# 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 + 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.