Binary numeral system ($N={\sum }_j{d}_j{2}^j,d_j\in \{0,1\}$)

The binary numeral system is positional notation with radix two. Each binary digit is zero or one, and its place value is a power of two. A one includes that weight in the sum; a zero skips it. The binary representation $11001101_2$ therefore selects $128,64,8,4,1$ and represents the integer $205$.

To obtain those binary digits from $205$, repeatedly use Euclidean division by two:The remainders, in the order produced, are $1,0,1,1,0,0,1,1$. Read them in reverse order to get $11001101_2$. The final zero is the quotient that stops the conversion, not an additional leading binary digit. Starting with zero itself needs the special single-digit representation $0_2$.

A useful independent check reads the binary digits from left to right. Start with $v=0$ and, for each binary digit $d$, replace $v$ by $2v+d$. For $11001101_2$, the intermediate values are $1,3,6,12,25,51,102,205$. This reconstructs the integer without reversing a list and explains why appending a zero on the right doubles the value.

Binary is well suited to computation because each binary digit has just two possibilities. A stored bit can encode either possibility; groups of bits encode larger integers. Practical integer arithmetic can use a radix that is itself a power of two, grouping several binary digits into each stored numeral digit.^[ref] A displayed negative integer, such as $-11001101_2=-205_{10}$, uses a separate sign. A computer's particular signed storage convention is an additional encoding choice, so the displayed sign should not be confused with an extra binary place.

A binary digit is a zero or one in the base-two positional representation of an integer. The sum $s_2(n)={\sum }_j{b}_j$ determines the Thue–Morse sequence and is the Hamming weight of the binary representation.