What is Binary to Decimal Conversion?

Binary to decimal conversion is the process of converting a number from its binary (base-2) representation to its decimal (base-10) representation. Binary numbers are commonly used in computing and digital electronics, while decimal numbers are more commonly used in everyday life.

Understanding Binary Numbers

A binary number is a number that is represented using only the digits 0 and 1. Each digit in a binary number represents a power of 2, with the rightmost digit representing $2^0$, the next digit to the left representing $2^1$, and so on. For example, the binary number 1011 represents the decimal number 11:

$$1 \times 2^3 = 8$$

$$0 \times 2^2 = 0$$

$$1 \times 2^1 = 2$$

$$1 \times 2^0 = 1$$

$$8 + 0 + 2 + 1 = 11$$

Converting Binary to Decimal

To convert a binary number to a decimal number, you can follow these steps:

1. Start with the rightmost digit of the binary number.
2. Multiply this digit by the corresponding power of 2.
3. Repeat steps 1 and 2 for each digit in the binary number.
4. Add the results of all the multiplications to get the decimal number.

For example, to convert the binary number 1011 to decimal, we would follow these steps:

1. Start with the rightmost digit, which is 1.
2. Multiply this digit by 2^0, which is 1.
3. Repeat steps 1 and 2 for each digit in the binary number: $1 \times 2^0 = 1$ $1 \times 2^1 = 2$ $0 \times 2^2 = 0$ $1 \times 2^3 = 8$
4. Add the results of all the multiplications to get the decimal number:

$$1 + 2 + 0 + 8 = 11$$

Therefore, the binary number 1011 is equal to the decimal number 11.

Binary to decimal conversion is a fundamental skill in computer science and digital electronics. By understanding how to convert between binary and decimal numbers, you can work with binary data and perform various operations on binary numbers.

Binary to Decimal Conversion Method

Binary numbers are base-2 numbers, which means they use only two digits, 0 and 1. Decimal numbers are base-10 numbers, which means they use ten digits, 0 through 9. To convert a binary number to a decimal number, you need to multiply each digit of the binary number by the corresponding power of 2 and then add the results together.

For example, to convert the binary number 1011 to decimal, you would:

1. Multiply the rightmost digit, 1, by $2^0$ = 1.
2. Multiply the next digit, 0, by $2^1$ = 2.
3. Multiply the next digit, 1, by $2^2$ = 4.
4. Multiply the leftmost digit, 1, by $2^3$ = 8.
5. Add the results together: 1 + 0 + 4 + 8 = 13.

So, the binary number 1011 is equal to the decimal number 13.

Steps for Binary to Decimal Conversion

Here are the steps for converting a binary number to a decimal number:

1. Write the binary number down.
2. Starting from the rightmost digit, multiply each digit by the corresponding power of 2.
3. Add the results together.
4. The final result is the decimal equivalent of the binary number.

Example

Let’s convert the binary number 1101 to decimal.

1. Write the binary number down: 1101
2. Starting from the rightmost digit, multiply each digit by the corresponding power of 2:
   • $1 \times 2^0 = 1$
   • $0 \times 2^1 = 0$
   • $1 \times 2^2 = 4$
   • $1 \times 2^3 = 8$
3. Add the results together: 1 + 0 + 4 + 8 = 13
4. The final result is the decimal equivalent of the binary number: 13

Conclusion

Binary to decimal conversion is a simple process that can be done by following the steps outlined above. With a little practice, you’ll be able to convert binary numbers to decimal numbers quickly and easily.

Binary to Decimal Conversion Formula

Binary numbers are base-2 numbers, which means they use only two digits, 0 and 1. Decimal numbers are base-10 numbers, which means they use ten digits, 0 through 9. To convert a binary number to a decimal number, you need to multiply each digit of the binary number by the corresponding power of 2 and then add the results together.

For example, to convert the binary number 1011 to decimal, you would:

1. Multiply the rightmost digit, 1, by $2^0$ = 1.
2. Multiply the next digit, 0, by $2^1$ = 2.
3. Multiply the next digit, 1, by $2^2$ = 4.
4. Multiply the leftmost digit, 1, by $2^3$ = 8.
5. Add the results together: 1 + 0 + 4 + 8 = 13.

So, the binary number 1011 is equal to the decimal number 13.

General Formula

The general formula for converting a binary number to a decimal number is:

decimal number = (b_n * 2ⁿ) + (b_n-1 * 2^n-1) + … + (b₁ * 2¹) + (b₀ * 2⁰)

where:

• b_n is the leftmost digit of the binary number
• b_n-1 is the next digit to the right of b_n
• …
• b₁ is the rightmost digit of the binary number
• b₀ is the digit to the right of b₁

Example

To convert the binary number 110101 to decimal, you would use the following formula:

decimal number = (1 * 2⁵) + (1 * 2⁴) + (0 * 2³) + (1 * 2²) + (0 * 2¹) + (1 * 2⁰)

decimal number = (32) + (16) + (0) + (4) + (0) + (1)

decimal number = 53

So, the binary number 110101 is equal to the decimal number 53.

Conclusion

Binary to decimal conversion is a simple process that can be done by hand or with a calculator. The general formula for converting a binary number to a decimal number is:

decimal number = (b_n * 2ⁿ) + (b_n-1 * 2^n-1) + … + (b₁ * 2¹) + (b₀ * 2⁰)

where:

• b_n is the leftmost digit of the binary number
• b_n-1 is the next digit to the right of b_n
• …
• b₁ is the rightmost digit of the binary number
• b₀ is the digit to the right of b₁

Binary to Decimal Conversion Table

Binary numbers are base-2 numbers, which means they use only two digits, 0 and 1. Decimal numbers are base-10 numbers, which means they use ten digits, 0 through 9.

