How to Convert Binary to Decimal

In the world of technology, IT scientists are constantly using different methods to convert every single data entry to a language for digital computers to process. Correspondingly, these methods work as an interpreter facilitating the communication between man and machine.
Nowadays, computers have become a huge part of people’s lives, therefore it's important for you to comprehend the basics.
Below, you will learn more about different number systems and learn specifically how to convert binary to decimal.

Number Systems

Number systems are different mathematical systems that people use to express numbers in a way that a computer can understand. There are mainly four types of number systems which are:
  • The decimal number system (base 10)
  • The binary number system (base 2)
  • The octal number system (base 8)
  • The hexadecimal number system (base 16)

How to Convert Binary to Decimal

There’s a base for each number system, and that base represents the combination of numbers used for that system.
The decimal number system is the most basic of the four systems. It's the system used in everybody’s daily routine. It's a base-10 system, which means that it uses ten digits (0-9) to represent the desired quantity.
Contrarily, the binary system uses only two digits (0 and 1), hence the name ‘base-2’. This system is used most commonly by computer languages. Each digit in the binary system is called a ‘bit’.
There are several known methods to convert a binary number to a decimal value.
Below are two of the simplest ways that you can easily apply.

Method 1: Positional Notation

In this method, you multiply each digit by the base number, which is 2 to the power of the corresponding value. Starting from the far right with the power value of 0 and moving to the left with the power increasing by 1 for each digit. Finally, add all the results of each digit together.

Example A: Convert the Binary Number 10110010 Into Decimal

  1. Write down the number (110010) 2
  2. Multiply each digit by 2 to the power of (0,1,2,...) consequently from right to left 1×25+1×24+0×23+0×22+1×21+0×20
  3. Add the resulting numbers 32+16+0+2+0 = 50

Example B: Convert the Binary Number 100111 Into Decimal

  1. Write down the number (1100111) 2
  2. Multiply each digit by 2 to the power of (0,1,2,...) consequently from right to left 1×26+1×25+0×24+0×23+1×22+1×21+1×20
  3. Add the resulting numbers 64+32+0+0+4+2+1 = 103

The Binary Point

The binary point is almost the same as the decimal point. It's the point dividing the integer part of the number from the fraction part. So, any digit falling to the left of the binary point has a non-negative power.
On the other hand, any digit positioned to the right has a negative power. In this case, the power of the digits after the binary point decreases from left to right. By using the same method, you multiply each digit by 2 to the power of (-1,-2,-3,...).
For example, to convert the binary number 1010.11 to decimal, multiply each digit before the binary point by 2 to the power of (0,1,2,...) consequently from right to left. Then, multiply each digit after the binary point by 2 to the power of (-1,-2,-3,...) from left to right. Finally, add the resulting numbers.

Convert the Binary Number 10011.101 Into Decimal

  1. Write down the number (10011.101) 2
  2. Multiply each digit before the binary point by 2 to the power of (0,1,2,...) consequently from right to left 1×24+0×23+0×22+1×21+1×20
  3. Multiply each digit after the binary point by 2 to the power of (-1, -2, -3,...) from left to right 1×2-1+0×2-2+1×2-3
  4. Add the resulting numbers 16+0+0+2+1+0.5+0+0.125 = 19.625

Method 2: Doubling

Also known as double dabble. It's a simpler method that doesn't use powers, so you can easily convert large numbers with no need for pen and paper. You only need to keep track of the subtotal.
This time, start from the left and work your way to the right by doubling the previous total and adding it to the current digit. Keep going till the last digit.

Example A: Convert the Binary Number 101110 Into Decimal

  1. There's no previous total, so you always start with a zero.
  2. Double the zero and add 1, which is the current digit. 0×2+1=1
  3. Repeat the previous step, double the 1, and add 0, which is the current digit. 1×2+0=2. Your current total is now 2.
  4. Repeat. 2×2+1=5
  5. Again. 5×2+1=11
  6. Keep going until you run out of digits. 11×2+1=23
  7. One last time. 23×2+0=46

Example B: Convert the Binary Number 111011 Into Decimal

  1. Use the same method as the previous example and start with zero.
  2. Double the zero and add 1, which is the current digit. 0×2+1=1
  3. Repeat. 1×2+1=3
  4. Again. 3×2+1=7
  5. Keep going. 7×2+0=14
  6. Almost there. 14×2+1=29
  7. Finally, 29×2+1=59

Use Doubling to Convert Any Base to Decimal

You can convert any base to decimal using the doubling method effortlessly. Remember that it's called doubling because the base of the binary number is 2.
For example, to convert the number (356) 23 to decimal:
  1. Start with zero as usual
  2. Multiply it by 23 instead of 2
  3. Add the current digit 0×23+3=3
  4. Repeat. 3×23+5=74
  5. Finally, 74×23+6=1708

Conclusion

Binary to decimal conversion helps translate numbers to a form that the human mind can understand.
Over time, people used many variations of number systems. Those systems kept evolving over the course of thousands of years.
Though the systems used in modern computing might seem challenging to comprehend, once you fully grasp the topic, it won’t pose an issue for you.
You can always use a calculator for these types of conversions. But before you do that, it's helpful for you to grasp the process and learn the fundamentals.