Point (decimal point, floating point)

A term pertaining to the representation of a real number by a fraction, and to the way real numbers are represented in digital computers.

Consider a number system with radix (base) $q$, in which the representation of a real number $x$ is

\begin{equation}x=\sum_{k=-\infty}^n\alpha_kq^k,\label{1}\end{equation}

where the $\alpha_k$ are integers between 0 and $q-1$ (inclusive). In the representation of $x$ by a $q$-ary fraction

\begin{equation}x=(\alpha_n\dots\alpha_1\alpha_0.\alpha_{-1}\alpha_{-2}\dots)\label{2}\end{equation}

the point (sometimes referred to in this context as a $q$-ary point) separates the coefficients in \eqref{1} into those relating to non-negative powers of $q$ and those relating to negative powers.

Depending on the mode of representation of real numbers, digital computers may be divided into fixed-point and floating-point devices.

Fixed-point arithmetic assumes that all numbers have a modulus less than 1. A fixed number of digits is set aside to store the coefficients $\alpha_{-1},\alpha_{-2},\dots$. If an operation involving numbers with fixed point produces a number with modulus greater than 1, the program run is interrupted and an overflow signal is produced. To avoid this situation, the programmer must check in advance for possible overflow and prevent it by suitable scaling. An example of a computer with fixed-point arithmetic was the "Setun", which worked with the ternary number system. The difficulties of programming for fixed-point arithmetic make it clear why most modern electronic computers use floating-point arithmetic. A number in floating-point notation is written as

\[x=\pm q^p\sum_{k=1}^t\alpha_kq^{-k},\]

where $p$ is known as the order (or exponent) and $(\alpha_1\dots\alpha_n)$ the mantissa of $x$. To store the order and mantissa of a floating-point number, one usually sets aside a fixed number of digits (defined by the length of a machine word), which imposes limits on the order. A floating-point number with $\alpha_1\neq0$ is known as a normalized number. The results of arithmetic operations in a computer with floating-point arithmetic are usually automatically normalized by the arithmetic unit of the computer.

Comments

For other representations see Numbers, representations of. For an extensive discussion including arithmetic operations in floating-point arithmetic see [a1].

References

[a1]	D.M. Young, R.T. Gregory, "A survey of numerical mathematics" , Dover, reprint (1988) pp. 19–55 Zbl 0732.65002