By Rita Rain
Last updated: Aug 03, 2021

This binary addition calculator sums up the numbers represented with zeros and ones.

## What is the binary system?

To understand the binary system, let's first have a 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 stands for 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 in the same way, but we only use two digits and multiply them by powers of two. So the formula would be 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 its 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`

## How to use the binary addition calculator?

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

1. 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".

2. Enter the second binary number in the second row.

3. The binary addition calculator will display the result in the third field.

4. In `advanced mode` (at the bottom), you can make the binary addition calculator print the long addition of the binary numbers.

## What are the rules of binary addition?

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).

## Binary sum: Adding binary numbers

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

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

`carry:   1  1 1001+ 1101= 10110`

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).

## FAQ

Binary addition is the operation of summing numbers in binary form. It works like a "normal" (decimal) addition, but the number can have only zeros and ones as digits, so if the sum exceeds 1, you have to carry 1 to the next bit. For example, `101 + 101 = 1010`.

### How to solve binary addition?

1. To solve binary addition, perform long addition — place numbers vertically and add digits in columns going from right to left.

2. If the sum in a column equals 2, carry 1 to the next column (to the left). If the sum is 1 or 0, write it down and go to the next column.

3. Proceed this way until you sum all columns (including the carried numbers).

### How to detect an overflow in binary addition?

Check if the sum of binary numbers makes sense. If the sum of two negative numbers is positive or the sum of two positive numbers is negative, something went wrong. This bug means that the sum overflowed — that is, the binary representation of the result can't fit in the allocated number of bits.

Rita Rain
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