class 9 real numbers

Class 9 Mathematics Unit 1: Real Numbers Exercise 1.1 Solutions with Tips

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Looking for detailed solutions for Class 9 Mathematics, Unit 1: Real Numbers Exercise 1.1? This guide provides clear explanations, step-by-step solutions, and helpful tips for solving problems on rational and irrational numbers, properties, and number line representation. Perfect for CBSE and NCERT students! and all boards of punjab

Question 1: Identify as rational or irrational

Solutions:
(i) 2.353535→ Rational (repeating decimal)
(ii) 0.6‾ → Rational (repeating decimal)
(iii) 2.236067… → Irrational (non-terminating, non-repeating decimal; it’s 5
(iv) √7 → Irrational (not a perfect square)
(v) e → Irrational (Euler’s number is irrational)
(vi) π → Irrational (value of π\pi is non-terminating and non-repeating)
(vii) 5+√11 → Irrational (irrational part √11)
(viii) √3+√13 → Irrational (sum of two irrationals)
(ix) 15/4→ Rational (fraction of integers)
(x) (2−√2)(2+√2)→ Rational (product simplifies to 22−√(2)2=4−2=2)

Tips for Identifying Rational and Irrational Numbers:

  1. Rational Numbers: Can be expressed as a fraction p/q, where p,q are integers and q≠0. Includes terminating and repeating decimals.
  2. Irrational Numbers: Cannot be expressed as a fraction. Examples: √2,π,e.

Question 2: Represent numbers on the number line

Steps to Plot the Numbers: (i) √2: Approximate 2≈1.41. Locate 1.41 on the number line.
(ii) √3}: Approximate √3≈1.73. Locate 1.73.
(iii) 4 1/3: Convert to improper fraction 13/3≈4.33 . Locate 4.33.
(iv) −1/7: Locate just slightly left of 0 (approx. -0.14).
(v) 5/8 : Approximate 5/8=0.625. Locate 0.625.
(vi) 2 3/4 : Convert to improper fraction 11/4≈2.75. Locate 2.75.

Tips for Plotting on the Number Line:

  1. Approximate square roots and fractions to decimals for accurate placement.
  2. Use a ruler to mark equal divisions on the number line for precision.

Question 3: Express repeating decimals as rational numbers

Solutions: (i) 0.4‾: Let x=0.4‾.
Multiply by 10: 10x=4.4‾.
Subtract: 10x−x=4.4‾−0.4‾.
9x=4, so x=4/9.

(ii) 0.37‾: Let x=0.37‾.
Multiply by 100: 100x=37.37‾.
Subtract: 100x−x=37.37‾−0.37‾
99x=37, so x=37/99x.

(iii) 0.21‾: Let x=0.21‾.
Multiply by 100: 100x=21.21‾
Subtract: 100x−x=21.21‾−0.21‾
99x=21, so x=21/99=7/33

Tips for Converting Repeating Decimals:

  1. Identify the repeating part and assign the decimal to x.
  2. Multiply x by powers of 10 to shift the decimal point.
  3. Subtract the equations to eliminate the repeating part and solve for x.

Question 4: Name the property used

Solutions:
(i) (a+4)+b=a+(4+b): Associative Property of Addition
(ii) √2+√3=√3+√2: Commutative Property of Addition
(iii) x−x=0 : Existence of Additive Inverse
(iv) a(b+c)=ab+ac : Distributive Property
(v) 16+0=16: Existence of Additive Identity
(vi) 100×1=100: Existence of Multiplicative Identity
(vii) 4×(5×8)=(4×5)×8: Associative Property of Multiplication
(viii) ab=ba Commutative Property of Multiplication

Tips to Remember Properties:

  1. Commutative: Order doesn’t matter (e.g., a+b=b+a, ab=ba).
  2. Associative: Grouping doesn’t matter (e.g., (a+b)+c=a+(b+c).
  3. Distributive: Multiplication distributes over addition (e.g., a(b+c)=ab+ac.
  4. Identity: Adding 0 or multiplying by 1 keeps the number unchanged.
  5. Inverse: Adding the opposite or multiplying by the reciprocal gives a neutral result (e.g., x−x=0).

Question 5: Name the property used

Solutions:
(i) −3<−1  ⟹  0<2: Transitive Property of Inequality
(ii) If a<b, then 1/a>1/b: Reciprocal Property of Inequality
(iii) If a<b, then a+c<b+c: Addition Property of Inequality
(iv) If ac<bc and c>0, then a<b: Multiplication Property of Inequality (for c>0)
(v) If ac<bc and c<0, then a>b: Multiplication Property of Inequality (for c<0)
(vi) Either a>b or a=b or a<b : Trichotomy Law

Tips to Solve Inequalities:

  1. Reciprocal Inequality: Reverses when reciprocals are taken (for positive numbers).
  2. Multiplication Rule: Inequality flips if multiplied/divided by a negative number.
  3. Transitive Property: If a<b and b<c , then a<c.

Question 6: Insert two rational numbers

Solutions:
(i) Between 1/3 and 1/4:

  • Find a common denominator: 1/3=4/12 = , 1/4=3/12.
  • Insert fractions like 10/36 and 11/36.

(ii) Between 33 and 44:

  • Choose decimals like 3.2 and 3.8.

(iii) Between 3/5 and 4/5:

  • Insert 7/10 and 9/10.

Tips for Inserting Rational Numbers:

  1. Convert fractions to a common denominator for clarity.
  2. Use decimals for whole numbers and simple fractions for fractions.
  3. Always choose values between the given numbers.

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