# Prove that quaternions are associative

1. Sep 30, 2007

### proton

1. The problem statement, all variables and given/known data
prove that quaternions are associative. ie (qr)s = q(rs), where q,r,s are quaternions

This isnt really a HW problem since I'm just wondering if theres a simpler way to prove associativity than the method I tried below

2. Relevant equations
i^2 = j^2 = k^2 = -1
ij=k=-ji
jk=i=-kj
ki=j=-ik
q=(a,b,c,d), r=(e,f,g,h), s=(l,m,n,p)

3. The attempt at a solution

Its not so much I dont know how to do the problem as it is that my work is tedious and will take forever, unless theres a simpler method

heres my work:
qr = (a,b,c,d)(e,f,g,h) = ae + af*i^2 + ag *ij + ah*k + be*i + bf*i^2 + bg*ij + bh*ik + ce*j + cf*ji + cg *j^2 + ch*jk + de*k + df*k + dg*jk + dh*k^2 = (ae-bf-cg-dh, af+be+ch-dg, ag-bh+ce+df, ah+bg=cf+de)

(qr)s = (ae-bf-cg-dh, af+be+ch-dg, ag-bh+ce+df, ah+bg=cf+de)(l,m,n,p) =
(lae-lbf-lcg-ldh-maf-mbe-mch+mdg-nag+nbh-nce-ndf-pah-pbg+pcf-pde, ..........)
I got too tired to work out the i,j,k components

Theres got to be a simpler way to prove associativity

2. Sep 30, 2007

### HallsofIvy

Staff Emeritus
There may be but there doesn't have to be. "Associativity" is always the most tedious property to prove! You are going to have to learn not to get so tired so quickly.

3. Sep 30, 2007

### proton

it took me a FULL page to prove (qr)s =q(rs)

Last edited: Sep 30, 2007
4. Oct 1, 2007

### mcampbell

If one were to show quaternions could be represented as a matrix and the product defined as standard matrix multiplication, would associativity follow as a consequence of the fact that a matrix represents a linear transformation and the matrix product is functional compostion?

Last edited: Oct 1, 2007
5. Oct 1, 2007

### Hurkyl

Staff Emeritus
If you start with an associative algebra and then modding out by some relations... then the result is automatically an associative algebra.

6. Oct 1, 2007

### meopemuk

You can first prove the associativity for any triple of 4 basis quaternions 1,i,j,k, and then check that the same property holds for any real linear combination of them.

Eugene.