Binary to Decimal
Binary to decimal conversion is done to convert a number given in the binary number system to its equivalent value in the decimal number system. A number system is a format to represent numbers in a certain way. The binary number system is used in computers and electronic systems to represent data and it consists of only two digits which are 0 and 1. The decimal number system is the most commonly used number system around the world which is easily understandable to people. It consists of digits from 0 to 9. Binary to decimal conversion can be done in the simplest way by adding the products of each binary digit with its weight (which is of the form  binary digit × 2 raised to a power of the position of the digit) starting from the rightmost digit which has a weight of 2^{0}.
Binary to decimal conversion can be done by two methods  the positional notation method and the doubling method. Let us understand the different methods for binary to decimal conversion.
What is Binary to Decimal Conversion?
Binary to decimal conversion is done to represent a number given in Binary Number System to its equivalent in the Decimal Number System. A number system is very essential to represent numbers. Every number system has a base and the base of a number system is determined by the total number of digits used in the number system. For example, the binary number system has a base of 2 because it has only two digits to represent any number. Similarly, the decimal number system has a base of 10, as it has 10 digits to represent a number.
The conversion of numbers from binary to decimal is important as it helps to read numbers that are represented as a set of 0s and 1s.
Binary to Decimal Conversion Methods
Binary to decimal conversion is done to help read large binary numbers easily in a form that humans can understand. There are two methods to convert a number in binary to decimal number system.
 Positional Notation Method
 Doubling Method
Let us understand these binary to decimal conversion methods in detail.
Binary to Decimal Conversion Using Positional Notation Method
The positional notation method is one in which the value of a digit in a number is determined by a weight based on its position. This is achieved by multiplying each digit by the base (2) raised to the respective power depending upon the position of that digit in the number. The sum of all these values obtained for each digit gives the equivalent value of the given binary number in the decimal system.
Observe the following steps to understand the binary to decimal conversion. Let us consider the binary number \((101101)_{2}\). In any binary number, the rightmost digit is called the 'Least Significant Bit' (LSB) and the leftmost digit is called the 'Most Significant Bit' (MSB). For a binary number with 'n' digits, the least significant bit has a weight of 2^{0} and the most significant bit has a weight of 2^{n1}.
 Step 1: List out the powers of 2 for all the digits starting from the rightmost position. The first power would be 2^{0} and as we move on it will be 2^{1}, 2^{2}, 2^{3}, 2^{4}, 2^{5},... In the given example, there are 6 digits, therefore, starting from the rightmost digit, the weight of each position from the right is 2^{0},2^{1},2^{2},2^{3},2^{4},2^{5}.
 Step 2: Now multiply each digit in the binary number starting from the right with its respective weight based on its position and evaluate the product. Observe the figure shown below to relate to the step. Finally, sum up all the products obtained for all the digits in the binary number.
 Step 3: Now, express the binary number as a decimal number: \((101101)_{2}\) = \((45)_{10}\)
Binary to Decimal Conversion Using Doubling Method
As the name suggests, the process of doubling or multiplying by 2 is done to convert binary to decimal. Let us use the same example for converting the binary number \((101101)_{2}\) to decimal. Observe the following steps given below to understand the binary to decimal conversion using the doubling method.
 Step 1: Write the binary number and start from the leftmost digit. Double the previous number and add the current digit. Since we are starting from the leftmost digit and there is no previous digit to the leftmost digit, we consider the double of the previous digit as 0. For example in \((101101)_{2}\), the leftmost digit is '1'. The double of the previous number is 0. Therefore, we get ((0 × 2) + 1) which is 1.
 Step 2: Continue the same process for the next digit also. The second digit from the left is 0. Now, double the previous digit and add it with the current digit. Therefore, we get, [(1 × 2) + 0], which is 2.
 Step 3: Continue the same step in sequence for all the digits. The sum that is achieved in the last step is the actual decimal value. Therefore, the result of converting the binary number \((101101)_{2}\) to a decimal using the doubling method is \(45_{10}\).
Observe the figure given below to relate to the steps and understand how the doubling method works.
Binary to Decimal Formula
In the previous section, we understood the methods and their stepwise process to convert a binary to a decimal. Let us learn the general formula for converting a binary number to a decimal number now. Considering \(d_{n}\) to be the digits of a binary number consisting of 'n' digits, the formula to convert binary to decimal is given as,
Binary to Decimal Conversion Formula:
(Decimal Number)_{10} = \((d_{0}\) × 2^{0 })+ \((d_{1}\) × 2^{1 })+ \((d_{2}\) × 2^{2 })+ ..... + \((d_{n1}\) × 2^{n1})
where, \(d_{0}\), \(d_{1}\), \(d_{2}\) are the individual digits of the binary number starting from the rightmost position.
Let us see the application of the above binary to decimal formula and learn how to convert binary to decimal using the following example.
For example, let us convert \((1110)_{2}\), from binary to decimal using the formula. We start doing the conversion from the rightmost digit, which is '0' here.
(Decimal Number)_{10} = \((d_{0}\) × 2^{0 } )+ \((d_{1}\) × 2^{1 })+ \((d_{2}\) × 2^{2 })+ ..... \((d_{n1}\) × 2^{n1}),
= (0 × 2^{0}) + (1 × 2^{1}) + (1 × 2^{2}) + (1 × 2^{3})
= (0 × 2^{0}) + (1 × 2^{1}) + (1 × 2^{2}) + (1 × 2^{3})
= 0 + 2 + 4 + 8
= 14
Therefore, \((1110)_{2}\) = \((14)_{10}\)
Binary to Decimal Conversion Chart
The binary to decimal conversion of the first 20 decimal numbers is displayed in the chart given below.
Binary  Decimal 

0  0 
1  1 
10  2 
11  3 
100  4 
101  5 
110  6 
111  7 
1000  8 
1001  9 
1010  10 
1011  11 
1100  12 
1101  13 
1110  14 
1111  15 
10000  16 
10001  17 
10010  18 
10011  19 
10100  20 
Binary to Decimal Converter
In the above sections, we have learned different methods to convert binary to decimal. Check out this binary to decimal converter to verify the results for the manual calculations done to convert a number given in binary number system to its equivalent in decimal number system  Binary to Decimal Calculator
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Examples on Binary to Decimal

Example 1: Find the decimal value of the binary number \((11001011)_{2}\)_{ } using the positional notation method of binary to decimal conversion.
Solution:
By the positional notation of binary to decimal conversion, we multiply every digit in the binary number with its base raised to the power based on its position. This is done by starting from the rightmost digit and moving on to the left and summing up all the values.
In the binary to decimal conversion shown below, we start from the right and move towards the left.
\((11001011)_{2}\) = (1 × 2^{0})+ (1 × 2^{1})+ (0 × 2^{2})+ (1 × 2^{3}) + (0 × 2^{4}) + (0 × 2^{5}) + (1 × 2^{6}) + (1 × 2^{7})
= (1 × 1) + (1 × 2) + (0 × 4) + (1 × 8) + (0 × 16) + (0 × 32) + (1 × 64) + (1 × 128)
= 1 + 2 + 0 + 8 + 0 + 0 + 64 +128
= 203Therefore, \((11001011)_{2}\) = \((203)_{10}\).

Example 2: Using the doubling method of binary to decimal conversion, find the decimal value of \((10101101)_{2}\).
Solution:
For the binary to decimal conversion of a number using the doubling method, we use the following steps:
 Step 1: Write the binary number and start from the leftmost digit. Take the doubled value of the previous digit and add it with the current digit. Since in the binary number \(10101101_{2}\), '1' is the leftmost digit and there is no previous number, the doubled value is 0. Now, adding it with the current value which is 1, we get (0 × 2) + 1, which is 1.
 Step 2: Follow the same step as described in step 1 for all the numbers as you move on to the right.
 Step 3: Finally, the sum that is achieved in the last step is the decimal equivalent of the binary value.
The steps that were discussed above for the binary to decimal conversion are shown below:

Example 3: Using the binary to decimal conversion method, find out if (159)_{10 }is the decimal equivalent of \((10011111)_{2}\).
Solution:
To find if (159)_{10} is the decimal equivalent of \((10011111)_{2}\), let us do the conversion from binary to decimal.
In the conversion from binary to decimal shown below, we start from the right most digit and move to the left.
\((10011111)_{2}\) = (1 × 2^{0}) + (1 × 2^{1}) + (1 × 2^{2})+ (1 × 2^{3})+ (1 × 2^{4}) + (0 × 2^{5}) + (0 × 2^{6}) + (1 × 2^{7})
= (1 × 1) + (1 × 2) + (1 × 4) + (1 × 8) + (1 × 16) + (0 × 32) + (0 × 64) + (1 × 128)
= 1 + 2 + 4 + 8 + 16 + 0 + 0 + 128
= 159.Therefore, the binary to decimal conversion of \((10011111)_{2}\) results in \((159)_{10}\).
FAQs on Binary to Decimal
What is Binary to Decimal Conversion?
The process of converting a binary number to a decimal number is called binary to decimal conversion. For example, \((100)_{2}\) in binary when converted to a decimal number is (4)_{10}. Binary numbers are composed of only 0 and 1, whereas, decimal numbers are composed of digits from 0 to 9. The binary number system is also called the base2 number system and the decimal number system is known as the base10 number system.
How to Convert Binary to Decimal?
A number given in the binary number system can be converted to its equivalent in the decimal number system by either the position notation method or by using the doubling method.
What is the Value of \((1010)_{2}\) from Binary to Decimal?
The decimal value of \((1010)_{2}\) is number 10. To get this, we multiply each digit in the binary number by 2 raised to the power depending upon the position of the digit in the number, starting from the rightmost digit and moving towards the left. The rightmost digit is multiplied by 2^{0} and the next digit by 2^{1} and so on. Finally, we add up all the values and get the decimal value which is 10.
How to Convert a Number From Binary to Decimal Using the Positional Notation Method?
To convert a number from binary to decimal using the positional notation method, we multiply each digit of the binary number with its base, (which is 2), raised to the power based on its position in the binary number. The rightmost digit of the binary digit carries a position of 0, and as we move on to the left, it increases by 1. Finally, we sum up all the values to get the decimal equivalent. For example, to convert \((101)_{2}\) from binary to decimal using the positional notation method, the conversion step is as follows. \((100)_{2}\) = (0 × 2^{0}) + (0 × 2^{1}) + (1 × 2^{2}), which is equal to 0 + 0 + 4. Therefore, \((100)_{2}\) = (4)_{10}.
How to Convert a Number From Binary to Decimal Using the Doubling Method?
In the doubling method, we double every previous digit and add it to the current digit of the binary number, starting from the leftmost digit and moving towards the right. For example, to convert \((110)_{2}\) from binary to decimal, we use the steps given below. Here, since we start from the leftmost digit, there is no previous number to it. Therefore, we consider the doubled value of the previous number for the leftmost digit to be 0. The sum that is obtained in the final step is the decimal equivalent of the binary number.
 (0 × 2) + 1 = 1
 (1 × 2) + 1 = 3
 (3 × 2) + 0 = 6
 Therefore, \((110)_{2}\) = \((6)_{10}\)
What is the Formula to Convert a Number From Binary to Decimal?
The formula to convert a binary number to decimal is as follows. Considering \(d_{n}\) to be the digits of a binary number consisting of 'n' digits, \((\text{Decimal Number})_{10}\) = \((d_{0}\) × 2^{0 })+ \((d_{1}\) × 2^{1 })+ \((d_{2}\) × 2^{2 })+ ..... \((d_{n}\) × 2^{n}), where \(d_{0}\), \(d_{1}\), \(d_{2}\) are individual digits of the binary number starting from the right most position.
Can We Convert \((1111.1)_{2}\) from Binary to Decimal?
Yes, it is possible to convert \((1111.1)_{2}\) from binary to decimal. To do this, we first convert the integer part to decimal or a base10 number. Therefore, the decimal equivalent of \((1111)_{2}\) = (1 × 2^{0}) + (1 × 2^{1}) + (1 × 2^{2}) + (1 × 2^{3}) , which is equal to 1 + 2 + 4 + 8, which is 15. Now, we convert the fractional part which is 0.1 to a decimal or a base10 number. Since it is a fractional part, the decimal equivalent of 0.1 = 1 × 2^{1}, which is equal to 0.5. Now, we sum up both the values together, which is 15 + 0.5, or 15.5. Therefore, the binary to decimal conversion of \((1111.1)_{2}\) is \((15.5)_{10}\).
List out the Binary to Decimal Values of the First Ten Decimal Numbers.
The list given below depicts the binary and the corresponding decimal equivalents of the first ten decimal numbers.
(0)\(_2\) = (0)\(_{10}\)
(1)\(_2\) = (1)\(_{10}\)
(10)\(_2\) = (2)\(_{10}\)
(11)\(_2\) = (3)\(_{10}\)
(100)\(_2\) = (4)\(_{10}\)
(101)\(_2\) = (5)\(_{10}\)
(110)\(_2\) = (6)\(_{10}\)
(111)\(_2\) = (7)\(_{10}\)
(1000)\(_2\) = (8)\(_{10}\)
(1001)\(_2\) = (9)\(_{10}\)
Does Binary to Decimal and Binary to Hexadecimal Conversions Result in the Same Answer?
No, binary to decimal and Binary to Hexadecimal conversions result in different answers because decimal and hexadecimal are different number systems. The decimal number system uses digits from 0 to 9, while the hexadecimal number system uses 16 digits for representing a number, using numbers from 0  9, followed by A, B, C, D, E, F for the numbers from 10 to 15.
How to Convert 111001\(_2\) from Binary to Decimal Number System Using Conversion Formula?
In order to convert 111001\(_2\) from binary to decimal let us use the Binary to Decimal Formula, \((d_0 \times 2^0) + (d_1 \times 2^1) + (d_2 \times 2^2) + (d_3 \times 2^3) ...\)
⇒\(111001_2 = (1 \times 2^0) + (0 \times 2^1) + (0 \times 2^2) + (1 \times 2^3) + (1 \times 2^4) + (1 \times 2^5) = 57_{10}\)
What is 10101 Binary to Decimal?
10101 is equal to 21 when it is converted from binary to decimal. This can be done using the formula, \((d_0 \times 2^0) + (d_1 \times 2^1) + (d_2 \times 2^2) + (d_3 \times 2^3) + (d_4 \times 2^4)...\) ⇒\(10101_2 = (1 \times 2^0) + (0 \times 2^1) + (1 \times 2^2) + (0 \times 2^3) + (1 \times 2^4) = 21_{10}\)
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