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.