Notes on Rational Numbers

Definition of Rational Numbers

• Rational Numbers: Numbers that can be expressed as a fraction of two integers.
  • Numerator: Must be an integer.
  • Denominator: Must be an integer (cannot be zero).

Characteristics of Rational Numbers

• Can be written as decimals:
  • Terminating Decimals: Decimals that cut off (e.g., 0.5).
  • Repeating Decimals: Decimals that continue in a pattern (e.g., 0.333...).

Examples of Rational Numbers

1. Whole Numbers

   • Example: 6 can be written as 6/1, 12/2, or 36/6 (all fractions of integers).
   • Conclusion: All whole numbers are rational.

2. Negative Numbers

   • Example: -6 can be written as -6/1 or -8/3.
   • Conclusion: Negative integers are also rational.

3. Terminating Decimals

   • Example: 7.10 can be expressed as 7/10, 21/30, or 70/100.
   • Conclusion: Any terminating decimal is rational.

4. Mixed Numbers/Decimals

   • Example: 2.75 can be written as 275/100 or its simplified form 11/4.
   • Conclusion: Mixed numbers or decimals that terminate are rational.

5. Zero

   • Example: 0 can be expressed as 0/1, 0/25, or 0/100.
   • Conclusion: Zero is a rational number.
   • Note: A fraction cannot have a denominator of zero (e.g., 1/0 is undefined).

6. Repeating Decimals

   • Example: 0.333... can be expressed as 1/3 or 3/9.
   • Example: 0.181818... can be expressed as 18/99 or 2/11.
   • Conclusion: All repeating decimals are rational.

7. Fractions

   • Example: 1/4 is a fraction where both numerator and denominator are integers.
   • Decimal equivalent: 1/4 = 0.25 (terminating).
   • Conclusion: Fractions are rational numbers.

8. Square Roots

   • Example: √25 = 5, which is rational (can be expressed as 5/1).
   • Example of non-rational: √3 ≈ 1.732... (non-terminating, non-repeating).
     • This cannot be expressed as a fraction of two integers.
     • Conclusion: Non-perfect square roots lead to irrational numbers.

Conclusion

• Rational Number Criteria:

  • Can be expressed as a fraction of two integers.
  • Decimal representations that either terminate or repeat.

• Irrational Numbers: Cannot be expressed as a fraction of two integers (e.g., √3).

• Further examples of irrational numbers to be discussed in another video.