# 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.