a quaternion can be 4 or 8 or 20
kind herald
just like a quaternion
you can write them down multiple ways
one of the ways you can write them down is 1i + 2j +3k if they are 3D
harsh atlas
Crap I can't open khan academy website
harsh atlas
i j k
kind herald
see the connection?
harsh atlas
That's a quaternion with a being 0
Which is a number
Wait I think the little
Uhh
^ things
Make it not a quarternion?
kind herald
well yes, quaternions are just translations of vectors
kind herald
that w (the first number) is always sin(angle) + cos(angle)
harsh atlas
?
kind herald
and the other three numbers, or the vector, is the axis
kind herald
exactly
so are vectors
and matrices
well actually, my bad, they aren't systems exactly
harsh atlas
Vectors and matrices are completely different tho?
From quaternions and numbers
They're not number systems
kind herald
they are not number systems
harsh atlas
Quaternions are just like natural numbers and whole numbers and real numbers
kind herald
no they are not
harsh atlas
Yes they are
They are an extension
Of complex numbers, which are the extension of the real numbers
kind herald
harsh atlas
Show
harsh atlas
Quaternions are numbers
If speaking purely mathematically
And that's what I've been doing, I'm talking about pure mathematics
In compsci they're a bunch of different stuff
It's just funky numbers which are very cool
I should learn more about them
kind herald
nvm that article just said "quaternions are numbers" as a buzzword typa thing
kind herald
also, I wouldn't agree with you, event mathematically
because
no
they don't "help"
harsh atlas
they can be used to describe rotations in space
harsh atlas
But what they are is numbers
kind herald
no
harsh atlas
Like unity?
kind herald
"They have the ability to uniquely describe any three-dimensional rotation about an arbitrary axis and do not suffer from gimbal lock."
so the number 90
harsh atlas
That's their ability
kind herald
what rotation is that
kind herald
they are exactly like vectors, which describe a point or direction in a space
kind herald
No clue, convert from quaternions to euler angles
and you get a vector
tiny plume
are we talking about abstract maths? i love abstract maths
tiny plume
yes, they are
kind herald
like, 2190.0121 is a quaternion
harsh atlas
They are
harsh atlas
like, 2190.0121 is a quaternion
It is
kind herald
i need explanations
harsh atlas
Ok so
tiny plume
perhaps not in the most literal sense, since we do encode a complex number as two real numbers (and a quaternion as four real numbers)
tiny plume
think about how a rational number is a ratio of two integers
it has two numbers in it, I guess, but it's still a number
harsh atlas
2190.0121 + 0i + 0j + 0k
tiny plume
24/64 is a rational number
kind herald
you cannot "make" or compute should i say, a quaternion from a single number
kind herald
2190.0121 + 0i + 0j + 0k
exactly
but it can also be
2190.0121 + 0.1i + 0.1j + 0.1k
kind herald
`24/64` is a rational number
because you can calculate it
kind herald
it has two numbers in it, I guess, but it's still a **number**
it's a number system
tiny plume
no, a complex number is not a "number system"
harsh atlas
I think you misunderstand what a number is
tiny plume
it is an element in a number system
harsh atlas
Numbers go beyond just real numbers
tiny plume
"A number is a mathematical object used to count, measure, and label.", as per wikipedia's first paragraph on number systems
kind herald
no, a complex number is not a "number system"
yes it is
"In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted i, called the imaginary unit and satisfying the equation i^2 = -1"
is has two parts
a real part and an imaginary part
2/8 is not a number system, therefore is a number, because it has only one real part
tiny plume
3blue1brown has some very nice videos talking about quaternions (and their applications for rotations)
kind herald
3blue1brown has some very nice videos talking about quaternions (and their appli...
yes, I did already link it here
tiny plume
yes it is
"In mathematics, a complex number is an element of a number system tha...
you are using these terms completely incorrectly
harsh atlas
Also 90 is a rotation of 0 according to the converter I found on the internet
tiny plume
2/8 is, indeed, not a number system, because it's a number
kind herald
Also 90 is a rotation of 0 according to the converter I found on the internet
can you share
tiny plume
this has nothing to do with it only having one part, or only a real part
harsh atlas
tiny plume
a number is an element of a number system
harsh atlas
A number system is like
tiny plume
1 is a number because it is an element of the integers
harsh atlas
Whole numbers and Real Numbers
kind herald
2 + 1i is not a number, it's a number system, it belongs on the complex plane and was "misjudged" as an "imaginary" number for simplicities sake
tiny plume
2 + 1i is not a number, it's a number system, it belongs on the complex plane an...
again, this is objectively wrong
the complex numbers are a number system
harsh atlas
Natural numbers, Whole numbers, Rational numbers, Real numbers, Complex numbers, Quaternion numbers
tiny plume
2 + 1i is a number in the complex number system
harsh atlas
From smallest to biggest up to wuaternions
I think I didn't miss anything at least 😓
harsh atlas
Natural numbers, Whole numbers, Rational numbers, Real numbers, Complex numbers,...
