4 ms·

> Quaternions are hypercomplex numbers of the form I'm gonna guess "hypercomplex" means "involves imaginary numbers" due to the rest of this. > Where w, x, y,

> Quaternions are hypercomplex numbers of the form I'm gonna guess "hypercomplex" means "involves imaginary numbers" due to the rest of this. > Where w, x, y, and z are real and i^2 = j^2 = k^2 = -1 and ij = k, ji = -k, jk = i, kj = -i, ki = j, ik = -j. Why even use different letters for i, j, and k, if they're all the same thing? Which thing appears to be i, as in, the square root of -1, that is to say, the most well-known and basic imaginary number, if I'm reading this right. As for the second part, I can't even begin to unravel the significance. I know it must not be, but it just seems like arbitrary rules added for... some unknown reason. > Being u = (x, y, z) = xi + yj + zk a unitary vector, it is possible rotate any vector q by an angle theta around u by doing: > pqp' > where p = cos(theta/2) + sin(theta/2)u and p' = cos(theta/2) - sin(theta/2)u . Oooooh kay... 1) Where'd w go? Is this one of those things where there's a (situationally-defined) constant in the formula but we just pretend it doesn't exist most of the time (until it comes time to actually use the math to, like, do anything real)? Would we need to bring it back in to apply the rest of this? 2) "u =" is just defining something, fine, but (x, y, z) doesn't seem to equal the thing after it at all—I suppose this is a shorthand function notation, though it seems really weird to me to use equality to relate that. Am I right, or is this something else?, 3) A quaternion is... a point, then? Since we're rotating around it? 4) I've got a feeling that theta needs a direction in this hypercomplex space but don't see where it's coming from. Somewhere "off screen", in this explanation? Or is it there but I'm not seeing it?

