Numeral systems

Positional numeral system

The value of the number (integer, or rational that has terminating representation in the base $b$) whose base-$b$ representation is $\pm (\underset{\text{Integer Part}}{\underbrace{a_{k+1}\dots a_2{a}_1{a}_0}}.\underset{\text{Fractional Part}}{\underbrace{a_{-1}{a}_{-2}\dots a_{-l}}}{)}_b$ is given by: $n=\pm (a_{k+1}\times b^{k+1}+\cdots +a_0\times b^0)+(a_{-1}\times b^{-1}+a_{-2}\times b^{-2}+\cdots +a_{-l}\times b^{-l})$ where:

• $b$ is the basis (or radix)

• $D=\{d_1,d_2,\dots ,d_b\}$ is the set of symbols ($\forall i:a_i\in D$)

• $k=\lceil {\log }_{b}n\rceil$ is the number of digits to the left of the radix point (the integer part)

• $l$ is the number of digits to the right of the radix point (the fractional part)

• todo irrational numbers and rational numbers with non-terminating representation

Conversion

Decimal to any base

Converting the integral part

// integer n in base 10 to base b
while n>0
	divide n by b to get quotient and remainder;
	append remainder to the left of the result;
	n=quotient;

Converting the fractional part

// fractional part n in base 10 to base b
while n>0
	multiply n by b to get integer part and fractional part;
	append integer part to the right of the result;
	n=fractional part;

// note: the fractional part may never become zero, stop when the desired precision is reached

Binary–hexadecimal conversion

4 binary digits can be represented by 1 hexadecimal digit

Binary–octal conversion

3 binary digits can be represented by 1 octal digit

Octal–hexadecimal conversion

we can convert octal to binary and then binary to hexadecimal or vice versa