Those are number systems!
kind herald
Also 90 is a rotation of 0 according to the converter I found on the internet
90 as the real part of the quaternion is indeed (0, 0, 0, 0)
but 90 is not the quaternion
tiny plume
3 is a natural number, integer, rational number, real number, complex number, and quaternion number
it satisfies the rules for all of these number systems
kind herald
okay but then so are vectors? right?
tiny plume
indeed
perhaps the confusion comes from how you tend to store complex numbers as a Vector2 and quaternions as a Vector4
this is just an implementation detail
harsh atlas
Vectors are just a collection of numbers
tiny plume
it is the most convenient way to implement them on a computer
harsh atlas
Which is helpful when you wanna store a quarternion since it has 4 variables
No
Quaternions are numbers
tiny plume
quaternions are not numbers either
quaternions are not vectors
harsh atlas
You can conveniently store it in a vector but it is not one
kind herald
how many components/numbers does it have
tiny plume
the distinction between "a complex number" and "a vector of two real numbers" is, indeed, down to semantics
kind herald
how many components/numbers does it have
harsh atlas
It's a vector representing a quaternion
Just representing
What this actually is is
0i + 0j + 0k + 1
You can just store the values in a vector for convenience
Which is what programmers do
I imagine it would be pretty hard to store a quarternion in anything but a vector for programmers
tiny plume
this is nonsense
harsh atlas
Also it literally says it's a number!
tiny plume
i was hoping to have an interesting discussion in here, but i'm obviously just being trolled
kind herald
what
i didn't get a single source that confirms your information and you are the one getting trolled 💀
i had to resort to ai, yes, it's funny
harsh atlas
"a quaternion cannot be represented as a single number"
"a quaternion is a number.."
It's dropping the "real"
And even then it's wrong
A quarternion is a 4 dimensional number
This is what it's right about
It's a number
In 4 dimensions
x y z and w
kind herald
so is a vector
it's a number too then
because it represents a singular point in those dimensions
harsh atlas
No for god's sakes!
"In mathematics, the quaternion number system extends the complex numbers."
A vector is a bunch of numbers
A quarternion is one number with a bunch of variables
A vector is not one number
A quarternion is one number
One can be represented with another, but just represented
harsh atlas
See? A number
harsh atlas
i used wolfram alpha to compute a quaternion
You can't "compute it" because it's not real
You're misunderstanding the meaning of "number"
Numbers go beyond real numbers
harsh atlas
The norm is just the geometrical mean pretty sure
kind herald
as in, it does not have to be written as a function but a single number
harsh atlas
It just squares all the components, adds them up and sqrts them pretty sure
Well yes
kind herald
except it does
the 0 before i
the 2 before j
and the 5 before k
those are variables, because they vary
so is the 2
harsh atlas
So does 2x + 3y
kind herald
exactly
harsh atlas
It's still one number except that one has variables
kind herald
and what is that?
harsh atlas
those are variables, because they vary
No?
You literally put them in
It'd be a variable if it was w + xi + yj + zk
But then you put in 2 instead of w, 1 instead of x, -2 instead of y and 5 instead of z
It's now not a variable because you put them in
It became a constant
And i j and k are not variables
harsh atlas
2 + i - 2j + 5k is one number
This has no variables
kind herald
they are also constants yea
harsh atlas
It's a single number
kind herald
but a quaternion is a single rotation along an axis by a certain amount of degrees
meaning, it consists of a vector
and a real number
harsh atlas
A quarternion is a 4 dimensional number that can represent rotation
kind herald
well not really a vector but yea
harsh atlas
It represents rotation
It's not rotation
It just represents and has the ability to do so
Just like complex numbers have the ability to represent alternating current
But you wouldn't call 2 + 4i "just alternating current" or wtv
That's applied mathematics
In pure mathematics which is again, what I'm talking about
A quarternion is one 4 dimensional number
harsh atlas
Just a bunch of numbers
kind herald
WELL
harsh atlas
a vector is 3 1d numbers?
