# Maths - Quaternion Notations - Superset of complex numbers

A complex number may be expressed as the sum of a real and imaginary part as follows:

a + i b

A quaternion adds two additional and independent imaginary parts as follows:

a + i b + j c + k d

So this adds two extra dimensions which square to a negative number, giving a total of:

• One dimension which squares to a positive number (real part)
• Three dimensions which square to a negative number (3 imaginary parts)

## Representing Rotations

It does superficially look like quaternions extend the way that complex numbers represent rotations, but I don't think quaternion rotation is an extension the way complex numbers represent rotations, they are completely different. I think it is just a coincidence that they both happen to represent rotations. (if it is valid to use the word 'coincidence' in mathematics). For instance:

• The two dimensions in complex numbers (real and imaginary) can represent coordinates of the objects being rotated. The four dimensions of quaternions have no direct relationship to the 3 dimensions of the objects being rotated.
• In complex numbers rotation is done using complex exponent, in quaternions its done using the 'sandwich' multiplication.
• In complex numbers 'i' represents 90 degree rotation, in quaternions 'i' represents 180 degree rotation.

 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. Quaternions and Rotation Sequences. Specific to this page here:

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