Binary Calculator

A binary calculator works with numbers written in base two, the positional number system used by digital computers. Instead of ten decimal digits, binary notation uses only zero and one. Each position represents a power of two: the rightmost bit represents one, then two, four, eight, sixteen, and so on. For example, 1010 equals eight plus two, or ten in decimal. This tool keeps the input and primary result in binary while also displaying a decimal interpretation for verification. Arithmetic operations follow familiar rules adapted to base two. Addition carries when a column reaches two. Subtraction borrows one power of two from the next column. Multiplication forms shifted partial products because multiplying by a bit either contributes zero or an unchanged copy of the first operand. Integer division repeatedly compares and subtracts shifted divisors, producing a quotient and remainder. The calculator performs unsigned integer arithmetic and displays the complete result without imposing a fixed machine word size. Bitwise operations treat corresponding digits as logical values. AND produces one only when both bits are one. OR produces one when either or both bits are one. XOR produces one when the bits differ. If the inputs have different lengths, their leading sides are effectively padded with zeros. These operations are essential for masks, permissions, graphics, network protocols, embedded systems, cryptography, and efficient manipulation of packed data. The arithmetic and bitwise controls intentionally share the same inputs so related results are easy to compare. Given 1010 and 0011, addition returns 1101, AND returns 0010 without insignificant leading zeros, OR returns 1011, and XOR returns 1001. Multiplication returns 11110. Integer division of 1010 by 11 returns quotient 11 with remainder 1, corresponding to decimal ten divided by three. Subtraction in this calculator requires the first number to be at least as large as the second. That choice avoids silently assuming a signed representation. Negative binary numbers normally use a fixed-width convention such as two's complement, and the same bit pattern can mean different values at different widths. Likewise, division requires a nonzero divisor. The remainder is always nonnegative and smaller than the divisor. Exact arbitrary-length integer operations avoid floating-point rounding for large inputs, although extremely long strings can still be impractical for a browser interface. Leading zeros are accepted and do not affect numerical value. Fractional binary points are not supported. When modeling real hardware, select a word width separately and consider overflow or truncation, because processors often retain only a fixed number of low-order bits.