Yeah
harsh atlas
It's multiple numbers
harsh atlas
Yes
kind herald
like an array
kind herald
but a quaternion "computes" it
kind herald
well
harsh atlas
Wdym computes it
harsh atlas
A vector is an array a matrix is an array of arrays of sorts
harsh atlas
Quaternion? Yeah
harsh atlas
Stage 1, denial
harsh atlas
Good luck
harsh atlas
ill continue searching about it
You'll probably find just the applications of quaternions tho
There are a lot of them
Very useful numbers
Google "quaternions in pure mathematics" you should find something
Good luck
kind herald
if you are still here
did we say, complex numbers are numbers? or did we say something else
harsh atlas
did we say, complex numbers are numbers? or did we say something else
They are
Quaternions are an extension of them
I mean they are literally called complex numbers
kind herald
a quaternion rotates a 3D point in space in regards to the origin
mathematically that means that it rotates a 3D direction (since a direction is already sort of defined by a origin, or usually, it's considered to be (0, 0))
meaning that you need the quaternion form to output a 3D point, which is a vector, which we said is not a number
meaning that a quaternion mathematically results in a vector of size of 3
a quaternion cannot output/result in a single number because that would mean you can get a vector (or compute a vector) from a single number
expanding on the wikipedia article (yea i know, not the greatest source):
https://en.wikipedia.org/wiki/Quaternion#:~:text=A quaternion is an expression of the form
In the picture below, i,** j** and k are unit vectors
Unit vectors: https://en.wikipedia.org/wiki/Unit_vector#:~:text=In mathematics%2C a,pronounced "v-hat"
Furthermore
Hamilton describes a quaternion q = a + bi + cj + dk as consisting of a scalar part and a vector part where the latter is sometimes called as the imaginary part.
This part right here took A LONG LONG TIME
"More properly, the field of real numbers is isomorphic to a subset of the quaternions. The field of complex numbers is also isomorphic to three subsets of quaternions."
The field of real numbers being the a, b, c and d
i was writting all of this
for nothing
"More properly, the field of real numbers is isomorphic to a subset of the quaternions. The field of complex numbers is also isomorphic to three subsets of quaternions"
which means
harsh atlas
that's how it can be represented and used
kind herald
" the field of real numbers is isomorphic to a subset of the quaternions, what we mean is that there is a one-to-one correspondence between the two systems. Specifically, every real number can be represented as a quaternion of the form a + 0i + 0j + 0k. This means that the real numbers and a subset of the quaternions have the same structure, even though the individual elements are different."
which means
harsh atlas
it can be used to rotate a point
harsh atlas
they always are
kind herald
this was a ride
kind herald
i mean sure
harsh atlas
yea
harsh atlas
but real numbers are also quaternions
kind herald
they can just be written as a "function"
harsh atlas
just like whole numbers are also rational numbers
harsh atlas
they can just be written as a "function"
yup
kind herald
isomorphism is fun
kind herald
yea thanks
harsh atlas
quaternions are fun little numbers
kind herald
no clue what that even means lmao
basically
imagine two systems
and a function in them
that returns the same output, respective to those systems
"In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping."
basically, "equal" things
kind herald
holy shiiit
more stuff
a vector can also be represented as a quaternion
which is so counterintuitive, but WHAT DOES IT MEAN
harsh atlas
a vector can also be represented as a quaternion
oh ye i said that already
it's very funny
kind herald
does that mean that a quaternion is both a single scalar value and a vector at the same time?
because, saying that it can be represented as both is ambigous, so shouldn't quaternions just be quaternions? just like vectors are vectors or arrays, single numbers are scalars or values? can't quaternions be classified more correctly as rotations or instructions of some sort?
harsh atlas
does that mean that a quaternion is both a single scalar value and a vector at t...
no they're numbers
they have components and you can put these components into a vector
or smth my brain is not working anymore
in pure maths theyre nothing but numbers
kind herald
but a quaternion can equal its vector part
"Hamilton describes a quaternion q = a + bi + cj + dk as consisting of a scalar part and a vector part where the latter is sometimes called as the imaginary part."
"More properly, the field of real numbers is isomorphic to a subset of the quaternions. The field of complex numbers is also isomorphic to three subsets of quaternions"
harsh atlas
vector part is just the imaginary part
kind herald
subset of the quaternions are isomorphic or equal to the field of real numbers
while three subsets are equal to the field of complex numbers
kind herald
vector part is just the imaginary part
yes, just represented as bi + cj + dk which is still mathematically three unit vectors
harsh atlas
unit vectors?
kind herald
yes
kind herald
i is the "x"
j is the "y"
k is the "z"
in other words i can be written as (1, 0, 0), j as (0, 1, 0) and k as (0, 0, 1)
the numbers before are the "amount" of how much each of those "axis" are influencing the rotation