Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Everywhere that I've looked, it seems to be assumed that $i^{2} = j^{2} = k^{2} = - 1$, along with the other rules of quaternion multiplication. However - for my homework - I'm being asked to show these rules are valid. Can someone point me in the right direction for going about doing this?

I've been given are the values of $I, J, K$ on the standard basis of $R^{4}$ (i.e. $I(e_1)=e_2)$. Using these rules I see how to show that it's a group and how to complete the rest of the question - I just don't know how to prove that $i^{2}$, etc. are calculated.


share|cite|improve this question
What definition of the quaternions are you working with? Ordinarily I would take $i^2=j^2=k^2=ijk=-1$ as the definition and prove that the resulting vector space has division ring structure. – user7530 Jan 21 '13 at 1:00
What is the exact definition of quaternions that you are given? – Shaun Ault Jan 21 '13 at 1:00
Can you use your rules to show that $I(I(e_i)) = -e_i$ for every $i$? – user7530 Jan 21 '13 at 1:24
That's a fantastic idea! It looks like it'll help with all of the multiplication. Thanks for your help! – Luke8ball Jan 21 '13 at 1:32

Hint: The method suggested by @user7530 in the comments is a good one. Here's another way to go. With this definition the quaternions are matrices. Find the matrix representations of $i$, $j$, and $k$ and simply multiply the matrices to find the required result, keeping in mind that the matrix representation of $1$ is the unit matrix.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.