Binary to Decimal

Understanding Binary and Decimal Systems:

Binary System:
Binary is a base-2 numbering system, meaning it only uses two digits, 0 and 1. Each digit's place in a binary number represents a power of 2. For example, the rightmost digit represents \(2^0\), the next represents \(2^1\), then \(2^2\), and so on.

Decimal System:
Decimal, on the other hand, is a base-10 numbering system that uses digits from 0 to 9. A power of ten is represented by each digit's location. The rightmost digit represents \(10^0\), the next represents \(10^1\), then \(10^2\), and so forth.

Converting Binary to Decimal:

Use these procedures to convert a binary number to a decimal value:

1. Write Down the Binary Number:
   Start by writing down the binary number you want to convert. For example, let's take the binary number 10110.

2. Identify the Place Values:
   Identify the place values of each digit in the binary number. The rightmost digit represents \(2^0\), the next represents \(2^1\), then \(2^2\), and so on.

3. Multiply and Add: 
   Add up all of the results after multiplying each binary number digit by the corresponding power of two.

4. Example: Converting Binary 10110 to Decimal:
   \[1 \times 2^4 + 0 \times 2^3 + 1 \times 2^2 + 1 \times 2^1 + 0 \times 2^0\]

5. Calculate:
   \[16 + 0 + 4 + 2 + 0 = 22\]

Therefore, 13 is the binary number 1101's decimal equivalent.

 Example Conversion:

Let's take another example:  
Binary number: 1101

\[1 \times 2^3 + 1 \times 2^2 + 0 \times 2^1 + 1 \times 2^0\]  
\[8 + 4 + 0 + 1 = 13\]

Therefore, 13 is the binary number 1101's decimal equivalent. 

Conclusion:

Converting binary numbers to decimal numbers involves understanding the place values of each digit in the binary system and performing the necessary multiplication and addition to find the decimal equivalent. Practice with various binary numbers will help reinforce this conversion process.

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