1. ## [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?

2. Originally Posted by ronaldo_07
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.

3. Done it thanks