Convert Decimal to Binary Using Repeated Division

Converting a positive decimal (base-10) integer into its binary (base-2) equivalent is a fundamental computing concept that can be accomplished quickly using the repeated division-by-2 method. This article explains the step-by-step algorithm for performing this conversion, demonstrates the process with a concrete example, and explains how to correctly read the resulting binary digits.

The Repeated Division Algorithm

The repeated division method involves dividing the given decimal number by 2 successively until the quotient becomes zero, while recording the remainder at each step.

Follow these steps:

  1. Divide by 2: Divide the positive decimal integer by 2 using integer division.
  2. Record the Remainder: Note the remainder of the division. In base-2 arithmetic, the remainder will always be either 0 or 1.
  3. Update the Dividend: Take the integer quotient from the division and use it as the new number to divide.
  4. Repeat: Repeat steps 1 through 3 until the integer quotient reaches 0.
  5. Construct the Binary Number: Write down the remainders in reverse order—starting from the last remainder calculated (the Most Significant Bit, or MSB) to the first remainder calculated (the Least Significant Bit, or LSB).

Step-by-Step Example: Convert Decimal 29 to Binary

To convert the decimal number 29 to binary:

  1. \(29 \div 2 = 14\) with a remainder of 1 (LSB)
  2. \(14 \div 2 = 7\) with a remainder of 0
  3. \(7 \div 2 = 3\) with a remainder of 1
  4. \(3 \div 2 = 1\) with a remainder of 1
  5. \(1 \div 2 = 0\) with a remainder of 1 (MSB)

The division stops because the quotient is now 0.

Reading the Result

Collect the remainders in reverse order (from bottom to top):

\[\text{Remainders: } 1, 1, 1, 0, 1\]

Therefore, the decimal number 29 equals \(11101_2\) in binary notation.

Why This Method Works

The binary number system is positional and based on powers of 2 (\(2^0, 2^1, 2^2, 2^3, \dots\)). Each division by 2 strips away the lowest power of 2 from the number, exposing whether that specific power is present (1) or absent (0) via the remainder. Reading the remainders from last to first correctly places each bit in its corresponding positional weight from highest to lowest.