In modern mathematical language, quaternions form a four-dimensional associative normed division algebra over the real numbers, and therefore also a domain. In fact, the quaternions were the first noncommutative division algebra to be discovered. The algebra of quaternions is often denoted by H (for Hamilton), or in blackboard bold by H{\displaystyle \mathbb {H} } (Unicode U+210D, ℍ). It can also be given by the Clifford algebra classifications Cl0,2(ℝ) ≅ Cl+3,0(ℝ). The algebra ℍ holds a special place in analysis since, according to the Frobenius theorem, it is one of only two finite-dimensional division rings containing the real numbers as a proper subring, the other being the complex numbers. These rings are also Euclidean Hurwitz algebras, of which quaternions are the largest associative algebra. Further extending the quaternions yields the non-associative octonions, which is the last normed division algebra over the reals (the extension of the octonions, sedenions, has zero divisors and so cannot be a normed division algebra).

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