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.

Binary to decimal conversion can be done by two methods - the positional notation method and the doubling method. Let us learn how to convert binary to decimal using some binary to decimal conversion examples.

Binary to decimal conversion is done to represent a number given in the 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. Let us learn about the different binary to decimal conversion methods and examples.

Binary to decimal conversion is done to help read large binary numbers easily in a form that humans can understand. But how to convert binary to decimal? There are two methods to convert a number from 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. The steps to convert binary to decimal are as follows:

• Step 1: Multiply each digit starting from the rightmost digit by the powers of 2. Here, we start with 2⁰ and increase the exponent by 1 as we move onto the left side.
• Step 2: 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 exponents of 2 for all the digits starting from the rightmost position. The first power would be 2⁰ and as we move on to the left side 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 right to left is 2⁰, 2¹, 2², 2³, 2⁴, and 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.

• Step 3: Finally, sum up all the products obtained for all the digits in the binary number, which gives the decimal equivalent of the given bu=inary number. i.e., 101101₂ = 45₁₀

☛ Also Check: Binary to Decimal Calculator

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. This method involves the following steps to convert bin to dec. 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 to 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.

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

We use the following conversion formula to convert the binary number d_n-1...d₂d₁d₀ with n digits into decimal:

(Decimal Number)₁₀ = (d₀ × 2⁰) + (d₁ × 2¹) + (d₂ × 2² )+ ..... + d_n-1 × 2^n-1)

Let us see the application of the above binary to decimal formula and learn how to convert binary to decimal using the following example.

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

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

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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, 100₂ in binary when converted to a decimal number is 4₁₀. 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 base-2 number system and the decimal number system is known as the base -10 number system.

How to Convert Binary to Decimal?

To convert binary into decimal:

• Multiply the digits of the binary number from right to left by the exponents 2⁰, 2¹, 2², 2³, ...
• Add all the products and the resultant sum would give the decimal form of the given binary number.

What is the Value of 1001 Binary to Decimal?

The decimal value of binary number 1001 is number 9. 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 2⁰ and the next digit by 2¹ and so on. i.e., 1001₂ = 1 · 2⁰ + 0 · 2¹ + 0 · 2² + 1 · 2³ = 9. Thus, the decimal value is 9.

How to Convert a Binary Number to Decimal Number 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 100₂ from binary to decimal using the positional notation method, the conversion step is as follows. 100₂ = (0 × 2⁰) + (0 × 2¹) + (1 × 2²), which is equal to 0 + 0 + 4. Therefore, 100₂ = 4₁₀.

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 left-most digit and moving towards the right. For example, to convert 110₂ from binary to decimal, we use the steps given below. Here, since we start from the left-most digit, there is no previous number to it. Therefore, we consider the doubled value of the previous number for the left-most 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, 110₂ = 6₁₀

What is the Formula to Convert Binary Number to Decimal?

The formula to convert a binary number to decimal is as follows. Considering the binary number d_n-1...d₂d₁d₀, the decimal equivalent is (d₀ × 2⁰) + (d₁ × 2¹ )+ (d₂ × 2² )+ ..... (d_n-1 × 2^n-1).

Can we Convert 1111.1₂ from Binary to Decimal?

Yes, it is possible to convert 1111.1₂ from binary to decimal. To do this, we first convert the integer part to decimal or a base-10 number. Therefore, the decimal equivalent of 1111₂ = (1 × 2⁰) + (1 × 2¹) + (1 × 2²) + (1 × 2³) , 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 base-10 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 1111.1₂ is 15.5₁₀.

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, (d₀ × 2⁰) + (d₁ × 2¹) + (d₂ × 2²) + (d₃ × 2³) + (d₄ × 2⁴) .....⇒ 10101₂ = (1 × 2⁰) + (0 × 2¹) + (1 × 2²) + (0 × 2³) + (1 × 2⁴) = 21₁₀

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.

0₂ = 0₁₀
1₂ = 1₁₀
10₂ = 2₁₀
11₂ = 3₁₀
100₂ = 4₁₀
101₂ = 5₁₀
110₂ = 6₁₀
111₂ = 7₁₀
1000₂ = 8₁₀
1001₂ = 9₁₀

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 111001₂ from Binary to Decimal Number System Using Conversion Formula?

In order to convert 111001₂ from binary to decimal let us use the Binary to Decimal Formula, (d₀ × 2⁰) + (d₁ × 2¹) + (d₂ × 2²) + (d₃ × 2³) ....., where d₀ is the last digit ⇒ 111001₂ = (1 × 2⁰) + (0 × 2¹) + (0 × 2²) + (1 × 2³) + (1 × 2⁴) + (1 × 2⁵) = 57₁₀

What is 11111 Binary to Decimal?

1111 binary to decimal is equal to 31. This can be done using the formula, (d₀ × 2⁰) + (d₁ × 2¹) + (d₂ × 2²) + (d₃ × 2³) + (d₄ × 2⁴) + (d₅ × 2⁵) .....⇒ 1111₂ = (1 × 2⁰) + (1 × 2¹) + (1 × 2²) + (1 × 2³) + (1 × 2⁴) = 1 + 2 + 4 + 8 + 16 = 31₁₀.