Kind of extends the complex numbers.
Some facts that make them stand out:
- one of the only three real associative division algebras in addition to the real numbers and complex numbers, according to the classification of associative real division algebras
- the simplest non-commutative division algebra. Contrast for example with complex numbers where multiplication is commutative
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A quaternion is with real coefficients, where . They form a noncommutative real division algebra. The norm squared is , and every nonzero quaternion has inverse its conjugate divided by that squared norm. The unit quaternions form the Lie group .