Hypercomplex Numbers

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 ℝ

n

, 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.