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OtB_BlueBerry

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

Parker

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

OtB_BlueBerry

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

• $\mathbb{H}$ is a ring
• (1) $\mathbb{H}$ is an abelian group under addition
• 0 is the additive identity
• Every element has an additive inverse
• (2) $\mathbb{H}$ is a monoid under multiplication
• Multiplication is noncommutative, e.g. $ij=k\neq -k=ji$