Codage des nombres réels (les virgules)

Introduction

This tutorial focuses on the encoding of real numbers, particularly how to represent them in different numeral systems such as binary, octal, and hexadecimal. Understanding these concepts is crucial for various applications in computer science, programming, and digital electronics.

Step 1: Understanding Real Number Representation

Real numbers can be represented in various numeral systems. This step covers the basics of how to encode real numbers in different formats.

• Binary Representation:

  • Real numbers in binary are represented using the base 2 system.
  • A binary number consists of bits (0s and 1s).

• Octal Representation:

  • Uses base 8, where digits range from 0 to 7.
  • Each octal digit represents three binary digits.

• Hexadecimal Representation:

  • Uses base 16, with digits from 0 to 9 and letters A to F.
  • Each hexadecimal digit corresponds to four binary digits.

Step 2: Encoding Real Numbers in Binary

To encode a real number in binary, follow these steps:

1. Separate the number into integer and fractional parts.

   • Example: For the number 12.75, the integer part is 12, and the fractional part is 0.75.

2. Convert the integer part to binary:

   • Divide the integer by 2 and record the remainder.
   • Continue dividing the quotient by 2 until it reaches zero.
   • Collect the remainders in reverse order.
   • Example:
     • 12 / 2 = 6 remainder 0
     • 6 / 2 = 3 remainder 0
     • 3 / 2 = 1 remainder 1
     • 1 / 2 = 0 remainder 1
     • Result: 1100

3. Convert the fractional part to binary:

   • Multiply the fractional part by 2.
   • The integer part of the result is the next binary digit.
   • Repeat with the new fractional part until you reach the desired precision.
   • Example:
     • 0.75 * 2 = 1.5 → 1
     • 0.5 * 2 = 1.0 → 1
     • Result: 0.11

4. Combine both parts:

   • Final binary representation of 12.75 is 1100.11.

Step 3: Encoding Real Numbers in Octal and Hexadecimal

Follow these steps to convert the binary representation into octal and hexadecimal.

1. Convert from binary to octal:

   • Group the binary digits into sets of three, starting from the decimal point.
   • Convert each group to its octal equivalent.
   • Example: 1100.11 → 001 100 . 110 → 3.6 in octal.

2. Convert from binary to hexadecimal:

   • Group the binary digits into sets of four.
   • Convert each group to its hexadecimal equivalent.
   • Example: 1100.11 → 1100 . 1100 → C.C in hexadecimal.

Step 4: Decoding Real Numbers

To decode a number from binary, octal, or hexadecimal back to decimal:

1. Binary to Decimal:

   • For each bit, multiply it by 2 raised to the power of its position (starting from 0 for the rightmost bit).
   • Sum the results for both integer and fractional parts.
   • Example:
     • Binary 1100.11 translates to:
       • 1×2^3 + 1×2^2 + 0×2^1 + 0×2^0 + 1×2^-1 + 1×2^-2 = 12.75 in decimal.

2. Octal to Decimal:

   • Multiply each digit by 8 raised to the power of its position and sum the results.

3. Hexadecimal to Decimal:

   • Multiply each digit by 16 raised to the power of its position and sum the results.

Conclusion

Understanding how to encode and decode real numbers in various numeral systems is essential for computer programming and digital systems. This guide provided a step-by-step approach to converting real numbers into binary, octal, and hexadecimal formats, as well as decoding them back into decimal. Next steps could involve practicing with different numbers or exploring more complex encoding techniques used in computing.