User:Abcxyz/Sandbox/Real Numbers/Definition:Real Multiplication

Multiplication/Axiomatic Definition
Let $\left({\R, +, \times, \le}\right)$ denote the real numbers, as axiomatically defined as a Dedekind complete totally ordered field.

The binary operation $\times$ is called multiplication.

Multiplication/Construction from Cauchy Sequences
Let $\R$ denote the set of real numbers, as constructed from Cauchy sequences.

Multiplication, denoted $\times$, is the binary operation on $\R$ defined as:
 * $\left[{\!\left[{\left\langle{x_n}\right\rangle}\right]\!}\right] \times \left[{\!\left[{\left\langle{y_n}\right\rangle}\right]\!}\right] = \left[{\!\left[{\left\langle{x_n \times y_n}\right\rangle}\right]\!}\right]$

Multiplication/Construction from Dedekind Cuts
Let $\R$ denote the set of real numbers, as constructed from Dedekind cuts.

From Identity for Real Addition, we have that $\left({\R, +}\right)$ has an identity $0^*$.

Let $\le$ denote the ordering on $\R$.

We have that $\le$ is a total ordering.

Multiplication, denoted $\times$, is the binary operation on $\R$ defined as:
 * $\displaystyle \alpha \times \beta = \begin{cases}

\left\{{pq: p \in \alpha, \, p > 0, \, q \in \beta}\right\} & :\alpha > 0^* \\ 0^* & :\alpha = 0^* \\ -\left({\left({-\alpha}\right) \times \beta}\right) & :\alpha < 0^* \end{cases}$ where $-\alpha$ denotes the inverse of $\alpha$ for $+$.

The existence of such inverses is proved in Inverses for Real Addition.