Understanding Decimal to Binary Conversion for 29

Converting a decimal number (base 10) into a binary number (base 2) is a fundamental concept in computer science. The most common method for integer conversion is the division-by-2 method. This involves repeatedly dividing the decimal number by 2 and recording the remainder at each step. The binary equivalent is then formed by reading the remainders from bottom to top.

Step-by-Step Conversion of 29 to Binary

Let's convert the decimal number 29 into its binary representation using the division-by-2 method:

1. Divide 29 by 2. The quotient is 14, and the remainder is 1.
2. Divide 14 by 2. The quotient is 7, and the remainder is 0.
3. Divide 7 by 2. The quotient is 3, and the remainder is 1.
4. Divide 3 by 2. The quotient is 1, and the remainder is 1.
5. Divide 1 by 2. The quotient is 0, and the remainder is 1.

We stop when the quotient becomes 0.

Summarizing the Division Steps

Forming the Binary Number

To get the binary equivalent of 29, read the remainders from the bottom up:

• The last remainder is 1 (Most Significant Bit or MSB).
• The next remainder up is 1.
• The next remainder up is 1.
• The next remainder up is 0.
• The first remainder is 1 (Least Significant Bit or LSB).

Reading from bottom to top, the remainders are 1, 1, 1, 0, 1. Therefore, the binary representation of decimal 29 is 11101.

We can write this as $29_{10} = 11101_2$.

Verifying the Binary Result

To verify the binary number $11101_2$, we can convert it back to decimal. The place values in binary are powers of 2, starting from $2^0$ from the rightmost digit.

$11101_2 = (1 \times 2^4) + (1 \times 2^3) + (1 \times 2^2) + (0 \times 2^1) + (1 \times 2^0)$

$= (1 \times 16) + (1 \times 8) + (1 \times 4) + (0 \times 2) + (1 \times 1)$

$= 16 + 8 + 4 + 0 + 1$

$= 29_{10}$

The conversion is correct.

Comparing with Options

Comparing the calculated binary number 11101 with the given options:

• A. 10101
• B. 11110
• C. 11101
• D. 11001

The calculated binary representation 11101 matches option C.

Revision Table: Decimal to Binary Conversion Key Points

Additional Information: Other Conversion Methods

While division by 2 is common for integers, other methods exist or are used for different parts of a number:

• Subtraction Method: Repeatedly subtract the largest possible power of 2 less than or equal to the remaining number. Place a '1' in the binary position corresponding to the power subtracted and '0' elsewhere.
• For Fractional Parts: Use repeated multiplication by 2. The integer part of the result at each step forms the binary digits for the fractional part. Read the integer parts from top down.

Understanding binary conversion is essential for working with digital systems, data representation, and low-level programming.