Binary to Decimal Number Conversion

Converting Binary to Decimal Numbers

• Binary numbers (base-2) convert to decimal numbers (base-10).
• Binary digits consist of 0 or 1.

Conversion Process

• Begin from the rightmost digit of the binary number.
• Assign powers of 2 to each digit, starting with 2^0 on the right, increasing to the left.
• Multiply each binary digit by its corresponding power of 2.
• Add the results of each multiplication.
• Ignore any terms with a binary digit of 0; focus on digits that are 1.

Examples

• 1010 (binary) = (1 * 2^3) + (0 * 2^2) + (1 * 2^1) + (0 * 2^0) = 8 + 0 + 2 + 0 = 10 (decimal).
• 10010 (binary) = (1 * 2^4) + (0 * 2^3) + (0 * 2^2) + (1 * 2^1) + (0 * 2^0) = 16 + 0 + 0 + 2 + 0 = 18 (decimal).
• Useful Powers of 2:
  • 2^0 = 1
  • 2^1 = 2
  • 2^2 = 4
  • 2^3 = 8
  • 2^4 = 16
  • 2^5 = 32
  • 2^6 = 64
  • 2^7 = 128
  • 2^8 = 256

Example with Multiple 1s

• 11011 (binary) = (1 * 2^4) + (1 * 2^3) + (0 * 2^2) + (1 * 2^1) + (1 * 2^0) = 16 + 8 + 0 + 2 + 1 = 27 (decimal).

Practice Examples

• Example 1: 1100011 (binary)
  • = (1 * 2^6) + (1 * 2^5) + (0 * 2^4) + (0 * 2^3) + (0 * 2^2) + (1 * 2^1) + (1 * 2^0)
  • = 64 + 32 + 0 + 0 + 0 + 2 + 1 = 99 (decimal).
• Example 2: 01010110 (binary)
  • = (0 * 2^7) + (1 * 2^6) + (0 * 2^5) + (1 * 2^4) + (0 * 2^3) + (1 * 2^2) + (1 * 2^1) + (0 * 2^0)
  • = 0 + 64 + 0 + 16 + 0 + 4 + 2 + 0 = 86 (decimal).

Converting Fractional Binary Numbers

• For binary numbers that include a fractional part, digits to the right of the decimal point are assigned negative powers of 2, beginning with 2^-1.
• Use the same process as with whole binary numbers, but include the negative powers of 2.

Examples of Negative Powers of 2

• 2^-1 = 1/2
• 2^-2 = 1/4
• 2^-3 = 1/8
• 2^-4 = 1/16
• 2^-5 = 1/32
• 2^-6 = 1/64
• 2^-7 = 1/128

Fractional Binary Number Conversion Example

• 1100.101 (binary)
  • = (1 * 2^3) + (1 * 2^2) + (0 * 2^1) + (0 * 2^0) + (1 * 2^-1) + (0 * 2^-2) + (1 * 2^-3)
  • = 8 + 4 + 0 + 0 + (1/2) + 0 + (1/8)
  • = 12 + 0.5 + 0.125 = 12.625 (decimal).

Another Fractional Binary Number Conversion Example

• 10011.011 (binary) = (1 * 2^4) + (0 * 2^3) + (0 * 2^2) + (1 * 2^1) + (1 * 2^0) + (0 * 2^-1) + (1 * 2^-2) + (1 * 2^-3)
  • = 16 + 0 + 0 + 2 + 1 + 0 + (1/4) + (1/8)
  • = 19 + 0.25 + 0.125 = 19.375 (decimal).

Learn how to convert binary numbers (base-2) to decimal numbers (base-10). Each binary digit is multiplied by powers of 2, starting from 2^0 on the right. The results are then added to obtain the decimal equivalent.