Enter two binary numbers and an operation, or convert a single value between binary and decimal.
Convert a single value between the two number systems.
Binary addition follows the same place-value carrying logic as decimal addition, but with only two digits available (0 and 1) — 1 + 1 = 10 in binary (write 0, carry 1). Adding 1010 (10 in decimal) + 0110 (6) column by column with carries gives 10000 (16). This tool handles the carrying for you by converting each number to decimal, doing the arithmetic, and converting the result back to binary.
Each digit in a binary number represents a power of 2 based on its position, counting from the right starting at 2^0. Multiply each digit by its positional power of 2 and add them up: binary 1010 = 1×2³ + 0×2² + 1×2¹ + 0×2⁰ = 8 + 0 + 2 + 0 = 10 in decimal.
Two's complement is the scheme computers actually use internally to represent negative binary numbers, using a fixed bit width so that subtraction can reuse the same circuitry as addition. It's a bit more involved to read by eye than a plain sign, so this tool shows negative results in a simpler sign-magnitude style instead — a minus sign followed by the binary magnitude of the result — which is easier to follow for learning purposes, even though it isn't how hardware stores negative values internally.
Binary (base 2) underlies essentially all of digital computing. Every value a computer stores or processes — numbers, text, images, sound — ultimately reduces to sequences of 0s and 1s at the hardware level, because a transistor naturally represents two stable states, off and on. Understanding binary arithmetic is foundational to computer science, digital logic design, and low-level programming.
Worked example: adding binary 1010 (10) and 110 (6): 10 + 6 = 16, which is 10000 in binary. Subtracting the smaller from the larger the other way — 110 (6) minus 1010 (10) — gives 6 − 10 = −4, shown as −100 (the binary magnitude of 4, with a leading minus sign).