There are a number of notations and ways to think about quaternions:
We usually denote quaternions as entities with the form:
a + i b + j c + k d
Where a,b,c and d are scalar values and i,j and k are 'imaginary operators' which define how the scalar values combine. This is how we introduced them on this page.
|We can see that the above notation is a superset of complex numbers with two additional imaginary values.
We can think of quaternions as an element consisting of a scalar number together with a 3 dimensional vector. In other words we have combined the 3 imaginary values into a vector. We could denote it like this:
|As the product of two independent complex planes.
|As a special case of a clifford algebra
|As a division of vectors
|'Euler Parameters' which are just quaternions but with a different notation. It is shown as four numbers separated by commas instead of the usual notation with the imaginary parts denoted with i, j and k.
Euler Parameters tends to be used in older textbooks, I don't think its used much these days.
When we are using quaternions to represent rotations in 3 dimensions, then we restrict the quaternions to unit length and only use the multiplication operator, in this case there are other notations and ways to think about quaternions:
|As a quantity similar to axis-angle except that real part is equal to cos(angle/2) and the complex part is made up of the axis vector times sin(angle/2).
|As a 2x2 matrix whose elements are complex numbers, generated by Pauli matrices.
|As the equivalent of a unit radius sphere in 4 dimensions.
|As a spinor in 3 dimensions.
|These are all equivalent and in group theory are represented by the group SU(2).
|The group generated by H = <a,b | a² = b² = (ab)²>
|Even when normalised, there is still some redundancy when used for 3D rotations, in that the quaternions a + i b + j c + k d represents the same rotation as -a - i b - j c - k d. At least it does in classical mechanics. However in quantum mechanics a + i b + j c + k d and -a - i b - j c - k d represent different spins for particles, so a particle has to rotate through 720° instead of 360° to get back where it started.