marcodiego 5y ago
>> Quaternions are hypercomplex numbers of the form > >I'm gonna guess "hypercomplex" means "involves imaginary numbers" due to the rest of this. Think it is right. >> >> Where w, x, y, and z are real and i^2 = j^2 = k^2 = -1 and ij = k, ji = -k, jk = i, kj = -i, ki = j, ik = -j. > >Why even use different letters for i, j, and k, if they're all the same thing? Which thing appears to be i, as in, the square root of -1, that is to say, the most well-known and basic imaginary number, if I'm reading this right. They are not the same thing. All of these are equal to -1 when squared, but they are different when multiplied by any other of this set and multiplication between them is anti-commutative. >As for the second part, I can't even begin to unravel the significance. I know it must not be, but it just seems like arbitrary rules added for... some unknown reason. > These are not added for unknown reasons. They specify rotations of unitary vectors of a canonical base around one another. >> Being u = (x, y, z) = xi + yj + zk a unitary vector, it is possible rotate any vector q by an angle theta around u by doing: >> >> pqp' >> >> where p = cos(theta/2) + sin(theta/2)u and p' = cos(theta/2) - sin(theta/2)u . >> >Oooooh kay... 1) Where'd w go? The coordinate 'w' is the real part. For the vector 'u', such coordinate is 0. For 'p' and 'p'', it is cos(theta/2). >Is this one of those things where there's a (situationally-defined) constant in the formula but we just pretend it doesn't exist most of the time (until it comes time to actually use the math to, like, do anything real)? No. >Would we need to bring it back in to apply the rest of this? No. Just follow the rules to multiply hypercomplex numbers and the rotation will occur. >2) "u =" is just defining something, fine, but (x, y, z) doesn't seem to equal the thing after it at all—I suppose this is a shorthand function notation, though it seems really weird to me to use equality to relate that. Am I right, or is this something else?, The notation '(x, y, z)' is just a shorter notation for "xi + yj + zk". For this problem, the vector 'u' defines the axis of rotation. >3) A quaternion is... a point, then? Since we're rotating around it? You can see a quaternion as a point in R^4 since it has 4 coordinates. Actually it describes an axis of rotation, by its imaginary part, and the angle of rotation. >4) I've got a feeling that theta needs a direction in this hypercomplex space but don't see where it's coming from. Somewhere "off screen", in this explanation? Or is it there but I'm not seeing it? > The axis of rotation is actually the vector "u".
- handrous 5y ago
  Thank you, this was very helpful. > The axis of rotation is actually the vector "u". Ah, I thought the vector was what we were rotating.
  - marcodiego 5y ago
    I think "p" is better described as a point instead of a vector. So, a better writing could be: "[..]it is possible rotate any point q by an angle theta around the axis define by u [...]"
cardiffspaceman 5y ago
It seems to me that i, j, and k are three things that have the same property, not three identical things.
- handrous 5y ago
  TIL that not every imaginary number is just "i" with optional scaling applied. Was taught "the square root of negative one is i".
  - fault1 5y ago
    i wish i was taught in terms of euler' formula, which is probably the most useful: https://en.wikipedia.org/wiki/Euler%27s_formula https://en.wikipedia.org/wiki/Euler%27s_formula
  - Twisol 5y ago
    That's a heck of a great observation. In the complex numbers, every imaginary number is, indeed, just "i" scaled. But the quaternions are like three copies of the complex numbers glued together; you have multiple kinds of "imaginary", each with their own unit -- like "i", but now also copies of it, "j" and "k". To bring this somewhere more familiar, you can probably imagine the real number line as a physical line stretching off to infinity. This line has a point called "1". Now if we take two more copies of this line, they each have their own point called "1" -- a different one for each line. And if you stick all three of these lines together as a three-dimensional set of axes, you get a world in which you have three 1s coexisting. We just say "in the X direction" to be clear about which we're talking about, or group them together in (x, y, z) triples -- in which case X's 1 is called (1, 0, 0). The situation is the same for quaternions -- we glue one real line together with three copies of the imaginary line. So we get three different i's, and we give them different names to distinguish them.
  - mkl 5y ago
    Unfortunately that's not even correct for complex numbers: i is a square root of -1, and -i is the other. Complex numbers are really nice in that every non-zero complex number has two square roots, three cube roots, four fourth roots, etc. A better definitional statement is the other way around: i has the property that i^2 = -1.
AnimalMuppet 5y ago
No, i, j, and k are not the same thing. They are the three distinct square roots of -1. You thus wind up with a space with one real axis and three orthogonal imaginary axes. Why should -1 have three distinct imaginary roots? Well, why should it have one? Essentially, we just made up i, and we found out that the complex numbers had some really useful algebraic properties. The same is true of the quaternions. But why not two imaginary roots, or four, or 17? Those turn out not to have nice algebraic properties. The only other thing out there are the octonions, with 7 imaginary roots.
- the_only_law 5y ago
  And like that I’ve given up hope of ever having a good understanding of any mathematical field.
  - wrycoder 5y ago
    Try this version, which is expanded somewhat. I found it clearer: https://news.ycombinator.com/item?id=29516191 https://news.ycombinator.com/item?id=29516191
at_compile_time 5y ago
Sounds a lot like the first time I tried to understand quaternions. The explanation in geometric algebra as a scalar and bivector gives it an intuitive geometric sense that words like words like "hypercomplex" fail to. Others here have linked to that material.

