Binary to Decimal | Mathematics for JAMB PDF Download

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. Binary to decimal conversion can be done by two methods - the positional notation method and the doubling method.

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 from binary to decimal number system.

• Positional Notation Method
• Doubling Method

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. Let us understand this with the help of examples.
Example: Convert the binary number 101101₂ to a decimal number.
Solution: Observe the following steps to understand the binary to decimal conversion. In any binary number, the rightmost digit is called the 'Least Significant Bit' (LSB) and the left-most digit is called the 'Most Significant Bit' (MSB). For a binary number with 'n' digits, the least significant bit has a weight of 2⁰ and the most significant bit has a weight of 2^n-1.

Step 1: List out the powers of 2 for all the digits starting from the rightmost position. The first power would be 20 and as we move on it will be 2¹, 2², 2³, 2⁴, 2⁵,... In the given example, there are 6 digits, therefore, starting from the rightmost digit, the weight of each position from the right is 2⁰, 2¹, 2², 2³, 2⁴, 2⁵.

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: 101101₂ = 45₁₀

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 101101₂ to decimal.
Example: Convert the binary number 101101₂ to decimal using doubling method.
Solution: 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 left-most digit. Double the previous number and add the current digit. Since we are starting from the left-most digit and there is no previous digit to the left-most digit, we consider the double of the previous digit as 0. For example, in 101101₂, the left-most 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 101101₂ to a decimal using the doubling method is 45₁₀

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 dn 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 = (d₀ × 2⁰)+ (d₁ × 2¹)+ (d₂ × 2²)+ ..... + d_n-1 × 2^n-1)
where, d₀, d₁, d₂ are the individual digits of the binary number starting from the right-most position.
Example: Convert 1110₂, from binary to decimal using the binary to decimal formula.
Solution: We start doing the conversion from the rightmost digit, which is '0' here.
(Decimal Number)₁₀ = (d₀ × 2⁰) + (d₁ × 2¹ )+ (d₂ × 2²)+ ..... (d_n-1 × 2^n-1),
= (0 × 2⁰) + (1 × 2¹) + (1 × 2²) + (1 × 2³)
= (0 × 2⁰) + (1 × 2¹) + (1 × 2²) + (1 × 2³)
= 0 + 2 + 4 + 8
= 14
Therefore, 1110₂ = 14₁₀

Binary to Decimal Conversion Chart

The binary to decimal conversion of the first 20 decimal numbers is displayed in the chart given below.

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 convert a number given in binary number system to its equivalent in decimal number system - Binary to Decimal Calculator