Converting a decimal number, such as 39, to binary involves expressing the number in base-2 form. Binary is a positional numeral system with a base of 2, using only two digits: 0 and 1. Here's a detailed guide on how to convert the decimal number 39 to binary.

Binary Conversion Algorithm:

1. Divide by 2:

Divide the decimal number by 2.

2. Record Remainder:

Note down the remainder.

3. Update Quotient:

Update the quotient to be the result of the division.

4. Repeat:

Repeat steps 1-3 until the quotient becomes 0.

5. Read Backwards:

The binary equivalent is the sequence of remainders read backward.

Conversion of 39 to Binary:

Let's apply this algorithm to convert the decimal number 39 to binary:

1. \( 39 \div 2 = 19 \) with a remainder of 1. Record 1.

2. \( 19 \div 2 = 9 \) with a remainder of 1. Record 1.

3. \( 9 \div 2 = 4 \) with a remainder of 1. Record 1.

4. \( 4 \div 2 = 2 \) with a remainder of 0. Record 0.

5. \( 2 \div 2 = 1 \) with a remainder of 0. Record 0.

6. \( 1 \div 2 = 0 \) with a remainder of 1. Record 1.

Reading the remainders backward, the binary representation of 39 is 100111.

Checking the Result:

To verify the conversion, we can convert the binary representation back to decimal:

\[ 1 \times 2^5 + 0 \times 2^4 + 0 \times 2^3 + 1 \times 2^2 + 1 \

times 2^1 + 1 \times 2^0 = 32 + 4 + 2 + 1 = 39 \]

The result matches the original decimal number, confirming the accuracy of the conversion.

Conclusion:

Converting decimal numbers to binary follows a systematic process of division and remainder recording. Understanding this process allows for accurate and efficient conversion between different number systems. In the case of 39, its binary representation is 100111.