Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Quaternions And Complex Numbers

  1. Sep 10, 2004 #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?

    Is the quaternion a+bi+0j+0k equal to the complex number a+bi ?
  2. jcsd
  3. Sep 10, 2004 #2

    matt grime

    User Avatar
    Science Advisor
    Homework Helper

    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.
  4. Sep 13, 2004 #3
    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
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook