Lecture Notes: Number Sets in Mathematics

Introduction

• Overview of different types of numbers and number sets.
• Real numbers as a broad category.

Real Numbers

• Includes almost any number: zero, whole numbers, decimals, repeating decimals, negative numbers, terminating decimals, fractions, square roots.
• Not real numbers: infinity.

Types of Numbers

Natural Numbers (Counting Numbers)

• Whole positive numbers starting from 1.
• Example: Counting apples, start with 1, 2, 3.
• Zero is not included.

Whole Numbers

• Similar to natural numbers but include 0.
• All positive whole numbers starting from 0.

Integers

• Whole numbers that can be positive or negative.
• Include all positive whole numbers and their opposites.
• Include 0.
• Example: 5 and -5, 6 and -6 are integers.
• Not an integer: -3.2 (as it’s not a whole number).

Rational Numbers

• Can be expressed as fractions or ratios.
• Includes integers, fractions, terminating decimals, repeating decimals.
• Examples:
  • 5 as 5/1.
  • Square root of 9 is 3, can be written as 3/1.
  • -8 as -8/1.
  • 1.2 as a fraction.
  • 0.333... as 1/3.
• Terminating decimals and repeating decimals are rational numbers.

Irrational Numbers

• Cannot be expressed as ratios.
• Do not terminate or repeat.
• Examples:
  • Pi (3.14159...) - non-terminating, non-repeating.
  • Square root of 2 and other square roots of primes.
  • Non-perfect squares like square root of 10, square root of 12.

Venn Diagram of Number Sets

• Rational and irrational numbers do not intersect.
• A number cannot be both rational and irrational.

Recap

• All real numbers include both rational and irrational numbers.
• Rational numbers include integers, whole numbers, and natural numbers.

Practice Questions

1. 6 is an integer

   • True; 6 is a whole number, and integers include whole numbers.

2. 0 is a counting number

   • False; counting starts from 1, not 0.

3. 0 is a whole number

   • True.

4. 2.3467 is a rational number

   • True; because it terminates.

5. -3.5 is an integer

   • False; it's a decimal, not a whole number.

6. Square root of 7 is an irrational number

   • True; square roots of primes are irrational.

7. Square root of 1 is an irrational number

   • False; it equals 1, can be expressed as 1/1, hence rational.

Conclusion

• Different types of numbers and their classifications.
• Understanding the distinctions and properties of number sets.