Binary to Decimal

• Introduction
• Understanding Binary to Decimal Conversion
• Methods of Conversion
• Conversion Using Positional Notation Method
• Conversion Using Doubling Method
• Formula for Binary to Decimal
• Table for Binary to Decimal
• Solved Examples
• Practice Problems
• Frequently Asked Questions

Introduction

Binary to decimal conversion is the process of changing a number from binary form, which uses only `0s` and `1s`, to its equivalent in the decimal system, which we commonly use. A binary number system is a number system where the numeric values are represented using only `2` numbers (hence the name binary), `0` and `1`. 

The decimal number system (also known as the base `10` numeral system) uses ten digits from `0` to `9` to represent numbers. The binary system is specifically used in computers, while the decimal system is what we're used to in mathematical calculations.

There are two main ways to do this conversion: one involves the position of each digit, and the other involves doubling values.

Understanding Binary to Decimal Conversion

Binary to decimal conversion is a way to change binary numbers into decimal numbers. A number system helps us represent numbers, and its base is determined by the number of digits it has. For instance, binary has a base of `2` since it only uses `0` and `1`, while decimal has a base of `10` because it uses digits from `0` to `9`. This conversion is crucial as it allows us to understand numbers written in `0s` and `1s`. It's like translating between two different languages, but with numbers!

Binary System: The Binary System uses only two numbers, `0` and `1`, to write numbers. Its base is `2`. Ancient Indian, Chinese, and Egyptian people used this system for different things. Nowadays, we use the binary system in electronics and computer programming.

Decimal System: The Decimal Number System is what we use every day. Its base is `10` and uses the numbers `0` through `9`.

Methods of Conversion

Converting from binary to decimal is essential for making large binary numbers readable for humans. There are two primary methods to do so:

`1`. Positional Notation Method

`2`. Doubling Method

Now, let's take a closer look at these methods for converting binary numbers to decimal.

Conversion Using Positional Notation Method

In the positional notation method, the value of each digit in a number is determined by its position. To convert binary numbers to decimal, follow these steps:

Step `1`: Write down the binary number. 

Step `2`: Then, assign powers of `2` to each digit from right to left, starting with `0`.

Step `3`: Multiply each binary digit with its corresponding power of `2`, moving from right to left.

Step `4`: Add up all these products to find the decimal equivalent.

Let's illustrate this with an example.

Example: Convert the binary number `1011` to a decimal number using the positional notation.

Solution: When converting from binary to decimal, it's important to consider each digit in the binary number from right to left.

Step `1`: Write down the binary number \( 1011_2 \).

Step `2`: Assign powers of `2` to each digit from right to left, starting with `0`. So, we have:

\( 1 \times 2^0  \quad \text{(for the rightmost digit)} \)

\( 1 \times 2^1  \quad \text{(for the second digit from the right)} \)

\( 0 \times 2^2  \quad \text{(for the third digit from the right)}  \)

\( 1 \times 2^3  \quad \text{(for the leftmost digit)} \)

Step `3`: Multiply each binary digit by its corresponding power of `2`:

\( 1 \times 2^0  = 1 \times 1 = 1 \)

\( 1 \times 2^1  = 1 \times 2 = 2 \)

\( 0 \times 2^2  = 0 \times 4 = 0 \)

\( 1 \times 2^3  = 1 \times 8 = 8 \)

Step `4`: Add up all these products:

\( 1 + 2 + 0 + 8 = 11 \)

So, the decimal equivalent of the binary number \( 1011_2 \) is \( 11_{10} \).

Conversion Using Doubling Method

In the doubling Method, you convert binary numbers to decimal by starting with a decimal value of `0` and doubling it for each binary digit encountered from left to right. To convert binary numbers to decimal, follow these steps:

Step `1`: Write down the binary number. 

Step `2`: Start with a decimal value of `0` and begin from the leftmost digit of the binary number and move towards the rightmost digit.

