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}

0^* & :\alpha = 0^* \\ \left\{{pq: p \in \alpha, \, p > 0, \, q \in \beta}\right\} & :\alpha > 0^*, \, \beta \ge 0^* \\ -\left({\alpha \times \left({-\beta}\right)}\right) & :\alpha > 0^*, \, \beta < 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.

Strictly Positive Real Numbers
Let $\left({\left({\R, \le}\right), \phi}\right)$ be the Dedekind completion of the ordered set $\left({\Q, \le}\right)$ of rational numbers.

We have that $\left({\Q, +, \le}\right)$ is an Archimedean totally ordered group.

By Bernoulli's Inequality, we have that:
 * $\forall q \in \Q_{>0}: q^n \ge 1 + n \left({q - 1}\right)$

Thus $\left({\Q_{>0}, \times, \le}\right)$ is an Archimedean totally ordered group.

We have that $\left({\R_{>0}, \le}\right)$ is Dedekind complete.

Let $\phi_+$ be the restriction of $\phi$ to $\Q_{>0} \times \R_{>0}$.

It is clear that the image $\phi_+ \left({\Q_{>0}}\right)$ is infimum-dense in $\left({\R_{>0}, \le}\right)$.

We have that, for all $x \in \R_{>0}$:

Thus $\phi_+ \left({\Q_{>0}}\right)$ is supremum-dense in $\left({\R_{>0}, \le}\right)$.

Hence, $\left({\left({\R_{>0}, \le}\right), \phi_+}\right)$ is the Dedekind completion of the ordered set $\left({\Q_{>0}, \le}\right)$.

By this theorem, there exists a unique binary operation $\times$ on $\R_{>0}$ such that:
 * $({1}): \quad \left({\R_{>0}, \times, \le}\right)$ is an ordered group
 * $({2}): \quad \phi_+$ is a group homomorphism from $\left({\Q_{>0}, \times}\right)$ to $\left({\R_{>0}, \times}\right)$

This binary operation $\times$ is called multiplication.

Notation
As is standard, we write $xy$ in place of $x \times y$.