Binary Addition Calculator

This binary addition calculator sums up the numbers represented with zeros and ones (you can generate them with the binary converter).

Read this text to learn more about the language of computers and how to add binary numbers without a binary math calculator. Once you get your head around adding binary numbers, you may want to learn how to subtract them with our binary subtraction calculator.

To understand the binary system, let's first look at the more familiar decimal system.

In the decimal system, we use ten digits (0-9), and, depending on their position, we multiply them by corresponding powers of ten. For example, let's break down the number 1934:

1934 = 1000 + 900 + 30 + 4

1934 = 1×1000 + 9×100 + 3×10 + 4×1

1934 = 1×10³ + 9×10² + 3×10¹ + 4×10⁰

If we made a formula for a 4-digit number, it could look like this:

a₃a₂a₁a₀ = a₃ × 10³ + a₂ × 10² + a₁ × 10¹ + a₀ × 10⁰

The symbol "a" here represents a digit from 0 to 9. The index next to "a" denotes the digit's place (we count from 0, from right to left).

Now, the binary system works similarly, but we only use two digits and multiply them by powers of two. So the formula would be the following:

a₃a₂a₁a₀ = a₃ × 2³ + a₂ × 2² + a₁ × 2¹ + a₀ × 2⁰

In this system, we multiply two digits (0 and 1) by 2 to the power corresponding to their place in the number.

For example, we would write "9" as "1001" because:

1001 = 1 × 2³ + 0 × 2² + 0 × 2¹ + 1 × 2⁰ = 8 + 0 + 0 + 1 = 9

Analogically:

• "2" in binary is "10" because 1 × 2¹ + 0 × 2⁰ = 2

• "3" in binary is "11" because 1 × 2¹ + 1 × 2⁰ = 2 + 1 = 3

• "4" in binary is "100" because 1 × 2² + 0 × 2¹ + 0 × 2⁰ = 4

There are four basic binary addition rules:

• 0 + 0 = 0
• 0 + 1 = 1
• 1 + 0 = 1
• 1 + 1 = 10 (write "0" in the column and carry 1 to the next bit)

The above equations work like in the decimal system, only here you need to carry 1 when the sum exceeds 1 (in the decimal system, we do it when it exceeds 9).

How are binary numbers added together? You could always convert binary numbers to decimals, add them as usual, and restore the result to the binary form.

The other way is to use the above binary addition rules and perform a long addition. Let's take a look at this binary sum:

In the first column from the right, 1 + 1 gives us 2, so we need to carry 1 to the second column (1 placed in the second column equals two because 1 × 2¹ = 2) and write down 0 in the first column. In the second column, 0 + 0 = 0, so we just write the 1 we carried over. In the third column from the right, 0 + 1 = 1. In the fourth column, we have 1 + 1, so again, we write 0 and carry 1. As there's nothing left to add, we write down "1" at the beginning of the result (to the left).

Applications of adding binary numbers are present in many error correction and detection techniques, like the parity bit and the Hamming distance!

To make the binary math calculator perform the addition of binary numbers, follow the steps below:

1. Select the binary representation you want to use. You can also enter a custom binary representation by selecting "Other" on the binary representation drop-down options.

2. Then, input the first number in the first field of the binary addition calculator. Remember to use only zeros and ones. You don't need to enter leading zeros, e.g., for "00001111," you can input just "1111".

3. Finally, enter the second binary number in the second row.

4. The binary addition calculator will display the results in a table under its Results section.

5. You can also tick the Display long addition equation checkbox if you're curious about it, and tick on the Show carry bits checkbox for full long addition reference. Due to the limited space of our binary addition calculator, it can only display equations for binary inputs of at most 16-bits.