Step `3`: For each digit encountered:

• Double the current decimal value.
• If the binary digit is `1`, add `1` to the doubled decimal value.

Step `4`: Continue this process for each digit until you reach the rightmost digit.

Step `5`: The final value obtained after the doubling and adding process is the decimal equivalent of the given binary number.

Let's illustrate this with an example.

Example: Convert the binary number `1011` to a decimal number using the doubling method.

Solution: When converting from binary to decimal using the doubling method, it's important to consider each digit in the binary number from left to right.

\( \text{Step 1:}  \quad \text{Write down the binary number } 1011_2. \)

\( \text{Step 2:}  \quad \text{Start with a decimal value of } 0. \)

\( \text{Step 3:}  \quad \text{Begin from the leftmost digit } 1. \)

\( \text{Step 4:}  \quad \text{Double the current decimal value (0) to get } 0 \times 2 = 0. \text{ Since the binary digit is 1, add 1 to the doubled value, resulting in } 0 + 1 = 1. \)

\( \text{Step 5:}  \quad \text{Move to the next digit } 0. \)

\( \text{Step 6:}  \quad \text{Double the current decimal value (1) to get } 1 \times 2 = 2. \text{ Since the binary digit is 0, no addition is needed, so the value remains } 2. \)

\( \text{Step 7:}  \quad \text{Move to the next digit } 1. \)

\( \text{Step 8:}  \quad \text{Double the current decimal value (2) to get } 2 \times 2 = 4. \text{ Since the binary digit is 1, add 1 to the doubled value, resulting in } 4 + 1 = 5. \)

\( \text{Step 9:} \quad \text{Move to the next digit } 1. \)

\( \text{Step 10:} \quad \text{Double the current decimal value (5) to get } 5 \times 2 = 10. \text{ Since the binary digit is 1, add 1 to the doubled value, resulting in } 10 + 1 = 11. \)

\( \text{Step 11:}  \quad \text{The final value obtained is } 11, \text{ which is the decimal equivalent of the binary number } 1011_2. \)

Formula for Binary to Decimal

We can formularize the positional notation method to transform a binary number into a decimal using binary to decimal formula. In the positional notation method, we execute a series of multiplication operations on each digit of the binary number, moving from right to left. Each digit is multiplied by increasing powers of `2`, beginning with `0`, and then we sum up all the results to obtain the decimal equivalent.

We use the following conversion formula to convert the binary number `d_{n-1},...,d_2,d_1,d_0` with `n` digits into decimal:

$$(\text{Decimal Number})_{10} = (d_0 \times 2^0) + (d_1 \times 2^1) + (d_2 \times 2^2) + \ldots + (d_{n-1} \times 2^{n-1})$$

Let's illustrate this with an example.

Example: Convert the binary number `1101` to a decimal number using the conversion formula.

Solution: Start the conversion process from the rightmost digit, indicated as `1` here.

\( (\text{Decimal Number})_{10} = (d_0 \times 2^0) + (d_1 \times 2^1) + (d_2 \times 2^2) + \ldots + (d_{n-1} \times 2^{n-1}) \)

\( = (1 \times 2^0) + (0 \times 2^1) + (1 \times 2^2) + (1 \times 2^3) \)

\( = (1 \times 1) + (0 \times 2) + (1 \times 4) + (1 \times 8) \)

\( = 1 + 0 + 4 + 8 \)

\( = 13 \)

Therefore, `1101_2 = 13_{10}`

Table for Binary to Decimal

The conversion of the initial `15` decimal numbers into binary is presented in the table below.

Solved Examples

Example `1`. Convert the binary number `11001` to a decimal number using the positional notation.

Solution:  

\( \text{Binary: }  11001_2 \)

\( \text{Decimal }  = (1 \times 2^0) + (0 \times 2^1) + (0 \times 2^2) + (1 \times 2^3) + (1 \times 2^4) \)

\( = (1 \times 1) + (0 \times 2) + (0 \times 4) + (1 \times 8) + (1 \times 16) \)

\( = 1 + 0 + 0 + 8 + 16 \)

\( = 25_{10} \)

Example `2`. Convert the binary number `11010` to a decimal number using the positional notation.

