Number and algebra and their dual

For the first post in this series, see here.

(1) Set theory and logic, (2) number and algebra, and (3) space and time are three foundational topics that have dual approaches. Let us begin with the standard approaches to these three topics, and then define duals to each of them. In some ways, the original and the dual may be used together.

(2) Number and algebra

The basic rules of algebra are as follows: addition and multiplication are commutative and associative; multiplication distributes over addition; addition and multiplication have identities and inverses with one exception: there is no multiplicative inverse for zero.

An idea of infinity comes from taking the limit of a number as its value approaches zero: ∞ ∼ 1/x as x → 0. Infinity can be partially incorporated via limits.

Dual: reciprocal numbers

An additive dual can be defined by negating every number. A more interesting dual comes from taking the multiplicative dual of every number. This latter case can be called reciprocal numbers with reciprocal arithmetic  in which operations take place in the denominator.

The reciprocal isomorphism relates every number x in series to its reciprocal dual by H(x) := 1/y. The dual of zero is ∞.

Reciprocal algebra is the multiplicative inverse of ordinary algebra. There is a sense in which reciprocal arithmetic counts down rather than up. Zero in reciprocal numbers is like infinity in ordinary numbers. Larger reciprocal numbers correspond to smaller ordinary numbers. Smaller reciprocal numbers correspond to larger ordinary numbers.