My understanding of quaternions begins with the 'Brougham Bridge Equations" i.e.

where i, j, and k are all 'independent' square roots of -1.

I seem to be unable to follow the logic of this statement. Taking 'independent' to mean the three symbols i,j,k are not equal to each other, I then stated the following:

Dividing both sides by i, I get

From this, we can infer that 'i' is greater than j or k.

However,

Dividing both sides by j, we get

This implies, then, that j is greater than i or k.

Repeating the same, this time with k^2,

But this then implies that all three values are greater than each other, i.e

which is logically impossible.

Have I grasped the wrong end of the stick somewhere? Where have I gone wrong?

Thanks for any help!