From the decimal system to the basis \(b\)

For the conversion of a number \(N=\) from the decimal system into the system with the basis of \(b=\) , the number must be divided by the basis of \(b\) for multiple times.

Back to the decimal system

For the conversion of a number \(N_b=\) from the number system with the basis \(b=\) into the decimal system, the coefficients have to be multiplied and added up with the respective potency.

Two's Complement Representation

For the represenation of negative numbers, without a sign's help, various formats can be applied. (see WebApp about negative numbers) In computer centers, usually the two's complement is used, since it can be realized very simply. Formally, a number in the 2's complement can be written as follows: $$ N = a_{n-1} \dots a_1 a_0 = - a_{n-1} \cdot 2^{n-1} + \sum\limits_{i=0}^{n-2} a_i \cdot 2^i $$ Usually, the number of Bits (n = ) used and the available value range \([-2^{n-1}, 2^{n-1}-1]\) i.e. \(\) is determined. The total amount of a number \(N=\) is firstly converted into its binary represenation (1111011). The 2's complement is obtained by inverting all bits and then adding an LSB (least significant bit).

The backwards calculation is done according to the above-equation through summing up the potencies \(a_{i}\cdot b^i\) für \(i=0,\dots,n-2\) and (if applicable) subtraction of the highest potency \(\).

Fixed point format (pure fractional numbers)

A number \(x\) with \(|x| \leq 1\) can be written according to the following formula in the binary system: $$ x = a_{n-1}\bullet a_{n-2}\dots a_1 a_0 \\ = -a_{n-1}\cdot 2^0 + \sum\limits_{i=1}^{n-1}a_{n-1-i}\cdot 2^{-i} $$ Typically the number of bits \(n = \) and hence the resolution is fixed. In that way, numbers can be mapped in the range analogous to the 2's complement \([-1, 1-2^{-(n-1)}]\) i.e. \([-1, 0.875]\). The conversion of a number \(x = \) is done by a repeated multiplication by 2.