Decimal to Hexadecimal

Converting decimal numbers to hexadecimal is an essential skill in computer science and digital electronics. Hexadecimal (base-16) numbers use sixteen digits: 0-9 and A-F, where A represents 10, B represents 11, C represents 12, D represents 13, E represents 14, and F represents 15. This article will explore methods to convert both whole decimal numbers and decimal fractions to hexadecimal with detailed explanations and examples.

Converting Whole Decimal Numbers to Hexadecimal

To convert a whole decimal number to hexadecimal, we use the method of repeated division by 16. Here’s the step-by-step process:

1. Divide the decimal number by 16.
2. Record the remainder. If the remainder is greater than 9, use the corresponding hexadecimal digit (A-F).
3. Update the decimal number to the quotient obtained from the division.
4. Repeat steps 1-3 until the quotient is 0.
5. The hexadecimal number is the sequence of remainders read from bottom to top.

Example 1: Converting 254 to Hexadecimal

1. Divide 254 by 16:

   • Quotient: 15
   • Remainder: 14 (E in hexadecimal)

2. Divide 15 by 16:

   • Quotient: 0
   • Remainder: 15 (F in hexadecimal)

Reading the remainders from bottom to top, we get FE.

So, $254_{10} = FE_{16}$.

Converting Decimal Fractions to Hexadecimal

To convert a decimal fraction to hexadecimal, we use the method of repeated multiplication by 16. Here’s how it works:

1. Multiply the decimal fraction by 16.
2. Record the integer part of the result. If the integer part is greater than 9, use the corresponding hexadecimal digit (A-F).
3. Update the decimal fraction to the fractional part obtained from the multiplication.
4. Repeat steps 1-3 until the fractional part is 0 or you reach the desired level of precision.
5. The hexadecimal fraction is the sequence of integer parts read in order.

Example 2: Converting 0.6875 to Hexadecimal

1. Multiply 0.6875 by 16:
   • Result: 11.0
   • Integer part: 11 (B in hexadecimal)
   • Fractional part: 0

The hexadecimal fraction is the sequence of integer parts, which gives us 0.B.

So, $0.6875_{10} = 0.B_{16}$.

General Conversion Method for Mixed Numbers

For numbers with both whole and fractional parts, convert each part separately and then combine the results.

Example 3: Converting 254.6875 to Hexadecimal

1. Convert the whole number part (254) to hexadecimal:

   • $254_{10} = FE_{16}$

2. Convert the fractional part (0.6875) to hexadecimal:

   • $0.6875_{10} = 0.B_{16}$

Combine the two parts to get the hexadecimal representation: $254.6875_{10} = FE.B_{16}$

Precision Considerations

In practice, converting decimal fractions to hexadecimal might not always yield an exact result, especially for non-terminating decimal fractions. The process can be stopped after a few iterations to get an approximate value with the desired precision.

Example 4: Converting 0.1 to Hexadecimal (Approximate)

1. Multiply 0.1 by 16:

   • Result: 1.6
   • Integer part: 1
   • Fractional part: 0.6

2. Multiply 0.6 by 16:

   • Result: 9.6
   • Integer part: 9
   • Fractional part: 0.6

The process can continue indefinitely, yielding a repeating hexadecimal fraction. For practical purposes, we might stop here. The hexadecimal fraction is approximately 0.19.

So, $0.1_{10} \approx 0.19_{16}$.

Conclusion

Converting decimal numbers to hexadecimal involves different methods for whole numbers and fractions. Whole numbers are converted using repeated division by 16, while fractions are converted using repeated multiplication by 16. Understanding these processes is crucial for working with digital systems and computer programming. With practice, you can accurately convert any decimal number to its hexadecimal equivalent.