Maths - Quaternion Notations - Product of two complex planes

There is an alternative way to think of quaternions, imagine a complex number:

n1 + i n2

but this time make n1 and n2 (the real and imaginary parts) to be themselves complex numbers (but with a different imaginary part at right angles to the first), so,

If we substitute them into the first complex number this gives,

(a + jc) + i (b + jd)

since i*j = k (see under multiplication) this can be rearranged to give the same form as above.

a + i b + jc + kd



There are a number of notations and ways to think about quaternions:

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:

These are all equivalent and in group theory are represented by the group SU(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 degrees instead of 360 degrees to get back where it started.

metadata block
see also:


Correspondence about this page

Book Shop - Further reading.

Where I can, I have put links to Amazon for books that are relevant to the subject, click on the appropriate country flag to get more details of the book or to buy it from them.


cover us uk de jp fr ca Quaternions and Rotation Sequences.

Terminology and Notation

Specific to this page here:


This site may have errors. Don't use for critical systems.

Copyright (c) 1998-2017 Martin John Baker - All rights reserved - privacy policy.