# Binary Converter

By Bogna Szyk and Philip Maus
Last updated: Mar 30, 2021

The binary converter is a handy tool that will enable you to quickly perform a conversion of numbers. You will be able to use it both as a binary to decimal converter and as a decimal to binary calculator. Read on to learn what is the binary system, how to convert the numbers and how to use this calculator to obtain correct results.

## What is the binary system?

We normally operate in a decimal system. That means that we use ten different digits, from 0 to 9. If you write down a number, for example 345, each consecutive digit of this number corresponds to a different power of ten. That means that:

• Digit 5 corresponds to 10⁰ (5 is equal to `5 * 10⁰`).
• Digit 4 corresponds to 10¹ (40 is equal to `4 * 10¹`).
• Digit 3 corresponds to 10² (300 is equal to `3 * 10²`).

In the binary system, there are only two available digits: 0 and 1. That means that the digits of every number, instead of corresponding to powers of 10, correspond to powers of 2. For example, the number 1111 in binary system can be analyzed in the following way:

• The last 1 corresponds to 2⁰ (1 is equal to `1 * 2⁰ = 1`).
• The second-to-last 1 corresponds to 2¹ (10 is equal to `1 * 2¹ = 2`).
• Third-to-last 1 corresponds to 2² (100 is equal to `1 * 2² = 4`).
• The first 1 corresponds to 2³ (1000 is equal to `1 * 2³ = 8`).

After adding all of these numbers, we get `1 + 2 + 4 + 8 = 15`. Number 1111 in the binary system corresponds to 15 in the decimal system.

## Converting from decimal to binary

You can use a very simple and efficient algorithm to convert numbers from the decimal to binary system.

1. Take your initial number. Divide it by 2.
2. Note down the remainder. It is going to be equal to 0 or 1. This will be the last digit of the binary number (the rightmost one).
3. Take the quotient. It is your new "initial number".
4. Keep repeating the above steps, each time adding the remainder to the left of previously obtained digits.

For example, for number 19, we would have the following steps:

1. 19/2 = 9, remainder 1
2. 9/2 = 4, remainder 1
3. 4/2 = 2, remainder 0
4. 2/2 = 1, remainder 0
5. 1/2 = 0, remainder 1

Reading from the bottom to the top, 19 corresponds to 10011 in binary system. Check this result with the binary converter!

## Converting from binary to decimal

It is similarly easy - all you have to do is reverse the algorithm explained above:

1. Take the leftmost digit of your initial number. Multiply it by 2.
2. Add the next digit of the binary number. The sum will be your new "initial number".
3. Keep repeating these steps, each time first multiplying by 2 and then adding the last digit.

For example, for the binary number 110011, we would have the following steps:

1. `1 * 2 = 2`
2. `(2 + 1) * 2 = 6`
3. `(6 + 0) * 2 = 12`
4. `(12 + 0) * 2 = 24`
5. `(24 + 1) * 2 = 50`
6. `50 + 1 = 51`

110011 corresponds to 51 in decimal system. Check this result with the binary converter!

## Converting negative numbers

We are used to simply adding a minus symbol in front of the number if we want to express a negative number in the decimal system. But the binary system does not allow the minus symbol. So how can we express negative numbers in binary?

The general concept to express negative numbers in the binary system is the signed notation. This means that the first bit indicates the sign of the number: `0` means positive, `1` is a negative value. The signed notation has two representations:

• The one's complement of a negative number in binary is achieved by switching all digits of the opposite positive number to opposite bit values.

• The two's complement of a negative number in binary is achieved by switching all digits of the opposite positive number to opposite bit values and adding 1 to the number.

Let's look at an example to better understand the one's and two's complement. We want to convert the number -87 in decimal system into an 8-bit binary system.

1. Find the binary representation of the positive number 87 in the decimal system: `0101 0111`.
2. Switch all digits to the opposite: `1010 1000`. This is the one's complement.
3. Add 1 to the one's complement to get the two's complement: `1010 1001`.

With these representations, you can make applications like binary subtraction without any problems.

## How to use the binary converter?

Finally, we can talk about how to use the binary converter. For example, we will transform the number -87 from the decimal to the binary system.

1. Choose the number of bits. For our example, 8 bits are a good choice since they allow for a range from -128 to 127.

2. Enter your decimal value in the input field in the decimal to binary section. The calculator displays the result:

• The binary value of the positive opposite of our number, so 87: `0101 0111`.
• The one's complement: `1010 1000`.
• The two's complement: `1010 1001`.

The one's and two's complement are calculated as described above, flipping all digits for the opposite number and adding 1 for the two's complement.

The converter also allows the inverse conversion from the binary to the decimal system. Simply type in your binary number in the according field, and see the result in decimal displayed below.

Bogna Szyk and Philip Maus
Binary representation
8-bit
Decimal to binary
You can enter a decimal number between -128 and 127.
Decimal
Binary to decimal
You can write a binary number with no more than 8 digits. You don't have to input leading zeros.
Binary
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