9.Floating-point_Number_Systems.md

Floating-point Number Systems

A finite floating point number system is a finite subset of $R$.

Floating-point number system is defined by the four natural numbers: 

• $b\ge 2$, the radix, (2 or 10)
• $p\ge 1$, the precision (the number of of digits in the significand)
• $e_{min}$, the largest possible exponent
• $e_{max}$, the smallest possible exponent, (shall be  for all formats)

Notation:

$F(b,p, e_{min}, e_{max})$

The set $F(b,p, e_{min}, e_{max})$ of real numbers represented by this system consists of all floating-point numbers of the form:

$s \in \{ 0, 1 \} $, $d_i \in \{ 0, ... , b - 1 \}$ for all $i$, $e \in \{ e_{min}, ... , e_{max} \}$

represented in radix $b$:

$\pm d_0.d_1 ... d_{p-1} \times b^e$.

Representation of the decimal number $0.1$ with $b = 10$:

$1.0 \times 10^{-1}, 0.1 \times 10^0, 0.01 \times 10^1, ...$

Different representation due to choise of exponent