Types of Numbers

In this lecture, we explore the different types of numbers, categorized into rational and irrational numbers.

Rational Numbers

Rational numbers include:

Integers

• Definition: Whole numbers, positive or negative, including zero.
• Examples: 5, 13, 412, -11, -92, 0.

Fractions

• Defined as parts of a whole number.
• Structure: Written as one integer over another (e.g., 3/8).
• Practical Example: Dividing a pizza into 8 slices and eating 3.
• Terms:
  • Numerator: The top number (e.g., 3).
  • Denominator: The bottom number (e.g., 8).

Terminating Decimals

• Definition: Decimals with a limited number of places.
• Examples: 0.5, 0.625.

Recurring Decimals

• Definition: Decimals that continue indefinitely.
• Notation: Use a dot over the recurring digit(s).
• Examples:
  • 0.6 recurring (0.666...) is written as 0.6̇.
  • 0.123 recurring is written as 0.1̇23̇.
• Fraction Equivalents:
  • 0.5 = 1/2
  • 0.625 = 5/8
  • 0.6 recurring = 2/3
  • 0.123 recurring = 123/999

Irrational Numbers

• Definition: Numbers with decimals that continue forever without repeating.
• Example Types:
  • Surds: Square roots of non-square numbers (e.g., √2 ≈ 1.414213...)
  • Pi (π): Begins with 3.14159 and continues indefinitely.
• Key Feature: Cannot be expressed as a simple fraction.

Summary

• Rational Numbers: Have limited decimal places, can be written as fractions.
• Irrational Numbers: Decimals continue indefinitely without a pattern, must be rounded when written down.

Conclusion

Understanding these number types is crucial for mathematics. Rational numbers are simpler to work with due to their fractional nature, whereas irrational numbers require approximation methods for calculations.