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.
Decimal place values are powers of 10; binary place values are powers of 2. The default values on this page, 1010 and 110, break down as:
Every operation on this page runs the same way under the hood: convert both binary strings to the decimal numbers they represent, do ordinary arithmetic, then convert the answer back. 10 + 6 = 16, and 16 in binary is 10000 — which is what this calculator returns for its own default inputs.
Doing this by hand instead follows the same column rules as decimal arithmetic, just with a carry that triggers at 2 instead of 10: 1+1=10 (write 0, carry 1), exactly like 9+1 carries in decimal. It is slower to do by hand precisely because there are only two digits to work with, so more columns are needed to represent the same value — ten in decimal is one digit, 1010 in binary is four.
A transistor is reliably read as one of two states — on or off, high voltage or low — and building hardware that reliably distinguishes ten distinct voltage levels for base-10 arithmetic is far harder than building one that distinguishes two. Base 2 is not a stylistic choice; it is the number system that matches what the physical hardware can actually do without error, and every other representation in computing (decimal displays, hexadecimal in web colours, binary in a network protocol) is binary underneath, formatted differently for whoever is reading it.
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).
Working with other machine representations? The Hex Calculator covers base 16, the Base64 Converter turns binary data into text-safe characters, and the IP Subnet Calculator is bitwise masking applied to network addresses. For a non-positional system entirely, see the Roman Numeral Converter.