Explore the complete solution of Exercise 1.1 from the new Math 9th syllabus, designed by the Punjab Curriculum and Textbook Board (PCTB), Lahore. This guide simplifies the latest class 9 math new book solution to help students excel in their studies.
Question 1: Identify each as Rational or Irrational
(i) \( 2.353535 \)
This is a repeating decimal.
A repeating decimal is a rational number.
Answer: Rational.
(ii) \( 0.\overline{6} \)
A repeating decimal can always be expressed as a fraction.
\( 0.\overline{6} = \frac{2}{3} \), which is a ratio of two integers.
Thus, \( 0.\overline{6} \) is rational.
(iii) \( 2.236067… \)
This is a non-terminating, non-repeating decimal (\( \sqrt{5} \)).
Non-terminating, non-repeating decimals are irrational numbers.
Answer: Irrational.
(iv) \( \sqrt{7} \)
The square root of 7 is not a perfect square.
Hence, \( \sqrt{7} \) is an irrational number.
Answer: Irrational.
(v) \( e \)
The mathematical constant \( e \) (Euler’s number) is known to be a transcendental number.
Transcendental numbers are irrational numbers.
Answer: Irrational.
(vi) \( \pi \)
The mathematical constant \( \pi \) is a transcendental number.
Transcendental numbers are irrational numbers.
Answer: Irrational.
(vii) \( 5 + \sqrt{11} \)
The sum of a rational number (5) and an irrational number (\( \sqrt{11} \)) is always irrational.
Answer: Irrational.
(viii) \( \sqrt{3} + \sqrt{13} \)
The sum of two irrational numbers (\( \sqrt{3} \) and \( \sqrt{13} \)) remains irrational.
Answer: Irrational.
(ix) \( \frac{15}{4} \)
This is a fraction of two integers, and it is in the form \( \frac{p}{q} \) (where \( q \neq 0 \)).
Hence, it is a rational number.
Answer: Rational.
(x) \( (2 – \sqrt{2})(2 + \sqrt{2}) \)
Using the formula \( (a – b)(a + b) = a^2 – b^2 \):
\[(2 – \sqrt{2})(2 + \sqrt{2}) = 2^2 – (\sqrt{2})^2 = 4 – 2 = 2.\]
Since 2 is an integer, it is a rational number.
Answer: Rational.
Question 2: Represent on the Number Line
(i) \( \sqrt{2} \)
Approximate value: \( \sqrt{2} \approx 1.41 \).
It lies between 1 and 2 on the number line.
(ii) \( \sqrt{3} \)
Approximate value: \( \sqrt{3} \approx 1.73 \).
It lies between 1 and 2 on the number line.
(iii) \( 4\frac{1}{3} \)
Convert to improper fraction
\[4\frac{1}{3} = \frac{13}{3} \approx 4.33.\]
It lies between 4 and 5 on the number line.
(iv) \( -2\frac{1}{7} \)
Convert to improper fraction:
\[-2\frac{1}{7} = -\frac{15}{7} \approx -2.14.\]
It lies slightly less than -2 on the number line.
(v) \( \frac{5}{8} \)
Approximate value
\[\frac{5}{8} = 0.625.\]
It lies between 0 and 1 on the number line.
(vi) \( 2\frac{3}{4} \)
Convert to improper fraction:
\[2\frac{3}{4} = \frac{11}{4} = 2.75.\]
It lies between 2 and 3 on the number line.
Question 3: Express as Rational \( p/q \)
(i) \( 0.\overline{4} \)
\[x = 0.\overline{4}\]
\[10x = 4.\overline{4}\]
\[10x – x = 4\]
\[9x = 4\]
\[x = \frac{4}{9}\]
(ii) \( 0.\overline{37} \)
\[x = 0.\overline{37}\]
\[100x = 37.\overline{37}\]
\[100x – x = 37\]
\[99x = 37\]
\[x = \frac{37}{99}\]
(iii) \( 0.\overline{21} \)
\[x = 0.\overline{21}\]
\[100x = 21.\overline{21}\]
\[100x – x = 21\]
\[99x = 21\]
\[x = \frac{21}{99} = \frac{7}{33}\]
Question 4: Name the property used
(i) \( (a + 4) + b = a + (4 + b) \)
Property: Associative property of addition.
(ii) \( \sqrt{2} + \sqrt{3} = \sqrt{3} + \sqrt{2} \)
Property: Commutative property of addition.
(iii) \( x – x = 0 \)
Property: Additive inverse property.
(iv) \( a(b + c) = ab + ac \)
Property: Distributive property.
(v) \( 16 + 0 = 16 \)
Property: Additive identity property.
(vi) \( 100 \times 1 = 100 \)
Property: Multiplicative identity property.
(vii) \( 4 \times (5 \times 8) = (4 \times 5) \times 8 \)
Property: Associative property of multiplication.
(viii) \( ab = ba \)
Property: Commutative property of multiplication.
Question 5: Name the property used
(i) \( -3 < -1 ⇒ 0 < 2 \)
\[ -3+ 3< -1+3 \]
\[ 0< 2 \]
Property: Additive property of inequality.
(ii) If \( a < b \) then \( \frac{1}{a} > \frac{1}{b} \)
Property: Reciprocal property of inequality.
(iii) If \( a < b \), then \( a + c < b + c \)
Property: Addition property of inequality.
(iv) If \( ac < bc \) and \( c > 0 \), then \( a < b \)
Property: Multiplication property of inequality.
(v) If \( ac > bc \) and \( c < 0 \), then \( a > b \)
Property: Multiplication property of inequality (with negative \( c \)).
(vi) Either \( a > b \) or \( a = b \) or \( a < b \)
Property: Trichotomy property of real numbers.
6) Insert two rational numbers between each of the given pairs.
(i) Rational numbers between \( \frac{1}{3} \) and \( \frac{1}{4} \)
Average of \( \frac{1}{3} \) and \( \frac{1}{4}\) is
\[\frac{\frac{1}{3} + \frac{1}{4}}{2} = \frac{7}{24} \]
Average of \( \frac{1}{3} \) and \( \frac{7}{24} \) is
\[ \frac{\frac{1}{3} + \frac{7}{24}}{2} = \frac{17}{72} \]
(ii) Rational numbers between 3 and 4
Average of 3 and 4 is
\[ \frac{3+4}{2} = 3.5 \]
Average of 3 and 3.5 is
\[ \frac{3+3.5}{2} = 3.25 \]
(iii) Rational numbers between \( \frac{3}{5} \) and \( \frac{4}{5} \)
Average of \( \frac{3}{5} \) and \( \frac{4}{5}\) is
\[ \frac{\frac{3}{5} + \frac{4}{5}}{2} = \frac{7}{10} \]
Average of \( \frac{3}{5} \) and \( \frac{7}{10}\) is
\[ \frac{\frac{3}{5} + \frac{7}{10}}{2} = \frac{13}{20} \]
