Binary number

A binary number is a number represented in base two.

After adding $k$ least-significant binary digits, the numerical sum of the two processed input suffixes equals the written $k$ result digits plus $carry{2}^k$, with $carry\in \{0,1\}$. Initially both suffixes are empty. For the next column, write $(x_k+y_k+carry)\mathrm{mod}2$ and pass $\lfloor (x_k+y_k+carry)/2\rfloor$ to the next column. Multiplying these column quantities by $2^k$ preserves the equation. After all columns, the remaining carry supplies the extra leading digit, proving correctness without converting either full input to a machine integer.

For an $n$-bit binary string $A=a_{n-1}\cdots a_0$, the unsigned interpretation assigns positive positional weights $2^i$ to its bits. Its values range from zero to $2^n-1$. The same stored bits have two's complement value $\mathrm{bin}(A)-a_{n-1}{2}^n$, so interpreting a machine word is a separate choice from storing its bits.

The number of binary digits in the usual representation of a positive integer $n$: $\lfloor {\log }_{2}n\rfloor +1$. The bit length of zero is conventionally zero in integer programming interfaces.

Every real number in $[0,1]$ is a sum ${\sum }_{j\ge 1}{\epsilon }_j{2}^{-j}$ with binary digits ${\epsilon }_j\in \{0,1\}$.

A base-two positional fraction with a binary point separating nonnegative and negative powers of two. For example, $10.101_2=2+1/2+1/8=21/8$. A finite expansion is a dyadic rational: with $F$ fractional places, its value is an integer divided by $2^F$. Conversely, any nonnegative reduced fraction whose denominator is a power of two has a finite binary expansion. Leading integer zeroes and trailing fractional zeroes do not change its value.

A representation $n={\sum }_{i\ge 0}{a}_i{2}^i$ with $a_i\in \{0,1,2\}$ and only finitely many nonzero coefficients. If $h(n)$ counts these representations, then $h(0)=1$, $h(2m+1)=h(m)$, and $h(2m)=h(m)+h(m-1)$ for $m\ge 1$. The integer parity of $n$ determines whether the coefficient of $2^0$ must be 1 or may be 0 or 2.

A dyadic rational has the form $m/2^k$. Its terminating binary expansion has an alternative expansion ending in infinitely many $1$ digits.