Lecture Notes: Properties of Whole Numbers and Irrational Numbers

Overview

• Objective: Practice exercise on whole numbers, focusing on properties and understanding of irrational numbers.
• Concepts: Consecutive integers, irrational numbers, perfect squares.

Key Concepts

Consecutive Integers

• Defined as numbers that follow each other in order: 1, 2, 3, etc.
• In this exercise, focus on finding irrational numbers between two consecutive integers.

Irrational Numbers

• Numbers that cannot be simplified by head and have non-terminating, non-recurring decimals.
• Example: Square root of 11.

Exercise: Determining Boundaries for Irrational Numbers

Example: Square Root of 11

• Calculation: Square root of 11 ≈ 3.3166
• Boundaries: Lies between 3 and 4.
• Method: Use perfect squares 9 and 16 to establish that √11 is greater than √9 (3) and less than √16 (4).

Steps to Determine Boundaries

1. Identify the irrational number (e.g., √11).
2. Use a calculator for approximation.
3. Determine between which two consecutive integers the number lies.
4. Use perfect squares to confirm the boundary integers.

Additional Examples

• Square Root of 7:

  • Lies between √4 (2) and √9 (3).
  • Conclusion: Between 2 and 3.

• Square Root of 15:

  • Lies between √9 (3) and √16 (4).
  • Conclusion: Between 3 and 4.

• Negative Square Roots:

  • Example: -√11
  • Lies between -4 and -3.
  • Special note: Negative values are reversed in order of size compared to positive values.

• Pi/2 Calculation:

  • Pi/2 ≈ 1.57
  • Lies between 1 and 2.

• -3 Pi Calculation:

  • -3 Pi ≈ -9.42
  • Lies between -10 and -9.

Important Tips

• Use perfect squares to find boundary integers for square roots.
• Understand that negative numbers have reversed order in terms of size.
• Practice: Solve as many problems as possible to reinforce understanding and familiarity.

Conclusion

• Revisions are crucial for mastering the basics and handling complex problems.
• Keep practicing and revisiting these concepts to enhance mathematical skills.