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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

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    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
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