Rational Numbers: Properties and Applications

Rational Numbers

Rational numbers, also known as the countable numbers or the computable numbers, are numbers that can be expressed as the quotient or fraction p/q of two integers, where p and q are integers and q ≠ 0. In other words, rational numbers are numbers that can be expressed in the form p/q, where p and q are integers. This means that every rational number can be represented as a simple fraction.

Properties of Rational Numbers

1. Addition: The sum of two rational numbers is always a rational number. For example, if a and b are two rational numbers, then a + b is also a rational number.

2. Subtraction: The difference between two rational numbers is always a rational number. For example, if a and b are two rational numbers, then a - b is also a rational number.

3. Multiplication: The product of two rational numbers is always a rational number. For example, if a and b are two rational numbers, then a * b is also a rational number.

4. Division: The division of two rational numbers is always a rational number. For example, if a and b are two rational numbers (where b ≠ 0), then a / b is also a rational number.

5. Ordering: Rational numbers can be ordered. For example, if a and b are two rational numbers, we can say that a < b, a > b, or a = b.

6. Density: Rational numbers are dense. This means that between any two rational numbers, there are infinitely many other rational numbers.

7. Completeness: The set of rational numbers is complete. This means that every non-empty subset of rational numbers that is bounded from above has a supremum (least upper bound).

8. Archimedean Property: The set of rational numbers has the Archimedean property. This means that for any positive real number x and any positive integer n, there exists a positive rational number greater than x and less than x + 1/n.

9. Cancellation: The cancellation property holds for rational numbers. This means that if a, b, and c are rational numbers such that a + c = b + c, then a = b.

Rational numbers have a wide range of applications in various fields, including mathematics, physics, engineering, and computer science. They are essential for performing calculations and solving problems that involve measurement and comparison.