Notes on Rational Numbers

Introduction

• Chapter number: 9
• Title: Rational Numbers
• Purpose: Discuss concepts to aid in answering related questions.

What are Rational Numbers?

• Defined as numbers that can be expressed in the form ( P/Q ) where ( Q \neq 0 ).
• Example:
  • ( 6 ) is a rational number because it can be written as ( 6/1 ).

Types of Rational Numbers

• Positive Rational Numbers: e.g., ( 2/3 ) (no negative sign).
• Negative Rational Numbers: e.g., ( -2/3 ).

Equivalent Rational Numbers

• Defined similarly to equivalent fractions.
• Example:
  • Multiplying both numerator and denominator by the same number.
  • For ( 2/3 ), multiplying by 6 gives ( 12/18 ).

Simplest Form

• Also referred to as standard or lowest form.
• Process:
  • Reduce ( 15/12 ) to simplest form:
    • Find common factors:
      • ( 15 = 3 \times 5 ), ( 12 = 3 \times 4 ) → ( 5/4 )
  • Example:
    • ( 40/50 ) reduces to ( 4/5 ) after dividing by 5.

Finding Rational Numbers Between Two Rational Numbers

• To determine rational numbers between two given numbers:
  • Make denominators the same.
• Example: Find rational numbers between ( -4/5 ) and ( -2/3 ).
  • Convert both to the same denominator, i.e., 15:
    • ( -4/5 ightarrow -12/15)
    • ( -2/3 ightarrow -10/15)
  • Rational numbers between: ( -11/15 ).
  • For more rational numbers, increase the common denominator.

Plotting Rational Numbers on a Number Line

• Example: Plot ( 3/4 ) on the number line.
• Steps:
  • Identify that ( 3/4 ) is positive, so it lies between ( 0 ) and ( 1 ).
  • Divide the segment into 4 parts (because denominator is 4).
  • Mark the rational number accordingly.

Comparing, Adding, and Subtracting Rational Numbers

• Similar to fractions:
  • Example: Compare ( 2/3 ) and ( 5/2 ).
    • Make denominators same (e.g., 6).
  • Addition/Subtraction:
    • Use common denominators to perform operations.
  • Example Calculation: ( 2/3 + 5/2 \rightarrow 4/6 + 15/6 = 19/6 )
  • Record signs appropriately when subtracting.

Multiplying and Dividing Rational Numbers

• Multiplication:
  • Directly multiply numerators and denominators (e.g., ( 9/2 \times -7/4 = -63/8 )).
• Division:
  • Involve reciprocal
  • Example: ( 3/13 ÷ 4/65 \rightarrow 3/13 imes 65/4 \rightarrow 15/4 )

Conclusion

• All concepts discussed will help in solving exercises based on rational numbers.
• Prepare well for exercises that will be provided.

Closing

• Mansi will return with a new chapter in the next session.