Binary to Decimal Formula

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Binary to Decimal Converter is a free online tool to convert binary to decimal. Converting between binary to decimal is a common task in everyday life. Here, Tam Tai Duc provides a free user-friendly, and efficient online binary decimal Conversion tool to simplify this process and ensure accuracy. It is a fast, easy-to-use general-purpose calculator that can be used in any field such as computer science etc. Furthermore, it also helps students and working professionals to solve a wide range of day-to-day problems.

What is Binary to Decimal?

Binary To Decimal Conversion is used to convert the binary value to the decimal values. Binary numbers are the numbers that have a base of 2 and are used in computer programming. Whereas Decimal numbers are the numbers that have a base of 10 and used in normal day to day operations.

What is Binary System?

Binary System is the system of writing number using only two numbers that are, 0 and 1. The base of binary number is 2. This system was first used by ancient Indian, Chinese, and Egyptian people for various purposes. The binary number system is used in electronic and computer programmings.

What is Decimal System?

The Decimal Numbers System is the number system that is used by us in our daily lives. The base of Decimal numbers is 10 and it uses 10 digits that are, 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.

How to use Binary to Decimal Calculator?

We can easily use the binary-to-decimal calculator by following the steps discussed below,

Step 1: Enter the given value in the binary input field.

Step 2: Click on the convert button to convert the binary value into the decimal value.

Step 3: The value shown as the result is the required value in the decimal form.

Binary to Decimal Formula

To convert a binary number to decimal we need to perform a multiplication operation on each digit of a binary number from right to left with powers of 2 starting from 0 and add each result to get the decimal number of it. 

Decimal Number = nth bit × 2n-1

Binary to Decimal Formula

n = bnq + bn-1qn-2 +………+ b2q2 +b1q1 +b0q0 + b-1q-1 + b-2q-2 

Where,

  • N is Decimal Equivalent
  • b is the Digit
  • q is the Base Value

How to Convert Binary to Decimal

You just have to follow the below steps to convert binary numbers to their decimal equivalent.

Step 1: Write the binary number and count the powers of 2 from right to left (starting from 0).

Step 2: Write each binary digit(right to left) with corresponding powers of 2 from right to left, such that MSB or the first binary digit will be multiplied by the greatest power of 2.

Step 3: Add all the products in the step 2

Step 4: The answer is our decimal number.

This can be better explained using the below examples.

Binary to Decimal Conversion

Binary to Decimal conversion is achieved using the two steps that are,

  • Positional Notation Method
  • Doubling Method

Now let’s learn about them in detail.

Method 1: Using Positions

Binary to Decimal Conversion can be achieved using the example added below.

Example 1: Let’s consider a binary number 1111. We need to convert this binary number to a decimal number.

As mentioned in the above paragraph while converting from binary to decimal we need to consider each digit in binary number from right to left.

By this way, we can do binary to decimal conversion.

Note: We represent any binary number with this format (xxxx)and decimal in (xxxx)10 format.

Example 2: Convert (101010)= (?)10

We keep on increasing the power of 2 as long as number of digits in binary number increases.

Example 3: Convert (11100)= (?)10

Resultant Decimal number = 0+0+4+8+16 = 28

So (11100)= (28)10

Method 2: Doubling Method

To explain this method we will consider an example and try to solve that stepwise.

Example 1: Convert Binary number (10001)2 to decimal.

Similar to the above approach, In this approach also consider each digit but from left to right and performs step-wise computations on it.

10001

Step-1 First we need to multiply 0 with 2 and add the 1st digit in binary number.

0 x 2 + = 0 + 1 = 1

Step-2 Now use the result of above step and multiply with 2 and add the second digit of binary number.

10001

1 x 2 + 0 = 2 + 0 = 2

The same step 2 is repeated until there will be no digit left. The final result will be the resultant decimal number.

10001

2 x 2 + 0 = 4 + 0 = 4

10001

4 x 2 + 0 = 8 + 0 = 8

10001

8 x 2 + 1 = 16 + 1 = 17

So we performed step 2 on all remaining numbers and finally, we left with result 17 which is a decimal number for the given binary number.

So (10001)= (17)10

Example 2: Convert (111)2 to decimal using doubling approach.

111

0 x 2 + = 0 + 1 = 1

111

1 x 2 + 1 = 2 + 1 = 3

111

3 x 2 + 1 = 6 + 1 = 7

The final result is 7 which is a Decimal number for 111 binary numeral system. So (111)= (7)10

These are the 2 approaches that can be used or applied to convert binary to decimal.

How to Read a Binary Number?

Binary numbers are read by separating them into separate digits. Each digit in binary is represented using 0 and 1 and they are the powers of 2 starting from left hand side and then the power is gradually increased from 0 to (n-1).

Binary to Decimal Conversion Table

The given binary to decimal conversion table will help you to convert binary to decimal.

Decimal NumberBinary Number
00
11
210
311
4100
5101
6110
7111
81000
91001
101010
111011
121100
131101
141110
151111
1610000
1710001
1810010
1910011
2010100
2110101
2210110
2310111
2411000
2511001
2611010
2711011
2811100
2911101
3011110
3111111
32100000
641000000
12810000000
256100000000

Conclusion

In conclusion, the Binary to Decimal Calculator is a free online tool prepared by GeekforGeeks that converts the given value of the binary number system into the value of a decimal number system . It is a fast and easy-to-use tool that helps students solve various problems.

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 2n-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

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.

Binary to Decimal Conversion Formula

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

(Decimal Number)10 = (d0 × 20) + (d1 × 21) + (d2 × 2)+ ….. + dn-1 × 2n-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 = (d× 20) + (d1 × 21) + (d2 × 2)+ ….. (dn-1 × 2n-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.

