Binary Arithmetic Calculator

Binary Arithmetic Calculator

Binary Arithmetic Calculator — Add, Subtract, Multiply, Bitwise Operations

This binary arithmetic calculator performs addition, subtraction, and multiplication on binary numbers, as well as bitwise AND, OR, XOR, and NOT operations. Results are shown in binary, decimal, and hexadecimal. For addition and subtraction, a step-by-step carry/borrow table is displayed so you can follow each bit operation.

Binary Addition

Binary addition follows the same column-by-column process as decimal addition, but with only two digits. The rules are: 0+0=0, 0+1=1, 1+0=1, 1+1=10 (write 0, carry 1). The carry propagates just like tens carry in decimal arithmetic. For example, 1010 + 0110 = 10000 (10 + 6 = 16 in decimal).

Binary Subtraction

Binary subtraction uses borrowing: 0−0=0, 1−0=1, 1−1=0, 0−1 requires a borrow from the next higher bit. In real computers, subtraction is typically performed using two's complement representation: negate a number (flip all bits and add 1), then add — this avoids the need for a separate subtraction circuit.

Bitwise Operations

AND: output bit is 1 only if both input bits are 1. Used to mask and clear bits. OR: output is 1 if either input is 1. Used to set bits. XOR (exclusive OR): output is 1 if inputs differ. Used in encryption, checksums, and toggle operations. NOT: inverts all bits within the bit-width of the input. All these operations are executed directly in hardware in a single clock cycle.

Why Binary Arithmetic Matters

Every calculation your processor performs — from adding two integers to comparing strings — ultimately reduces to binary arithmetic and bitwise logic. Understanding these fundamentals is essential for low-level programming, embedded systems, network protocol implementation, cryptography, and computer architecture courses.

Frequently Asked Questions

What is two's complement?
Two's complement is the standard method for representing negative integers in binary. To negate a number: invert all bits (one's complement) then add 1. For example, +5 in 4-bit binary is 0101; −5 is 1010+1 = 1011. Most modern processors use two's complement for signed integer arithmetic.
What does XOR do in cryptography?
XOR is a fundamental operation in symmetric encryption. XORing plaintext with a key produces ciphertext; XORing ciphertext with the same key recovers the plaintext. XOR is also used in parity checks, error detection codes (CRC), and as a building block in hash functions.
How does binary multiplication work?
Binary multiplication works like long multiplication in decimal, but simpler: multiply by each bit of the multiplier. If the multiplier bit is 1, write the multiplicand shifted left by the bit position; if 0, write zeros. Then add all the partial products. In hardware this is implemented with shift-and-add circuits.
Can I use this for signed negative binary numbers?
This calculator uses unsigned binary arithmetic. For signed (negative) operations, use two's complement representation manually: enter the two's complement binary form and the result will also be in unsigned binary. Subtract the result from 2^n to convert back if needed.