Lec 02 - Rational Numbers

Introduction

This tutorial aims to provide a comprehensive understanding of rational numbers, their properties, and how to work with them. By following these steps, you will learn to define rational numbers, compare them, calculate their greatest common divisor, and appreciate their density on the number line.

Step 1: Define Rational Numbers

Rational numbers are defined as numbers that can be expressed as the quotient of two integers, where the denominator is not zero. Here’s how to identify and express rational numbers:

• A rational number can be in the form of a/b, where:
  • a is an integer (the numerator).
  • b is a non-zero integer (the denominator).

Examples of Rational Numbers

• 1/2
• -3/4
• 5 (which can be expressed as 5/1)
• 0 (which can be expressed as 0/1)

Step 2: Order and Compare Rational Numbers

To order and compare rational numbers, follow these steps:

1. Convert to Common Denominator: If you have different denominators, convert them to a common denominator.

   • For example, to compare 1/2 and 1/3:
     • Common denominator: 6
     • Convert: 1/2 = 3/6 and 1/3 = 2/6

2. Compare Numerators: After converting, compare the numerators of the fractions.

   • Since 3/6 > 2/6, it follows that 1/2 > 1/3.

Practical Tip

When comparing rational numbers, it can be helpful to convert them to decimal form for a quick visual comparison.

Step 3: Find the Greatest Common Divisor

The greatest common divisor (GCD) of two integers is the largest integer that divides both numbers without leaving a remainder. Here’s how to find the GCD:

1. List Factors: Write down the factors of each integer.

   • Example: Factors of 12 are 1, 2, 3, 4, 6, 12.
   • Factors of 15 are 1, 3, 5, 15.

2. Identify Common Factors: Find the common factors from both lists.

   • Common factors of 12 and 15 are 1 and 3.

3. Select the Greatest Common Factor: The greatest common factor is the GCD.

   • For 12 and 15, the GCD is 3.

Alternative Method: Euclidean Algorithm

• For two integers a and b:
  • If b = 0, then GCD(a, b) = a.
  • Otherwise, GCD(a, b) = GCD(b, a mod b).

Step 4: Understand Density of Rational Numbers

Rational numbers are densely packed on the number line, which means between any two rational numbers, there exists another rational number. Here’s how to visualize this:

• Choose any two rational numbers, for example, 1/2 and 1/3.
• You can find a rational number between them, such as 5/12.

Key Insight

This density property implies that there are infinitely many rational numbers between any two distinct rational numbers.

Conclusion

In this tutorial, you learned to define rational numbers, compare them, find their GCD, and understand their density on the number line. Mastering these concepts forms the foundation for further mathematical studies and applications in various fields. As a next step, practice comparing different rational numbers and calculating the GCD for pairs of integers to reinforce your understanding.