marcodiego 5y ago

>> Quaternions are hypercomplex numbers of the form > >I'm gonna guess "hypercomplex" means "involves imaginary numbers" due to the rest of this. Think it is right. >> >> Where w, x, y, and z are real and i^2 = j^2 = k^2 = -1 and ij = k, ji = -k, jk = i, kj = -i, ki = j, ik = -j. > >Why even use different letters for i, j, and k, if they're all the same thing? Which thing appears to be i, as in, the square root of -1, that is to say, the most well-known and basic imaginary number, if I'm reading this right. They are not the same thing. All of these are equal to -1 when squared, but they are different when multiplied by any other of this set and multiplication between them is anti-commutative. >As for the second part, I can't even begin to unravel the significance. I know it must not be, but it just seems like arbitrary rules added for... some unknown reason. > These are not added for unknown reasons. They specify rotations of unitary vectors of a canonical base around one another. >> Being u = (x, y, z) = xi + yj + zk a unitary vector, it is possible rotate any vector q by an angle theta around u by doing: >> >> pqp' >> >> where p = cos(theta/2) + sin(theta/2)u and p' = cos(theta/2) - sin(theta/2)u . >> >Oooooh kay... 1) Where'd w go? The coordinate 'w' is the real part. For the vector 'u', such coordinate is 0. For 'p' and 'p'', it is cos(theta/2). >Is this one of those things where there's a (situationally-defined) constant in the formula but we just pretend it doesn't exist most of the time (until it comes time to actually use the math to, like, do anything real)? No. >Would we need to bring it back in to apply the rest of this? No. Just follow the rules to multiply hypercomplex numbers and the rotation will occur. >2) "u =" is just defining something, fine, but (x, y, z) doesn't seem to equal the thing after it at all—I suppose this is a shorthand function notation, though it seems really weird to me to use equality to relate that. Am I right, or is this something else?, The notation '(x, y, z)' is just a shorter notation for "xi + yj + zk". For this problem, the vector 'u' defines the axis of rotation. >3) A quaternion is... a point, then? Since we're rotating around it? You can see a quaternion as a point in R^4 since it has 4 coordinates. Actually it describes an axis of rotation, by its imaginary part, and the angle of rotation. >4) I've got a feeling that theta needs a direction in this hypercomplex space but don't see where it's coming from. Somewhere "off screen", in this explanation? Or is it there but I'm not seeing it? > The axis of rotation is actually the vector "u".

handrous 5y ago
Thank you, this was very helpful. > The axis of rotation is actually the vector "u". Ah, I thought the vector was what we were rotating.
- marcodiego 5y ago
  I think "p" is better described as a point instead of a vector. So, a better writing could be: "[..]it is possible rotate any point q by an angle theta around the axis define by u [...]"

cardiffspaceman 5y ago

It seems to me that i, j, and k are three things that have the same property, not three identical things.

handrous 5y ago
TIL that not every imaginary number is just "i" with optional scaling applied. Was taught "the square root of negative one is i".
- fault1 5y ago
  i wish i was taught in terms of euler' formula, which is probably the most useful: https://en.wikipedia.org/wiki/Euler%27s_formula https://en.wikipedia.org/wiki/Euler%27s_formula
- Twisol 5y ago
  That's a heck of a great observation. In the complex numbers, every imaginary number is, indeed, just "i" scaled. But the quaternions are like three copies of the complex numbers glued together; you have multiple kinds of "imaginary", each with their own unit -- like "i", but now also copies of it, "j" and "k". To bring this somewhere more familiar, you can probably imagine the real number line as a physical line stretching off to infinity. This line has a point called "1". Now if we take two more copies of this line, they each have their own point called "1" -- a different one for each line. And if you stick all three of these lines together as a three-dimensional set of axes, you get a world in which you have three 1s coexisting. We just say "in the X direction" to be clear about which we're talking about, or group them together in (x, y, z) triples -- in which case X's 1 is called (1, 0, 0). The situation is the same for quaternions -- we glue one real line together with three copies of the imaginary line. So we get three different i's, and we give them different names to distinguish them.
- mkl 5y ago
  Unfortunately that's not even correct for complex numbers: i is a square root of -1, and -i is the other. Complex numbers are really nice in that every non-zero complex number has two square roots, three cube roots, four fourth roots, etc. A better definitional statement is the other way around: i has the property that i^2 = -1.

AnimalMuppet 5y ago

No, i, j, and k are not the same thing. They are the three distinct square roots of -1. You thus wind up with a space with one real axis and three orthogonal imaginary axes. Why should -1 have three distinct imaginary roots? Well, why should it have one? Essentially, we just made up i, and we found out that the complex numbers had some really useful algebraic properties. The same is true of the quaternions. But why not two imaginary roots, or four, or 17? Those turn out not to have nice algebraic properties. The only other thing out there are the octonions, with 7 imaginary roots.

the_only_law 5y ago
And like that I’ve given up hope of ever having a good understanding of any mathematical field.
- wrycoder 5y ago
  Try this version, which is expanded somewhat. I found it clearer: https://news.ycombinator.com/item?id=29516191 https://news.ycombinator.com/item?id=29516191

at_compile_time 5y ago

Sounds a lot like the first time I tried to understand quaternions. The explanation in geometric algebra as a scalar and bivector gives it an intuitive geometric sense that words like words like "hypercomplex" fail to. Others here have linked to that material.