# Quaternions And Complex Numbers

1. Sep 10, 2004

### Kambiz_Veshgini

1.
Are the HAMILTON‘ian unit vectors i, j, k still valid beside the imaginary
unit i(Sqrt(-1))?
Can we expand quaternions using complex numbers?

2.
Is the quaternion a+bi+0j+0k equal to the complex number a+bi ?

2. Sep 10, 2004

### matt grime

1. what does valid mean. yes the quarternions can be realized as en extension of the complex numbers, though as i doesn't commute with j or k, there are several ways of doing this and different sources may adopt different ways.

3. Yes and no. a+bi+0j+0k=a+bi IN the quartenions.

3. Sep 13, 2004

### AntonB

Hi Kambiz,
one picture/representation of quaternions (i,j,k)
you can have is of them being traceless hermitian 2*2 matrices
over complex numbers.

(Then exponentiating combinations of them, you generate 2*2 unitary matrices, which we can map to ordinary rotations in 3 dimensions - in fact, I believe, it was Hamilton's obsession with `adding rotations' (in the manner that one might add vectors so effortlessly) that led him to write down the quaternionic algebra in the first place.)

A common basis for this 2*2 complex matrix representation of
quaternions is given by the Pauli matrices, used extensively in physics!

This is the lowest dimension representation of the quaternionic
algebra [sometimes called the spinor representation].

best, Anton.

Last edited: Sep 13, 2004