What is binary addition?
Binary addition combines two numbers written in base-2 (using only the digits 0 and 1) into a single base-2 result. Computers store and process all data in binary, so understanding binary arithmetic is fundamental to computer science, digital electronics, and programming. This calculator adds any two binary numbers and shows you both the binary sum and the equivalent decimal values, so you can check your work step by step.
How to use this calculator
Enter your first binary number in the top field and your second binary number in the bottom field. Use only the digits 0 and 1 — for example 1010 or 1101. Click calculate and you will instantly see the binary sum, plus the decimal value of each input and the decimal total for verification.
The formula explained
To add two binary numbers, we first convert each one to decimal. Each bit contributes its value times a power of two based on its position, counting from the right starting at \(2^0\). We add the two decimal values, then convert the total back into binary by repeatedly taking the remainder when dividing by 2.
$$\text{Sum}_2 = \left( \text{A} + \text{B} \right)_2$$
$$\text{where}\quad \left\{ \begin{aligned} \text{A}_{10} &= \sum_{i} d_i^{A} \cdot 2^{\,i} \\ \text{B}_{10} &= \sum_{i} d_i^{B} \cdot 2^{\,i} \end{aligned} \right.$$
Worked example
Add 1010 and 1101. The first equals \(8+0+2+0 = 10\) in decimal; the second equals \(8+4+0+1 = 13\). Their sum is \(10 + 13 = 23\). Converting 23 back to binary gives 10111. So $$1010_2 + 1101_2 = 10111_2$$
FAQ
Can I add binary numbers with leading zeros? Yes. Leading zeros do not change the value, so 0011 is treated the same as 11.
What characters are allowed? Only the digits 0 and 1. Any other character will be ignored or treated as invalid.
Why show the decimal values? Seeing the decimal equivalents makes it easy to verify the binary result and helps you learn how the two bases relate.
