Quaternions - Meaning and multiplication

In summary, quaternions are a base for associative division algebras, and their rules are explained by their use in 3-dimensional rotation.
  • #1
david.aloha
14
0
All right, I've been doing a lot of reading on quaternions, and while I think I understand how to use them, I'm still very confused as to why certain things are defined the way that they are. First question:
Why, when multiplying the imaginary portions of quaternions do you get these "rules":
i*i = j*j = k*k = -1
i*j = k, j*i = -k
j*k = i, k*j = -i
k*i = j, i*k = -j

I understand why i*i = -1, but why does an imaginary number times an imaginary number sometimes equal the negative of another imaginary number? I can't seem to find any underlying reason for this in anything I've read - it's just stated.

Second question:
Why use imaginary numbers at all if you're just defining a 3-vector plus a scalar (at least in terms of its use for 3D rotation which is the main context I've viewed it in)? What is complex or imaginary about it? The imaginary parts each define an axis in the same way as x, y, and z in a 3-vector - I'm just struggling to find what's different or special about adding the imaginary numbers (which, as I said above, also confuse me by only sometimes acting like imaginary numbers in multiplication).

I've spent a few days now looking through many different guides to quaternions, and none of them seem to answer these questions. Any help or clarification would be greatly appreciated. I'm the kind of guy who needs to know WHY something works, not just HOW it works, and this lack of why is driving me nuts.
 
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  • #2
We can define multiplication pretty much however we want to get an algebra. That particular definition is nice, because multiplication is associative and every nonzero element has an inverse.

My limited knowledge of history is that quaternions inspired the invention of the dot and cross products.
 
  • #3
Hurkyl said:
We can define multiplication pretty much however we want to get an algebra. That particular definition is nice, because multiplication is associative and every nonzero element has an inverse.
Which possibly makes it sound more arbitrary than it might otherwise be regarded - see also http://en.wikipedia.org/wiki/Frobenius_theorem_(real_division_algebras).
 
  • #4
While large parts of the link you gave were a bit beyond me, I think I understand the significance behind why Quaternions are they way they are as a sort of base for "associative division algebras" (along with real numbers and complex numbers). I was reading this letter from Hamilton on how he arrived at Quaternions and I'm seeing a little better the process behind how he came to those rules:
http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/QLetter/QLetter.pdf

I haven't read the entire letter yet (takes a long time when I'm really trying to formulate all of what he's saying) but I think I have a better understanding now than I did previously. Thanks for the responses.
 
  • #5


I can provide some insight into your questions about quaternions and their multiplication.

Firstly, the rules for multiplication of imaginary portions in quaternions are defined in a way that ensures consistency and allows for easier mathematical manipulation. This is similar to how we define the rules for multiplication of real numbers, with the commutative and distributive properties. In the case of quaternions, the rules you mentioned are a result of the algebraic properties of the imaginary units i, j, and k. These units are not arbitrary, but rather have specific properties that allow for the formation of a 4-dimensional algebraic system. The negative values in the multiplication rules are a result of the anti-commutative property of quaternions, where the order of multiplication matters.

To understand why imaginary numbers sometimes result in the negative of another imaginary number, it may be helpful to think of quaternions as a combination of two complex numbers. The first two components, a and b, represent the real and imaginary parts of a complex number, while the last two components, c and d, represent another complex number. When multiplied together, the real and imaginary parts of the first complex number combine with the real and imaginary parts of the second complex number, resulting in the four components of the quaternion. This is why the rules for multiplication may seem different from traditional complex numbers, as we are now dealing with two complex numbers instead of just one.

Secondly, the use of imaginary numbers in quaternions allows for a more efficient representation of 3D rotations. As you mentioned, the imaginary parts of quaternions define axes in a similar way as x, y, and z in a 3-vector. However, by adding the imaginary numbers, we are able to incorporate the concept of rotation in a more compact and elegant way. This is because quaternions are a 4-dimensional system, allowing for a more efficient representation of 3D rotations compared to traditional 3D vector rotations.

I understand your frustration with wanting to know the underlying reasons for the rules and concepts in quaternions. As a scientist, it is important to have a deep understanding of the principles behind our tools and methods. However, in this case, the rules and concepts of quaternions are a result of mathematical properties and principles that may not have a clear physical interpretation. Nonetheless, quaternions have proven to be a useful tool in many fields, including 3D computer graphics
 

1. What are quaternions?

Quaternions are a type of mathematical concept that extends the system of complex numbers to four dimensions. They are often used in computer graphics and 3D animation to represent rotations.

2. How are quaternions different from complex numbers?

Quaternions have four components (a, b, c, d) compared to the two components (a, bi) of complex numbers. Quaternions also have different multiplication rules, as they are non-commutative and non-associative.

3. What is the meaning of quaternion multiplication?

Quaternion multiplication is a way of combining two quaternions to produce a third quaternion. It is similar to multiplying complex numbers, where the product of two quaternions is a combination of their components and the imaginary unit i, j, and k.

4. How are quaternions used in computer graphics?

Quaternions are used in computer graphics to represent rotations in three-dimensional space. They are preferred over other rotation representations, such as Euler angles, because they avoid the problem of gimbal lock and provide smooth interpolation between rotations.

5. Can quaternions be used for other applications besides computer graphics?

Yes, quaternions have a wide range of applications in physics, engineering, and mathematics. They are used in quantum mechanics, signal processing, and even in robotic control systems.

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