Definition:Floating-Point Representation
Definition
Floating-point representation is a technique for representing a real number $x$ in a given number base $\beta$ by presenting it uniquely in the form:
- $x = f \times \beta^e$
where:
- $f$ is a real number such that $\dfrac 1 \beta \le \size f < 1$, expressed in decimal notation
- $e$ is an integer.
Base
The number $\beta$ is known as the base of the floating-point representation.
Mantissa
The real number $f$ is known as the mantissa of the floating-point representation.
Exponent
The integer $e$ is known as the exponent of the floating-point representation.
Computer Implementation
Implementation of floating-point representation of a real number:
- $x = f \times \beta^e$
is subject to the following considerations.
Both the mantissa $f$ and the exponent $e$ have a limited range:
In such a system there is a finite set of numbers which can be so implemented, and $x$ can be written as:
| \(\ds x\) | \(=\) | \(\ds \pm \sqbrk {0.f_1 f_2 \ldots f_t} \times \beta^e\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \pm \paren {\dfrac {f_1} \beta + \dfrac {f_2} {\beta^2} + \cdots + \dfrac {f_t} {\beta^t} } \times \beta^e\) |
where each digit $f_i$ satisfies $0 \le f_i \le \beta - 1$.
Examples
Example: $105.7$
$105.7$ is represented in floating-point representation as:
- $0.1057 \times 10^3$
Also see
- Results about floating-point representation can be found here.
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): floating-point representation
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): floating-point representation