To convert a binary number to a decimal number, you can use the following steps:

1. Write the binary number down.
2. Starting from the rightmost digit, multiply each digit by the corresponding power of 2.
3. Add up the results of step 2.

For example, to convert the binary number 1011 to decimal, we would:

1. Write the binary number down: 1011
2. Starting from the rightmost digit, multiply each digit by the corresponding power of 2:
   • $1\times2^0 = 1$
   • $1\times2^1 = 2$
   • $0\times2^2 = 0$
   • $1\times2^3 = 8$
3. Add up the results of step 2: 1 + 2 + 0 + 8 = 11

Therefore, the decimal equivalent of the binary number 1011 is 11.

Binary to Decimal Conversion Table

The following table shows the binary equivalents of the decimal numbers from 0 to 15:

Binary numbers are used in a variety of applications, including computing, telecommunications, and data storage. By understanding how to convert binary numbers to decimal numbers, you can better understand how these applications work.

Solved Examples of Binary to Decimal Conversion

Binary to decimal conversion is the process of converting a binary number (base 2) to its equivalent decimal number (base 10). Here are a few solved examples to illustrate the conversion process:

Example 1: Convert 11010110₂ to decimal

Step 1: Start from the rightmost bit and assign powers of 2 to each bit position.

$1 1 0 1 0 1 1 0$
$2^7 2^6 2^5 2^4 2^3 2^2 2^1 2^0$

Step 2: Multiply each bit by its corresponding power of 2.

$1 × 2^7 = 128$
$1 × 2^6 = 64$
$0 × 2^5 = 0$
$1 × 2^4 = 16$
$0 × 2^3 = 0$
$1 × 2^2 = 4$
$1 × 2^1 = 2$
$0 × 2^0 = 0$

Step 3: Add up the products to get the decimal equivalent.

128 + 64 + 0 + 16 + 0 + 4 + 2 + 0 = 214

Therefore, 11010110₂ = 214₁₀.

Example 2: Convert 1001110110₂ to decimal

Follow the same steps as in Example 1.

Step 1: Assign powers of 2 to each bit position.

$1 0 0 1 1 1 0 1 1 0$
$2^9 2^8 2^7 2^6 2^5 2^4 2^3 2^2 2^1 2^0$

Step 2: Multiply each bit by its corresponding power of 2.

$1 × 2^9 = 512$
$0 × 2^8 = 0$
$0 × 2^7 = 0$
$1 × 2^6 = 64$
$1 × 2^5 = 32$
$1 × 2^4 = 16$
$0 × 2^3 = 0$
$1 × 2^2 = 4$
$1 × 2^1 = 2$
$0 × 2^0 = 0$

Step 3: Add up the products to get the decimal equivalent.

512 + 0 + 0 + 64 + 32 + 16 + 0 + 4 + 2 + 0 = 630

Therefore, 1001110110₂ = 630₁₀.

Example 3: Convert 11111111₂ to decimal

Step 1: Assign powers of 2 to each bit position.

$1 1 1 1 1 1 1 1$
$2^7 2^6 2^5 2^4 2^3 2^2 2^1 2^0$

Step 2: Multiply each bit by its corresponding power of 2.

$1 × 2^7 = 128$
$1 × 2^6 = 64$
$1 × 2^5 = 32$
$1 × 2^4 = 16$
$1 × 2^3 = 8$
$1 × 2^2 = 4$
$1 × 2^1 = 2$
$1 × 2^0 = 1$

Step 3: Add up the products to get the decimal equivalent.

128 + 64 + 32 + 16 + 8 + 4 + 2 + 1 = 255

Therefore, 11111111₂ = 255₁₀.

These examples demonstrate the process of converting binary numbers to their decimal equivalents by multiplying each bit by its corresponding power of 2 and then adding the products.

Binary To Decimal Conversion FAQs

What is binary to decimal conversion?

Binary to decimal conversion is the process of converting a number from its binary representation (base 2) to its decimal representation (base 10).

Why is binary to decimal conversion important?

Binary to decimal conversion is important because it allows us to work with binary numbers in a more familiar format. Decimal numbers are the numbers we use in everyday life, so converting binary numbers to decimal numbers makes them easier to understand and work with.

How do you convert binary to decimal?

To convert a binary number to decimal, you can follow these steps:

1. Write down the binary number.
2. Starting from the rightmost bit, multiply each bit by its corresponding power of 2.
3. Add up the results of step 2 to get the decimal equivalent of the binary number.

Example of binary to decimal conversion

Let’s convert the binary number 101101 to decimal.

1. Write down the binary number: 101101

2. Starting from the rightmost bit, multiply each bit by its corresponding power of 2:

   $1\times2^0 = 1$
   $0\times2^1 = 0$
   $1\times2^2 = 4$
   $1\times2^3 = 8$
   $0\times2^4 = 0$
   $1\times2^5 = 32$
   

3. Add up the results of step 2:

   1 + 0 + 4 + 8 + 0 + 32 = 45

Therefore, the decimal equivalent of the binary number 101101 is 45.

Common mistakes in binary to decimal conversion

Here are some common mistakes that people make when converting binary numbers to decimal numbers:

• Forgetting to multiply the rightmost bit by $2^0$. This is a common mistake that can lead to an incorrect conversion.
• Multiplying the bits by the wrong power of 2. This can also lead to an incorrect conversion.
• Adding up the results of step 2 incorrectly. This can also lead to an incorrect conversion.

Conclusion

Binary to decimal conversion is a simple but important skill that can be used to work with binary numbers in a more familiar format. By following the steps outlined in this article, you can easily convert binary numbers to decimal numbers.