Machine number

A machine number is a number that is stored in the computer in binary form. A distinction is made between integers ( whole numbers ) and floating point numbers or real numbers ( real numbers ).

Due to the representation in a specifically specified storage format, a finite number of numbers is created. Signed integers with a length of n bits in two's complement can map the numbers from to , unsigned from 0 to . With a length of 8 bits (one byte ), the numbers from −128 to +127 can be stored with a sign, and from 0 to 255 without a sign. ${\displaystyle -2^{(n-1)}}$ ${\displaystyle +2^{(n-1)}-1}$ ${\displaystyle 2^{n}-1}$

Real numbers are stored as a combination of mantissa * exponent . Because of this, rounding errors occur if the result does not accidentally hit exactly one machine number again. How these numbers can be saved is partially standardized - e.g. B. IEEE 754 .

The reduction image

The reduction mapping maps an (exact) real number onto the computer representation . ${\displaystyle fl:\mathbb {R} \rightarrow \mathbb {R} }$ ${\displaystyle x}$ ${\displaystyle fl(x)}$

In principle, every real number can be represented uniquely in a power series representation for the base : ${\displaystyle x\in \mathbb {R} \setminus \{0\}}$ ${\displaystyle b\in \mathbb {N} \setminus \{1\}}$

${\displaystyle x=\pm \left(\sum _{j=1}^{\infty }d_{j}b^{-j}\right)b^{e}}$,

whereby the following restrictions (for uniqueness) should apply:

${\displaystyle {\begin{aligned}d_{j}&\in \mathbb {N} _{0}\quad {\text{mit}}\quad {\begin{cases}1\leq d_{1}<b\\0\leq d_{j}<b,\;j=2,3,\ldots \end{cases}}\\e&\in \mathbb {Z} .\end{aligned}}}$

In a computer display, mantissas of unlimited length cannot be stored, but only a finite mantissa with the mantissa length . There is also a restricted interval for the exponent : with . ${\displaystyle d_{j},j\in \mathbb {N} }$ ${\displaystyle d_{j},j=1,2,\ldots ,m}$ ${\displaystyle m\in \mathbb {N} }$ ${\displaystyle -r\leq e\leq R}$ ${\displaystyle r,R\in \mathbb {N} }$

The reduction map is defined as follows: ${\displaystyle fl\colon \mathbb {R} \rightarrow \mathbb {R} }$

${\displaystyle fl(x):=\pm \left\{{\begin{matrix}\left(\sum _{j=1}^{m}d_{j}b^{-j}\right)\cdot b^{e}&{\mbox{falls}}\quad d_{m+1}<{\frac {b}{2}}\\\left(\sum _{j=1}^{m}d_{j}b^{-j}+b^{-m}\right)\cdot b^{e}&{\mbox{falls}}\quad d_{m+1}\geq {\frac {b}{2}}.\end{matrix}}\right.}$

Remarks

The number can also be accurately represented by: . ${\displaystyle x=0}$ ${\displaystyle fl(0):=0}$

For with an exponent that is not in the interval , the reduction map is defined as follows: ${\displaystyle x\in \mathbb {R} ,x\neq 0}$ ${\displaystyle e}$ ${\displaystyle [-r,R]}$

${\displaystyle {\begin{aligned}fl(x):={\begin{cases}0,&\quad e<-r\\\operatorname {sgn}(x)\cdot \infty ,&\quad e>R.\end{cases}}\end{aligned}}}$

It is important to note here that is neither injective nor surjective . Namely, many numbers are represented with the same machine number (therefore not injective) or the picture of the function is only a finite set (therefore not surjective). ${\displaystyle fl}$ ${\displaystyle x}$

Another frequently used name for the reduction figure is: . ${\displaystyle fl(x)=rd(x)}$

They can be used to determine the absolute rounding error when coding a number .

See also

• the accuracy of the representation using machine numbers describes the machine accuracy
• Optimization of the number of machines for technical and economic processes

Web links

• Teaching material from Rolf Mantyk
• Film for a lecture by Frank Loose