[SOLVED] Proof

**ronaldo_07** · January 18th 2009, 06:30 PM

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

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

(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?

**ThePerfectHacker** · January 18th 2009, 07:10 PM

Originally Posted by ronaldo_07

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

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

(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, $ijk = -1 \implies i^2jk = - i \implies jk=i$ .
But then, $jk^2 = ik \implies ik = -j$ .
Also, $jk = i \implies j^2 k = ji \implies ji = -k$ .
Now, $ijk = -1 \implies ijk^2 = -k \implies ij = k$ .
Continue from here to get the other identities: $kj = -i, ki = j$
We see from here $ji \not = ij$ .
Thus, they are not commutative.

**ronaldo_07** · January 18th 2009, 07:46 PM

Done it thanks

Thread: [SOLVED] Proof

LinkBack

Thread Tools

Display

[SOLVED] Proof

Similar Math Help Forum Discussions

[SOLVED] direct proof and proof by contradiction

[SOLVED] Please help with proof

[SOLVED] set proof help

[SOLVED] Proof GCD and LCM

[SOLVED] proof

Search Tags