Hypercomplex Numbers

Part of the Undergraduate Texts in Mathematics book series (UTM)


This chapter is the story of a generalization with an unexpected outcome. In trying to generalize the concept of real number to n dimensions, we find only four dimensions where the idea works: n = 1, 2, 4, 8. “Numberlike” behavior in ℝn, far from being common, is a rare and interesting exception. Our idea of “numberlike” behavior is motivated by the cases n = 1, 2 that we already know: the real numbers ℝ and the complex numbers ℂ. The number systems ℝ and ℂ have both algebraic and geometric properties in common. The common algebraic property is that of being a field, and it is captured by nine laws governing addition and multiplication, such as ab = ba and a(bc) = (ab)c (commutative and associative laws for multiplication). The common geometric property is the existence of an absolute value, |u|, which measures the distance of u from O and is multiplicative: |uv| = |u||v|. In the 1830s and 1840s, Hamilton and Graves searched long and hard for “numberlike” behavior in ℝ n , but they came up short. Beyond ℝ and ℂ, only two hypercomplex number systems even come close: for n = 4 the quaternion algebra ℍ, which has all the required properties except commutative multiplication, and for n = 8 the octonion algebra \(\mathbb{O}\), which has all the required properties except commutative and associative multiplication. Despite lacking some of the field properties, ℍ and \(\mathbb{O}\) can serve as coordinates for projective planes. In this setting, the missing field properties have a remarkable geometric meaning. Failure of the commutative law corresponds to failure of the Pappus theorem, and failure of the associative law corresponds to failure of the Desargues theorem.


Projective Plane Multiplicative Property Commutative Multiplication Octonion Algebra Hypercomplex Number 
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© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of San FranciscoSan FranciscoUSA

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