Infinite resolution

We may reasonably claim that the next integer after $4$ is $5$. What is the next real number after $\pi$?
The notion of consecutive numbers doesn't quite make sense for $\mathbb{R}$. The real number line has a sort of "infinite resolution": for any real numbers $a < b$, there is a real number $c$ such that $a < c < b$.
This is a third issue.
Fix a base $\beta \geq 2$ and integers $p$, $m$, and $M$. The floating point system (FPS) $\mathbb{F}(\beta, p, m, M)$ is the set of all combinations of the form \begin{align*} \pm \bigg(\sum_{n=1}^{p} d_n \beta^{-n}\bigg)\times \beta^e, \end{align*} with each digit $d_n$ in $\{0, 1, \ldots, \beta -1\}$, and some integer exponent $m \leq e \leq M$.
Elements of an FPS are called floats.
In view of the decimal expansion theorem, the FPS $\mathbb{F}(\beta, p, m, M)$ is the set of all elements of the form \begin{align} \pm (0.d_1d_2\ldots d_{p})_{\beta}\times \beta^e, \end{align} with each digit $d_n$ in $\{0, 1, \ldots, \beta -1\}$, and an integer power $m \leq e \leq M$.
By convention, and to ensure uniqueness of representation, we always assume $d_1 \neq 0$ for any nonzero float.

Why floating?

In any FPS the point floats, literally.

Consider the following elements of an FPS with base $\beta = 10$ and precision $p=4$: \begin{align*} 0.1234 \times 10^{{\color{var(--emphColor)} -}{\color{var(--emphColor)} 2}} =& 0{\color{var(--emphColor)} .}001234 \\ 0.1234 \times 10^{{\color{var(--emphColor)} 1}} =& 1{\color{var(--emphColor)} .}234 \\ 0.1234 \times 10^{{\color{var(--emphColor)} 4}} =& 1234{\color{var(--emphColor)} .} \\ 0.1234 \times 10^{{\color{var(--emphColor)} 8}} =& 12,340,000{\color{var(--emphColor)} .} \end{align*}

Changing the exponent $e$ moves the decimal point: it floats to where we need it!

Let's take a closer look...

\begin{align} \pm \bigg(\sum_{n=1}^{p} d_n {\color{var(--emphColor)} \beta}^{-n}\bigg)\times {\color{var(--emphColor)} \beta}^e = \pm (0.d_1d_2\ldots d_{p})_{{\color{var(--emphColor)} \beta}}\times {\color{var(--emphColor)} \beta}^e \end{align}

The base $\beta$ controls the number of digits available: each $d_n$ is taken from the set $\{0, 1, \ldots, \beta - 1\}$.

Let's take a closer look...

\begin{align*} \pm \bigg(\sum_{n=1}^{{\color{var(--emphColor)} p}} d_n \beta^{-n}\bigg)\times \beta^e = \pm (0.d_1d_2\ldots d_{{\color{var(--emphColor)} p}})_{\beta}\times \beta^e \end{align*}

The precision $p$ determines expressiveness of the system; the maximum length of any digit string.

Let's take a closer look...

\begin{align*} \pm \bigg(\sum_{n=1}^{p} d_n \beta^{-n}\bigg)\times \beta^{\color{var(--emphColor)} e} = \pm (0.d_1d_2\ldots d_{p})_{\beta}\times \beta^{\color{var(--emphColor)} e} \end{align*}

The bounds $m$ and $M$ on the integer exponent $e$ determine limits of the system, since $m \leq {\color{var(--emphColor)} e} \leq M$.

Properties of $\mathbb{F}(\beta, p, m, {\color{var(--emphColor)} M})$

In contrast to the reals, every FPS is finite and therefore bounded. Every FPS has a largest element.

In particular, \begin{align*} \bigg|\pm\bigg(\sum_{n=1}^{p} d_n \beta^{-n}\bigg)\times \beta^e\bigg| \leq 0.(\beta-1)\cdots(\beta-1)_{\beta}\times\beta^{\color{var(--emphColor)} M}. \end{align*}

Properties of $\mathbb{F}(\beta, p, {\color{var(--emphColor)} m}, M)$

Perhaps more surprisingly, each FPS has a smallest nonzero element (in magnitude).

In particular, \begin{align*} 0.1_{\beta} \times \beta^{{\color{var(--emphColor)} m}} \leq \bigg|\pm \bigg(\sum_{n=1}^{p} d_n \beta^{-n}\bigg)\times \beta^e\bigg| \end{align*}