Rational Numbers

Introduction to Rational Numbers

Rational numbers are numbers that can be expressed as the quotient or fraction p/q, where p and q are integers and q ≠ 0. They include integers (both positive and negative), and whole numbers. The chapter begins by illustrating how we need rational numbers to solve certain types of equations that cannot be solved with natural numbers, whole numbers, or integers alone. For instance:

• Equation like x + 5 = 5 only gives the solution of 0, which is a whole number, whereas for x + 18 = 5, we need to use negative integers, leading us to consider rational numbers to solve equations of the form ax = b where a and b are real numbers.

Properties of Rational Numbers

1. Closure:

   • Addition: The sum of any two rational numbers is also a rational number.
   • Subtraction: The difference between any two rational numbers is also a rational number.
   • Multiplication: The product of any two rational numbers remains a rational number.
   • Division: Division among rational numbers is generally not closed if one of the numbers is zero, as division by zero is undefined. However, excluding zero, the division of two non-zero rational numbers is a rational number.

2. Commutativity:

   • Addition: a + b = b + a (this holds for rational numbers).
   • Multiplication: a × b = b × a (also holds true).
   • However, subtraction and division are not commutative.

3. Associativity:

   • Addition: (a + b) + c = a + (b + c).
   • Multiplication: (a × b) × c = a × (b × c).
   • Subtraction and division are not associative.

4. Additive Identity (0):

   • Adding zero to any rational number returns the number itself, showcasing that zero is the identity element for addition among rational numbers.

5. Multiplicative Identity (1):

   • Multiplying any rational number by 1 yields the number itself, indicating that 1 is the identity element for multiplication among rational numbers.

6. Distributive Property:

   • Multiplication distributes over addition: a(b + c) = ab + ac. This property is essential when simplifying expressions involving rational numbers.

Infinite Rational Numbers Between Two Rational Numbers

It is also noted that between any two rational numbers, there are infinitely many rational numbers. The concept of the mean can be used to determine a rational number that lies between two given rational numbers.

Conclusion

This chapter encapsulates the fundamental aspects of rational numbers and their operations, providing a foundational understanding necessary for tackling more complex mathematical problems involving rational expressions and equations.

1. Rational Numbers: Defined as p/q where p and q are integers and q ≠ 0.
2. Closure Property: Rational numbers are closed under addition, subtraction, and multiplication.
3. Commutativity: Addition and multiplication are commutative for rational numbers.
4. Associativity: Addition and multiplication are associative for rational numbers.
5. Identity Elements: 0 is the additive identity; 1 is the multiplicative identity.
6. Non-Closure in Division: Division by zero is undefined among rational numbers.
7. Distributive Property: a(b + c) = ab + ac.
8. Infinitude: Infinitely many rational numbers exist between any two rational numbers.