Solution: 

\( \text{Binary: }  11010_2 \)

\( \text{Decimal }  = (0 \times 2^0) + (1 \times 2^1) + (0 \times 2^2) + (1 \times 2^3) + (1 \times 2^4) \)

\( = (0 \times 1) + (1 \times 2) + (0 \times 4) + (1 \times 8) + (1 \times 16) \)

\( = 0 + 2 + 0 + 8 + 16 \)

\( = 26_{10} \)

Therefore, `11010_2 = 26_{10}`.

Example `3`. Convert the binary number `10110` to a decimal number using the doubling method.

Solution:  Binary Number: \(10110_2\)

Step `1`: Start with `0` and add the first digit.

\(0 \times 2 + 1 = 1\)

Step `2`: Use the result of the previous step, multiply it by `2`, and add the next digit.

\(1 \times 2 + 0 = 2\)

Step `3`: Repeat the process for the remaining digits.

\(2 \times 2 + 1 = 5\)

Step `4`:

\(5 \times 2 + 1 = 11\)

Step `5`:

\(11 \times 2 + 0 = 22\)

Therefore, `10110_2 = 22_{10}`.

Example `4`. Convert the binary number `100110` to a decimal number using the doubling method.

Solution: Binary Number: \(100110_2\)

Step `1`: Start with `0` and add the first digit.

\(0 \times 2 + 1 = 1\)

Step `2`: Use the result of the previous step, multiply it by `2`, and add the next digit.

\(1 \times 2 + 0 = 2\)

Step `3`: Repeat the process for the remaining digits.

\(2 \times 2 + 0 = 4\)

Step `4`:

\(4 \times 2 + 1 = 9\)

Step `5`:

\(9 \times 2 + 1 = 19\)

Step `6`:

\(19 \times 2 + 0 = 38\)

Therefore, `100110_2 = 38_{10}`.

Practice Problems

Q`1`. Convert the binary number \( 1010_2 \) to decimal using the positional notation.

1. `8`
2. `10`
3. `12`
4. `14`

Answer: b

Q`2`. Convert the binary number \( 11001_2 \) to decimal using the doubling method.

1. `17`
2. `19`
3. `23`
4. `25`

Answer: d

Q`3`. Convert the binary number \( 11110_2 \) to decimal using the doubling method.

1. `27`
2. `29`
3. `30`
4. `31`

Answer: c

Q`4`. Convert the binary number \( 100101_2 \) to decimal.

1. `35`
2. `37`
3. `41`
4. `45`

Answer: b

Q`5`. Convert the binary number \( 111111_2 \) to decimal.

1. `61`
2. `63`
3. `67`
4. `69`

Answer: b

Frequently Asked Questions

Q`1`. What is binary to decimal conversion?

Answer: Binary to decimal conversion is the process of converting a binary number (base-`2`) into its equivalent decimal number (base-`10`).

Q`2`. How do I convert a binary number to decimal?

Answer: To convert a binary number to decimal, you multiply each digit of the binary number by increasing powers of `2`, starting from the rightmost digit, and then sum up the results.

Q`3`. What is the significance of binary to decimal conversion?

Answer: Binary to decimal conversion is important in computer science and digital electronics, where binary numbers are commonly used to represent data and decimal numbers are more easily understood by humans.

Q`4`. What is the largest decimal number that can be represented by a given binary number?

Answer: The largest decimal number that can be represented by a binary number depends on the number of bits used in the binary representation. For example, an `8`-bit binary number can represent decimal numbers from `0` to `255`.

Q`5`. Is binary to decimal conversion reversible?

Answer: Yes, binary to decimal conversion is reversible. You can convert a binary number to decimal and then back to binary using the reverse process, known as decimal to binary conversion. However, ensure that you use the same number of bits to avoid loss of information.