Binary to Decimal Number Conversion

Converting Binary to Decimal Numbers

• Conversion from binary to decimal requires understanding that binary is base-2 and decimal is base-10.
• Binary digits are 0 or 1.
• Each binary digit is multiplied by 2 raised to the power of its position, starting from 0 on the right.

Conversion Process Explained

• Begin with the rightmost digit, which represents 2 to the power of 0.
• Moving left, the positions are 2 to the first power, 2 squared, 2 cubed, and so on.
• Digits with a value of 0 do not contribute to the decimal value.
• Focus on digits that are 1 and sum their calculated values. That is, 2 to the power of their position.

Example Conversions

• Binary number 1010 converts to decimal 10.
  • (0 * 2^0) + (1 * 2^1) + (0 * 2^2) + (1 * 2^3) = 0 + 2 + 0 + 8 = 10
• Binary number 10010 converts to decimal 18.
  • (0 * 2^0) + (1 * 2^1) + (0 * 2^2) + (0 * 2^3) + (1 * 2^4) = 0 + 2 + 0 + 0 + 16 = 18

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
• Knowing these powers facilitates quicker conversions.

Additional Examples

• Binary number 11011 converts to decimal 27.
  • (1 * 2^0) + (1 * 2^1) + (0 * 2^2) + (1 * 2^3) + (1 * 2^4) = 1 + 2 + 0 + 8 + 16 = 27

Practice Problems

• Convert the binary numbers 10110011 and 01010110 into decimal numbers.

Solutions to Practice Problems

• Binary 10110011 equals decimal 179.
  • (1 * 2^0) + (1 * 2^1) + (0 * 2^2) + (0 * 2^3) + (1 * 2^4) + (1 * 2^5) + (0 * 2^6) + (1 * 2^7) = 1 + 2 + 0 + 0 + 16 + 32 + 0 + 128 = 179
• Binary 01010110 equals decimal 86.
  • (0 * 2^0) + (1 * 2^1) + (1 * 2^2) + (0 * 2^3) + (1 * 2^4) + (0 * 2^5) + (1 * 2^6) + (0 * 2^7) = 0 + 2 + 4 + 0 + 16 + 0 + 64 + 0 = 86

Converting Fractional Binary Numbers

• For binary fractions, digits to the right of the decimal point represent negative powers of 2, starting with 2^-1.
• The pattern associates the first digit after the decimal with 2^-1, the second with 2^-2, and so on.

Example and Explanation

• The binary number 1100.101 is equivalent to 12.625 in decimal.
  • Digits to the left of the decimal are 2^0, 2^1, 2^2, 2^3 from right to left.
  • Digits to the right of the decimal are 2^-1, 2^-2, 2^-3 from left to right.
  • (0 * 2^0) + (0 * 2^1) + (1 * 2^2) + (1 * 2^3) + (1 * 2^-1) + (0 * 2^-2) + (1 * 2^-3) = 0 + 0 + 4 + 8 + 0.5 + 0 + 0.125 = 12.625

Negative Powers of 2

• 2^-1 = 1/2 = 0.5
• 2^-2 = 1/4 = 0.25
• 2^-3 = 1/8 = 0.125
• 2^-4 = 1/16
• 2^-5 = 1/32
• 2 to the negative power is the reciprocal of 2 to the positive power. For example, 2^-n = 1 / 2^n

Practice with Fractional Binary Numbers

• Convert the binary number 10011.011 into a decimal number by assigning the appropriate values to each digit.

Solution

• The binary number 10011.011 equals 19.375 in decimal form.
  • (1 * 2^0) + (1 * 2^1) + (0 * 2^2) + (0 * 2^3) + (1 * 2^4) + (0 * 2^- 1) + (1 * 2^-2) + (1 * 2^-3) = 1 + 2 + 0 + 0 + 16 + 0 + 0.25 + 0.125 = 19.375

Learn how to convert binary numbers to decimal numbers. Understand the base-2 system and the conversion process. Each binary digit is multiplied by 2 raised to the power of its position.