 More complicated numbers systems with generalizations of this type of multiplication in 4 and 8 dimensions can be defined. Kervaire (1958) and Bott &amp; Milnor (1958) independently proved that the only four division algebras built on the reals are \(\mathbb{R}\), \(\mathbb{C}\), \(\mathbb{H}\) and \(\mathbb{O}\), so the table gives a comprehensive view of all possible algebra’s that define a notion of extensiveness. For each of the number classes in the table a separate theory of information measurement, based on the properties of multiplication, can be developed. For the countable classes \(\mathbb{N}\), \(\mathbb{Z}\) and \(\mathbb{Q}\) these theories ware equivalent to the standard concept of information implied by the notion of Turing equivalence. Up to the real numbers these theories satisfy our intuitive notions of extensiveness of information. For complex numbers the notion of information efficiency of multiplication is destroyed. The quaternions lack the property of commutativity and the octonions that of associativity. These models are not just abstract constructions since the algebras play an important role in our descriptions of nature: