Converting Between Binary and Decimal

Converting Between Binary and Decimal

Understanding Binary

Binary is a base-2 number system, meaning it only uses two digits: 0 and 1. Each position in a binary number represents a power of 2, starting from the rightmost digit as 20, then 21, 22, and so on.

Converting Binary to Decimal

1. Identify the place values: Write down the powers of 2 for each digit in the binary number, starting from the rightmost digit.

   1011 (binary) = 2^3 2^2 2^1 2^0

2. Multiply the digit by its place value: Multiply each binary digit by its corresponding power of 2.

   1 * 2^3 = 8 0 * 2^2 = 0 1 * 2^1 = 2 1 * 2^0 = 1

3. Sum the results: Add the results from step 2.

   8 + 0 + 2 + 1 = 11

Therefore, 1011 (binary) is equal to 11 (decimal).

Converting Decimal to Binary

1. Find the largest power of 2 less than the decimal number: Identify the largest power of 2 that is smaller than the decimal number.

   Example: Decimal number 13 Largest power of 2 less than 13 is 2^3 = 8

2. Subtract the power of 2 from the decimal number: Subtract the largest power of 2 from the decimal number.

   13 - 8 = 5

3. Repeat steps 1 and 2 with the remaining value: Find the largest power of 2 less than the remaining value (5 in this case).

   Largest power of 2 less than 5 is 2^2 = 4 5 - 4 = 1

4. Continue repeating steps 1 and 2 until the remaining value is 0:

   Largest power of 2 less than 1 is 2^0 = 1 1 - 1 = 0

5. Write a '1' for each power of 2 used and a '0' for unused powers: In our example, we used 23, 22, and 20.

   13 (decimal) = 1101 (binary)

Example:

Binary to Decimal:

• 100101 (binary) = (1 * 25) + (0 * 24) + (0 * 23) + (1 * 22) + (0 * 21) + (1 * 20) = 32 + 4 + 1 = 37 (decimal)

Decimal to Binary:

• 29 (decimal):
  • 24 = 16 (less than 29)
  • 29 - 16 = 13
  • 23 = 8 (less than 13)
  • 13 - 8 = 5
  • 22 = 4 (less than 5)
  • 5 - 4 = 1
  • 20 = 1 (less than 1)
  • 1 - 1 = 0
  • Therefore, 29 (decimal) = 11101 (binary)