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Thread: [SOLVED] Proof

  1. #1
    Member ronaldo_07's Avatar
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    [SOLVED] Proof

    The quaternions are expressions of the form $\displaystyle a+bi+cj+dk$, where $\displaystyle a;b;c;d$ are
    real numbers. They are added by the “obvious rule”, and multiplication is based on the formulae:

    $\displaystyle i^2 = j^2 = k^2 = ijk = -1$:

    (a) Prove that the associative law for multiplication holds.
    (b) Prove that the distributive law holds.
    (c) Prove that the commutative law for multiplication does not hold.[/tex]

    How would I prove these?
    Last edited by ronaldo_07; Jan 18th 2009 at 06:52 PM.
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  2. #2
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    Quote Originally Posted by ronaldo_07 View Post
    The quaternions are expressions of the form $\displaystyle a+bi+cj+dk$, where $\displaystyle a;b;c;d$ are
    real numbers. They are added by the “obvious rule”, and multiplication is based on the formulae:

    $\displaystyle i^2 = j^2 = k^2 = ijk = -1$:

    (a) Prove that the associative law for multiplication holds.
    (b) Prove that the distributive law holds.
    (c) Prove that the commutative law for multiplication does not hold.[/tex]

    How would I prove these?
    I help you start, $\displaystyle ijk = -1 \implies i^2jk = - i \implies jk=i$.
    But then, $\displaystyle jk^2 = ik \implies ik = -j $.
    Also, $\displaystyle jk = i \implies j^2 k = ji \implies ji = -k$.
    Now, $\displaystyle ijk = -1 \implies ijk^2 = -k \implies ij = k$.
    Continue from here to get the other identities: $\displaystyle kj = -i, ki = j$
    We see from here $\displaystyle ji \not = ij$.
    Thus, they are not commutative.
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  3. #3
    Member ronaldo_07's Avatar
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    Done it thanks
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