Rational Numbers

Understanding Rational Numbers

  • Rational numbers are numbers that can be expressed as a ratio of two integers, where the denominator is not equal to zero.
  • Most numbers we use in daily life, like decimals and fractions, are rational numbers.
  • Rational numbers can be positive, negative, or zero.
  • Every integer is a rational number but every rational number is not an integer.

Floating-Point Representation of Rational Numbers

  • Rational numbers are typically represented as floating-point numbers in computers.
  • The floating-point number representation is, essentially, a form of scientific notation, adapted for computers.
  • A floating-point number comprises three parts: the sign bit, the exponent, and the mantissa or fraction.
  • Changes in the value of the exponent allow for representation of numbers in a wide range, from very large to very small.
  • However, not all rational numbers can be precisely represented due to the finite precision of the floating-point representation.

Encoding and Decoding Rational Numbers

  • Encoding a rational number requires representing its floating-point form in binary.
  • This involves separate encoding of the sign, the exponent, and the mantissa.
  • The exponent is usually represented using biased notation to handle both positive and negative exponents.
  • Decoding involves reversing the process to obtain the original number from its binary representation.

Importance of Rational Numbers in Computing

  • Rational numbers are used wherever quantities that aren’t whole numbers need to be represented, such as real-world measurements and calculations.
  • Many scientific and engineering applications require precise representation and manipulation of rational numbers.
  • Errors in representing rational numbers due to finite precision, known as round-off errors, can significantly affect the accuracy of computations.
  • Understanding rational numbers and their representation is critical to prevent and mitigate such errors.