# Digital technology and semiconductor circuit design

## Number representationWebApp

Number systems serve to display numbers by means of suitable digits and their systematic arrangement. In polyadic number systems, the valency of a symbol depends on its position. The composition of such a system can be represented formally with $$N = \sum\limits_{i=0}^n a_i \cdot b^i = a_n \cdot b^n + \dots + a_1 \cdot b^1 + a_0 \cdot b^0$$ whereby $$b$$ is the basis and $$a_i$$ are the digits of the displayed number $$N$$.

### 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.

The divisions' rests from bottom to top result in the number to be displayed:

### 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.

The sum is

### 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).

Number in 2's complement representation:

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 .

Die Summe ergibt:

### 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.

Reading from top to bottom, the decimal places of the binary representation are shown:
The conversion of the negative number occurs analogously to the 2's complement.