Binary Addition Calculator

How Binary Addition Works

Binary addition is the fundamental arithmetic operation of digital computers. Every addition performed by a CPU, from adding two integers to floating-point arithmetic, reduces to column-by-column binary addition with carry propagation.

The process is identical to decimal addition: you start from the rightmost (least significant) bit and work left, adding the two bits in each column plus any carry from the column to the right. The only difference is that the base is 2 rather than 10, so the digit "wraps around" after 1 instead of after 9.

The Four Addition Rules

Worked Example: 1101 + 1011

Using the default values (A = 1101, B = 1011):

1. Column 1 (rightmost): 1 + 1 = 10 (binary). Write 0, carry 1.
2. Column 2: 0 + 1 + carry 1 = 10 (binary). Write 0, carry 1.
3. Column 3: 1 + 0 + carry 1 = 10 (binary). Write 0, carry 1.
4. Column 4: 1 + 1 + carry 1 = 11 (binary). Write 1, carry 1.
5. Column 5: carry 1 only. Write 1, no further carry.

Result: 11000 in binary, which equals 24 in decimal (13 + 11 = 24). This matches the calculator output above.

Half Adders and Full Adders

Digital circuits implement binary addition using logic gates. A half adder adds two single bits and produces a sum bit and a carry bit. It can be built from an XOR gate (for the sum) and an AND gate (for the carry). A full adder adds three bits (A, B, and a carry-in) and produces a sum bit and a carry-out. Full adders are chained together to add multi-bit numbers: this is called a ripple-carry adder. Modern CPUs use faster variants such as carry-lookahead adders to avoid waiting for carries to propagate through each stage.

Binary, Decimal, and Hexadecimal

Binary (base 2) uses only 0 and 1. Each digit position represents a power of 2: from right to left, the positions represent 1, 2, 4, 8, 16, 32, 64, 128, and so on. Hexadecimal (base 16) is a compact notation for binary: each hex digit represents exactly four binary digits (bits). For example, the binary number 11000 is 24 in decimal and 18 in hexadecimal. Programmers often use hex as a shorthand when working with binary data.

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Sources and method: Binary addition algorithm as described in Mano, M. M. & Ciletti, M. D., Digital Design (5th ed., Pearson, 2013). Full adder truth table per IEEE standard combinational logic conventions. Decimal and hexadecimal conversions use standard positional notation (base 10 and base 16).

This calculator adds unsigned binary integers. It does not handle two's complement negative numbers, binary fractions, or floating-point representations. Inputs are limited to 32 binary digits each.