Conversion Binary <--> Decimal

 Conversion Binary - Decimal

Converting binary numbers to decimal numbers is a fundamental skill in computer science and mathematics. To convert a binary number to decimal, you can follow these steps:

Step 1: Write down the binary number you want to convert.

Step 2: Assign powers of 2 to each digit in the binary number, starting from the right (the least significant bit) and increasing by 1 for each digit to the left.

Step 3: Multiply each digit in the binary number by the corresponding power of 2.

Step 4: Sum up the results from Step 3 to obtain the decimal equivalent.

Let's illustrate this with an example:

Binary number: 1101

Step 1: Write down the binary number: 1101

Step 2: Assign powers of 2 (from right to left): 2^0, 2^1, 2^2, 2^3

Step 3: Multiply each digit by the corresponding power of 2:

- 1 * 2^0 = 1 * 1 = 1

- 0 * 2^1 = 0 * 2 = 0

- 1 * 2^2 = 1 * 4 = 4

- 1 * 2^3 = 1 * 8 = 8

Step 4: Sum up the results from Step 3: 1 + 0 + 4 + 8 = 13

So, the binary number 1101 is equivalent to the decimal number 13.

You can use this method to convert binary numbers of any length to their decimal equivalents. It's a valuable skill in computer science and programming when dealing with binary data or working with low-level hardware.

Another Example: Convert the binary fraction 0.1101 to decimal.

Here's how you can do it step by step:

Step 1. Write Down the Binary Fraction: Start with the binary fraction you want to convert: 0.1101.

Step 2. Assign Powers of 2: Begin from the left of the binary point (decimal point) and assign decreasing negative powers of 2 to each digit. Start with -1 for the rightmost digit after the binary point, then -2 for the next digit, and so on.

Step 3. Multiply and Sum:

   - Multiply each digit by the corresponding power of 2 and sum the results.

   - For the first digit (1 before the binary point):

     - 1 * 2^(-1) = 1/2

   - For the second digit (1 before the binary point):

     - 1 * 2^(-2) = 1/4

   - For the third digit (0 after the binary point):

     - 0 * 2^(-3) = 0

   - For the fourth digit (1 after the binary point):

     - 1 * 2^(-4) = 1/16

Step 4. Calculate the Decimal Equivalent: Sum all the results from step 3:

   - 1/2 + 1/4 + 0 + 1/16 = 8/16 + 4/16 + 0 + 1/16 = (8 + 4 + 0 + 1)/16 = 13/16

So, the binary fraction 0.1101 is equivalent to the decimal fraction 13/16. If you prefer a decimal approximation, you can convert 13/16 to a decimal by dividing the numerator (13) by the denominator (16):

13 ÷ 16 = 0.8125

Therefore, the binary fraction 0.1101 is approximately equal to 0.8125 in decimal.

This demonstrates how to convert a binary fractional number to a decimal fractional number using powers of 2 and addition. This process can be applied to binary fractions of any length.

Converting Decimal to Binary

Converting a decimal number to binary involves representing the decimal value in base-2 using binary digits (0 and 1). Here's how you can convert a decimal number to binary step by step:

Example: Convert the decimal number 27 to binary.

1. Division by 2:

   - Start by dividing the decimal number by 2.

   - Keep track of the remainders for each division, as they will be the binary digits.

Decimal Number: 27

Step 1: 27 ÷ 2 = 13 remainder 1 (Write down the remainder, which is 1)

2. Continue Division:

   - Take the quotient from the previous division (13) and divide it by 2.

   - Again, keep track of the remainders.

Step 2: 13 ÷ 2 = 6 remainder 1

Step 3: 6 ÷ 2 = 3 remainder 0

Step 4: 3 ÷ 2 = 1 remainder 1

Step 5: 1 ÷ 2 = 0 remainder 1

3. Write Down the Binary Digits:

   - The remainders obtained in reverse order (from the last division to the first) form the binary representation of the decimal number.

Binary Representation: 11011

So, the decimal number 27 is equivalent to the binary number 11011.

To verify the conversion, you can convert the binary number back to decimal by adding up the corresponding powers of 2:

1 * 2^4 + 1 * 2^3 + 0 * 2^2 + 1 * 2^1 + 1 * 2^0 = 16 + 8 + 0 + 2 + 1 = 27 (decimal)