Binary Addition Calculator
Enter two binary numbers (using only 0s and 1s) and this calculator adds them immediately, showing the binary sum, the decimal and hexadecimal equivalents, and a column-by-column carry table so you can follow every step of the addition by hand.
How binary addition works
Binary (base-2) addition follows the same column-by-column process as decimal long addition, but with only two digits: 0 and 1. Because there are only two symbols, a column sum of 2 cannot be written as a single digit, so it is expressed as 10 in binary - meaning "write 0 in the current column and carry 1 to the next column to the left". A sum of 3 (1 + 1 + 1 carry) similarly becomes 11 in binary - "write 1 and carry 1". The four base cases are: 0 + 0 = 0, 0 + 1 = 1, 1 + 0 = 1, and 1 + 1 = 10 (write 0, carry 1). With a carry in, the column sum can reach 3, which becomes 11 (write 1, carry 1).
Step-by-step worked example
To add 1101 (13 in decimal) and 1011 (11 in decimal): align the numbers by their rightmost bit, then work from right to left. Rightmost column: 1 + 1 = 10 - write 0, carry 1. Next column: 0 + 1 + 1 carry = 10 - write 0, carry 1. Next column: 1 + 0 + 1 carry = 10 - write 0, carry 1. Leftmost column: 1 + 1 + 1 carry = 11 - write 1, carry 1. The carry beyond the leftmost column gives one more 1. Reading the result bits from left to right: 11000, which equals 24 in decimal - confirming 13 + 11 = 24.
Binary, decimal and hexadecimal equivalents
Every binary number has an exact decimal equivalent found by multiplying each bit by its place value (2^0 = 1, 2^1 = 2, 2^2 = 4, 2^3 = 8, and so on from right to left) and summing the products. Hexadecimal (base-16) groups four binary bits at a time into a single hex digit (0-9 then A-F), making it a compact shorthand for long binary strings. For example, the 8-bit binary number 11001010 groups as 1100 (= C) and 1010 (= A), giving the hex value CA. This calculator shows all three representations so you can cross-check your work in any base.
Overflow and fixed bit-width arithmetic
Digital systems store numbers in fixed-width registers - 8-bit, 16-bit, 32-bit, and so on. When an addition result requires more bits than the register can hold, the extra leading bit is lost, a condition called overflow. For example, adding 11111111 (255) and 00000001 (1) in an 8-bit register gives 100000000 (256) but the leading 1 is dropped, leaving 00000000 (0). This calculator lets you select a bit width; if the result exceeds it the overflow flag is set and the display shows only the bits that fit. Understanding overflow is essential for low-level programming and digital circuit design.
Binary addition rules (all four cases)
| Bit A | Bit B | Carry in | Column sum | Result bit | Carry out |
|---|---|---|---|---|---|
| 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 1 | 1 | 0 |
| 1 | 0 | 0 | 1 | 1 | 0 |
| 1 | 1 | 0 | 2 | 0 | 1 |
| 0 | 0 | 1 | 1 | 1 | 0 |
| 0 | 1 | 1 | 2 | 0 | 1 |
| 1 | 0 | 1 | 2 | 0 | 1 |
| 1 | 1 | 1 | 3 | 1 | 1 |
These four rules cover every possible column in a binary addition. The carry is propagated to the next column to the left.
Frequently asked questions
How do I add two binary numbers by hand?
Align the two numbers so their rightmost bits line up, then add column by column from right to left, just as you would with decimal long addition. The only twist is that 1 + 1 = 10 in binary, not 2, so you write 0 in the current column and carry a 1 to the next column to the left. Keep applying the same rule until all columns are done. Any remaining carry becomes a new leading bit in the result.
What is a carry bit in binary addition?
A carry bit (also called carry-out) is the extra 1 that is passed to the next column to the left when a column sum is 2 or 3. In decimal addition you carry whenever a column reaches 10 or above. In binary the threshold is 2, because the only single-digit values are 0 and 1. The carry-in for each column is the carry-out from the column immediately to its right.
What does overflow mean in binary addition?
Overflow happens when the result of an addition requires more bits than the fixed width of the register or storage location. For example, in 8-bit arithmetic the largest value is 11111111 (255). Adding 1 produces 100000000 (256), but the 9th bit has no place to go and is discarded, leaving 00000000 (0). This is a common source of bugs in low-level programming. The overflow flag in this calculator warns you whenever the exact result does not fit within the selected bit width.
How do I convert a binary sum to decimal?
Write the binary result and number the bit positions from right to left starting at 0. Multiply each bit (0 or 1) by 2 raised to its position number, then add all the products. For example, 11000 is (1 x 16) + (1 x 8) + (0 x 4) + (0 x 2) + (0 x 1) = 16 + 8 = 24. This calculator does the conversion automatically for both the inputs and the sum.
What is the difference between binary and hexadecimal?
Both are positional numeral systems used in computing. Binary uses only the digits 0 and 1 (base 2), while hexadecimal uses 16 symbols: the digits 0-9 and the letters A-F (base 16). Hexadecimal is popular because exactly four binary bits map to one hex digit, making it a compact and human-readable shorthand for binary data. Programmers often write memory addresses and color codes in hex.
Can this calculator handle numbers larger than 8 bits?
Yes. The calculator operates on binary strings of any length up to 32 bits displayed (or longer in auto mode). JavaScript integers are precise up to 2^53, so results stay exact well beyond 32-bit values. If you choose a fixed bit-width display option, the calculator flags overflow but still shows you the full natural result in the steps panel.