BinaryDecimal
00
11
102
113
1004
1015
1106
1117
10008
10019
101010
101111
110012
110113
111014
111115
1000016
1000117
1001018
1001119
1010020

Binary to Decimal Conversion Examples

Example 1: Convert (111)2 to Decimal.

Solution:

We have (111)2 in binary

⇒ 1 ⨯ 22 + 1 ⨯ 21 + 1 ⨯ 20

= 4 + 2 + 1 = 7

Example 2: Convert (10110)2 to Decimal.

Solution:

We have (10110)2 in Binary

1 ⨯ 24 + 0 ⨯ 23 + 1 ⨯ 22 + 1 ⨯ 2+ 0 ⨯ 20

= 16 + 4 + 2 = 22

Example 3: Convert (10001)2 to Decimal.

Solution:

We have (10001)2 in Binary

⇒ 1 ⨯ 24 + 0 ⨯ 23 + 0 ⨯ 22 + 0 ⨯ 21 + 1 ⨯ 20

= 16 + 0 + 0 + 0 + 1 = 17

Example 4: Convert (1010)2 to Decimal.

Solution:

We have (1010)2 in Binary

⇒ 1 ⨯ 23 + 0 ⨯ 22 + 1 ⨯21 + 0 ⨯ 20

= 0 + 8 + 2 + 0 = 10

Example 5: Convert (10101101)to Decimal.

Solution:

Question 2: Convert the binary number 10100011 to decimal.

Solution:

Given binary number is 10100011

Using the conversion formula,

10100011 = (1 × 27) + (0 × 26) + (1 × 25) + (0 × 24) + (0 × 23) + (0 × 22) + (1 × 21) + (1 × 20)

= 128 + 0 + 32 + 0 + 0 + 0 + 2 + 1

= 163

Therefore, binary number 10100011 = 163 decimal number

Question 3: Convert the binary number 11101111 to decimal.

Solution:

Given binary number is 11101111

Using the conversion formula,

11101111 = (1 × 27) + (1 × 26) + (1 × 25) + (0 × 24) + (1 × 23) + (1 × 22) + (1 × 21) + (1 × 20)

= 128 + 64 + 32 + 0 + 8 + 4 + 2 + 1

= 239

Therefore, binary number 11101111 = 239 decimal number

Example 1: Find the decimal value of the binary number 11001011using 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= (1 × 20)+ (1 × 21)+ (0 × 22)+ (1 × 23) + (0 × 24) + (0 × 25) + (1 × 26) + (1 × 27)= (1 × 1) + (1 × 2) + (0 × 4) + (1 × 8) + (0 × 16) + (0 × 32) + (1 × 64) + (1 × 128)= 1 + 2 + 0 + 8 + 0 + 0 + 64 +128= 203

Answer: ∴ 110010112 = 20310

Example 2: Using the doubling method of binary to decimal conversion, find the decimal value of 101011012

Solution:

For the binary to decimal conversion of a number using the doubling method, we use the following steps:

  • Step 1: The binary number 101011012, ‘1’ is the left-most digit and there is no previous number, take the doubled value to be 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:

Answer: ∴ 101011012 = 17310

Example 3: Fill in the blanks with respect to binary to decimal conversion.a.) 10112 binary to decimal is __.b.) The binary number 101012 is equivalent to the decimal number __.

Solution:a.) 1011 binary to decimal is 11.b.) The binary number 10101 is equivalent to the decimal number 21.

Answer: (a) 11 (b) 21

Binary to Decimal(bn to dec) Conversion-FAQs

1. What is Binary System?

Binary describes a numbering system in which there exist only two possible values for each digit 0 and 1. Each digit is expressed with the help of two digits only 0 and 1.

2. What is Decimal System?

The decimal number system is the base 10 number system. It uses the digits from 0 to 9 to represent values. 

3. What is Binary to Decimal Conversion Formula?

Decimal Number = nth bit * 2n-1

Binary to Decimal Formula: n = bnq + bn-1qn-2 +………+ b2q2 +b1q1 +b0q0 + b-1q-1 + b-2q-2 

Where,

  • N is Decimal equivalent
  • b is the Digit
  • q is the Base value

4. How to Convert from Binary to Decimal?

To convert Binary to Decimal use the Binary to Decimal converter added above in the article.

5. What is the Binary Number 01010101 to a Decimal?

The binary number (01010101) in decimals is equal to 85, i.e. (01010101)2 = (85)10

6. What is 1001 0011 in Decimal?

The value of 1001 0011 in decimal is 204, i.e. (1001 0011)2 = (204)10

7. What is Binary Number for 3?

The binary Number for 3 is 11, i.e. (3)10 = (11)2

8. What is Binary Number for 6?

The binary Number for 6 is 11, i.e. (6)10 = (110)2

9. What is Binary Number for 8?

The binary Number for 8 is 11, i.e. (8)10 = (1000)2

10. What is Binary Number for 7?

The binary Number for 7 is 11, i.e. (7)10 = (111)2

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 dn-1…d2d1d0, the decimal equivalent is (d0 × 20) + (d1 × 2)+ (d2 × 2)+ ….. (dn-1 × 2n-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, (d× 20) + (d× 21) + (d× 22) + (d× 23) + (d× 24) …..⇒ 10101= (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, (d× 20) + (d× 21) + (d× 22) + (d× 23) ….., where d0 is the last digit ⇒ 111001= (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, (d× 20) + (d× 21) + (d× 22) + (d× 23) + (d× 24) + (d× 25) …..⇒ 1111= (1 × 20) + (1 × 21) + (1 × 22) + (1 × 23) + (1 × 24) = 1 + 2 + 4 + 8 + 16 = 3110.

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