# Binary Addition Calculator

Add two binary numbers with carry bits shown at each step. See the binary sum and decimal equivalents. Free online binary addition tool.

## What this calculates

Add two binary numbers together and see every carry bit along the way. This calculator shows the full binary sum plus the decimal equivalents so you can verify your work instantly.

## Inputs

- **Binary Number A** — Enter a binary number (only 0s and 1s).
- **Binary Number B** — Enter a binary number (only 0s and 1s).

## Outputs

- **Binary Sum** — formatted as text — The result of adding the two binary numbers.
- **Carry Bits** — formatted as text — The carry bits generated at each position during addition.
- **Decimal Value of A** — The decimal equivalent of Binary Number A.
- **Decimal Value of B** — The decimal equivalent of Binary Number B.
- **Decimal Sum** — The decimal equivalent of the binary sum.

## Details

Binary addition follows the same column-by-column approach you already know from decimal addition, but with only two digits: 0 and 1.

**Binary Addition Rules:**

- 0 + 0 = 0
- 0 + 1 = 1
- 1 + 0 = 1
- 1 + 1 = 10 (write 0, carry 1)
- 1 + 1 + 1 = 11 (write 1, carry 1)

**How It Works:**

Start from the rightmost bit and work left, just like decimal addition. When a column adds up to 2 or 3, you write the remainder and carry a 1 to the next column.

**Example:**

Adding 1011 (decimal 11) and 1101 (decimal 13):

| Carry | 1 | 1 | 1 | 0 |   |
|-------|---|---|---|---|---|
|       | 1 | 0 | 1 | 1 |   |
| +     | 1 | 1 | 0 | 1 |   |
| =   | 1 | 1 | 0 | 0 | 0 |

Result: 11000 (decimal 24). That checks out since 11 + 13 = 24.

Binary addition is the foundation of how every computer processor handles arithmetic. ALUs (arithmetic logic units) use chains of full adders to process numbers bit by bit.

## Frequently Asked Questions

**Q: What happens when you add 1 + 1 in binary?**

A: In binary, 1 + 1 = 10 (which is 2 in decimal). You write down 0 and carry 1 to the next column, just like carrying in decimal when 5 + 5 = 10.

**Q: How do computers add binary numbers?**

A: Computers use circuits called full adders. Each full adder takes two input bits and a carry-in bit, then produces a sum bit and a carry-out bit. Multiple full adders are chained together to handle numbers of any width, processing one column at a time from right to left.

**Q: Can binary addition produce a longer result than either input?**

A: Yes. When the final carry bit is 1, the result has one more digit than the longest input. For example, 1111 (15) + 0001 (1) = 10000 (16). This is called overflow in fixed-width systems and is something programmers watch for carefully.

**Q: How do I check if my binary addition is correct?**

A: Convert both binary numbers to decimal, add them normally, then convert the decimal sum back to binary. If it matches your binary result, the addition is correct. For example, 101 (5) + 110 (6) should give 1011 (11).

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Source: https://vastcalc.com/calculators/math/binary-addition
Category: Math
Last updated: 2026-04-08