Binary Addition Calculator
Step-by-Step Addition
Add binary numbers with clear step-by-step carry visualization. See how each bit-column adds up, when carries occur, and verify your results in decimal, hexadecimal, and octal.
5Number Systems
∞Precision
0msLatency
Interactive Tool Module
The Carry Operations
When adding `1 + 1` in binary, the result is `10` (a `0` with a carry of `1`). Click the Add +1 button below to watch how carries cascade sequentially through the logic gates.
← Carries
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FAQ
Frequently Asked Questions
What are the rules of binary addition?
Binary addition has four rules: 0 + 0 = 0, 0 + 1 = 1, 1 + 0 = 1, and 1 + 1 = 10 (which means 0 and carry 1 to the next position). When three 1s are added (1 + 1 + carry 1), the result is 11 (1 and carry 1). These are the only rules needed for all binary addition.
How do you add two binary numbers?
Align the binary numbers by their least significant bit (rightmost). Add column by column from right to left, applying the binary addition rules. When a column sums to 2 (10₂), write 0 and carry 1. When it sums to 3 (11₂), write 1 and carry 1. Example: 1101 + 1011 = 11000₂.
What is a carry in binary addition?
A carry in binary addition occurs when the sum of a column exceeds 1 (the maximum single binary digit). Since 1 + 1 = 10₂, the 0 stays in the current position and the 1 carries to the next higher position. This is identical in concept to carrying in decimal addition when a column exceeds 9.
Can binary addition cause overflow?
Yes. In fixed-width binary (e.g., 8-bit), if the result requires more bits than available, overflow occurs. For unsigned 8-bit addition, 11111111 + 00000001 = 100000000 (9 bits), but only 8 bits are stored, so the result wraps to 00000000. This is a common source of bugs in programming.
How is binary addition used in computers?
Binary addition is the most fundamental arithmetic operation in CPUs. Addition circuits (adders) perform all arithmetic: subtraction is done by adding the 2's complement, multiplication is repeated addition with shifting, and comparison is done via subtraction. Every calculation in a computer ultimately relies on binary addition.
What is a half adder and full adder?
A half adder is a logic circuit that adds two single bits, producing a sum and a carry. It uses an XOR gate for the sum and an AND gate for the carry. A full adder adds three bits (two inputs plus a carry-in), producing a sum and carry-out. Full adders are chained together to create multi-bit adders.