A radix \(\beta\) floating point number \(x\) is of the form
\(m\beta{e}\), where \(|m|<\beta\) is called the **significand**
and \(e\) is the **exponent**.

A floating point number is characterized by four integers:

- A
**radix**(base) \(\beta\ge2\) - A
**precision**\(p\ge2\) - \(e_{\textit{min}}\): The smallest possible exponent
- \(e_{\textit{max}}\): The largest possible exponent

In the IEEE 754, \(e_{\textit{min}}=1-e_{\textit{max}}\)

A finite floating point number \(x\) is such that \(x=M\beta^{e-p+1}\) where:

- \(|M|\le\beta^{p}-1\). It is called the
**integral significand**.- Note that this just means that \(|M|\) can be represented exactly in base \(\beta\) within \(p\) characters.

- \(e_{\textit{min}}\le e\le e_{\textit{max}}\)

The tuple \((M,e)\) need not be unique. The set of equivalent
\((M,e)\) is called a **cohort**.

The number \(\beta^{e-p+1}\) is called the **quantum** of \(x\).

There is another representation:

\begin{equation*}
x=(-1)^{s}m\beta^{e}
\end{equation*}

- \(m=|M|\beta^{1-p}\) and is called the
**normal significand**(or just the**significand**). It has one digit before the radix point.

Let \(x\) be any number. The **infinitely precise significand** of
\(x\) in \(\beta\) is the number:

\begin{equation*}
\frac{x}{\beta^{\lfloor\log_{\beta}|x|\rfloor}}
\end{equation*}