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In my university I started learning quaternions and don't quite understand them. Could I, please, get some help ?

Donatas Barkauskas
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  • What do you know what quaternions? Can you state the definition and tell us some basic properties? –  Jan 24 '18 at 03:03
  • Well as I know it's sort of a set of numbers simillar to complex numbers, but with more elements. I do not study quaternions in English, so it is pretty hard for me to explain, considering I'm just a beginner. – Donatas Barkauskas Jan 24 '18 at 03:08
  • I would suggest you ask yourself the same question in $\mathbb R$: try to find all real numbers so that $q^2=1$. Then think if the same arguments can be genearalized to the quaternion. –  Jan 24 '18 at 03:20
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    So basically, I should consider 1 and -1 as the real roots and then determine whether they are possible arguments to the quaternion? – Donatas Barkauskas Jan 24 '18 at 03:33
  • The square of the quaternion $t+xi+yj+zk$ is $(t^2-x^2-y^2-z^2)+2txi+2tyj+2tyk$. How can this equal $1$? – Angina Seng Jan 24 '18 at 03:58
  • @DonatasBarkauskas Sort of, but what you said contains a rather dangerous line of thought. "Roots" do not work nicely for noncommutative division rings. But you can still look at the expression $(q-1)(q+1)=0$ and draw conclusions multiplying inverses on left and right (if inverses exist for the $q$ you chose :) ) – rschwieb Jan 24 '18 at 16:04

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Hint:

$q=\pm 1$ are obviously solutions, so let's attempt to consider $q\neq \pm 1$ with that property.

We have $0=q^2-1=(q-1)(q+1)$. If $q\neq 1$, then $q-1\neq 0$, hence it has an inverse, call it $q'$. Multiply both sides of the equation on the left with $q'$. What results?

rschwieb
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