Free Binary Calculator

The binary calculator is a free tool that performs arithmetic, base conversion, and bitwise operations on binary numbers. It shows step-by-step work for every calculation, making it useful for computer science courses, programming, and understanding how computers process data at the lowest level.

Binary (base-2) is the language of computers. Every number, character, and instruction a computer processes is ultimately represented as a sequence of 0s and 1s. Understanding binary arithmetic and bitwise operations is fundamental to fields like systems programming, networking, embedded development, and digital electronics.

Arithmetic Mode: Enter two binary numbers and choose an operation (add, subtract, multiply, or divide). The result appears instantly in binary with decimal, octal, and hexadecimal equivalents. Expand the step-by-step section to see carries, partial products, or long division work.

Converter Mode: Type a number in any base (binary, decimal, octal, or hexadecimal) and see it converted to all four formats simultaneously. Use the quick presets for common values like 255, 1024, or 65535. Step-by-step explanations show positional values and grouping methods.

Bitwise Mode: Enter binary values and apply logical operations (AND, OR, XOR, NOT) or bit shifts. The visual bit display color-codes each bit position, making it easy to see how operations affect individual bits.

Bit Width: Toggle between 8-bit, 16-bit, and 32-bit modes. Enable the signed toggle to interpret results as two's complement signed integers.

Full binary arithmetic. Add, subtract, multiply, and divide binary numbers. Subtraction uses two's complement, multiplication shows partial products, and division shows long-division steps. Every operation includes the complete working process.

Multi-base conversion. Convert between binary, decimal, octal, and hexadecimal instantly. The converter shows how positional notation works, how to group bits for hex (4-bit nibbles) and octal (3-bit groups), and the repeated-division method for decimal-to-binary conversion.

Bitwise operations with visualization. Perform AND, OR, XOR, NOT, left shift, and right shift. Each bit is displayed as a colored box with its positional value, so you can see exactly which bits change during an operation.

Export results. Download your calculation as a CSV file or copy the full result with steps as JSON for use in documentation, homework, or development workflows.

The binary number system uses only two digits: 0 and 1. Each position represents a power of 2, just as each position in decimal represents a power of 10. The rightmost bit (least significant bit) is 2⁰ = 1, the next is 2¹ = 2, then 2² = 4, 2³ = 8, and so on. For example, binary 1101 = 8 + 4 + 0 + 1 = 13 in decimal.

Computers use binary because digital circuits have two states: on (1) and off (0). This simplicity makes hardware reliable and fast. A single binary digit is called a bit. Eight bits form a byte, which can represent 256 different values (0 to 255). Modern processors work with 32-bit or 64-bit numbers, enabling values up to 4 billion or 18 quintillion respectively.