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Quaternions form a (noncommutative) division ring (aka skew field). Fields are commutative division rings.


BlueBerry, would you mind explaining that in more detail? Specifically, why do quaternions form a noncummatative ring?


Quaternions ($\mathbb{H}$) is a noncommutative division ring

  • $\mathbb{H}$ is a ring
    • (1) $\mathbb{H}$ is an abelian group under addition
    • Addition is associative
    • 0 is the additive identity
    • Every element has an additive inverse
    • Addition is commutative
    • (2) $\mathbb{H}$ is a monoid under multiplication
    • Multiplication is associative
    • 1 is the multiplicative identity
    • (3) Multiplication is (both left and right) distributive over addition
  • Every nonzero element has a multiplicative inverse (hence "division" ring)
  • Multiplication is noncommutative, e.g. $ij=k\neq -k=ji$

The quaternions can be viewed as a 4D vector space over its center, which is the real numbers (a field). This vector space equipped with quaternion multiplication can then be classified as an associative division algebra (over the reals).


I think another way to interpret this is that since i,j,k encode rotation in different directions and the order of rotation matters in 3D, it makes sense that multiplication doesn't commute.