Chapter 1 of CBSE NCERT Class 9 Math covers number system. Concepts covered in exercise 1.5 include rational numbers, rationalizing irrational numbers by multiplying with the conjugate of the denominator to simplify the expressions.
Solutions to Question 1 and Question 3 are given in this page. For the other questions, click on the solution links given adjacent to the questions.
Question 1
Classify the following numbers as rational or irrational:
i) 2 - √5
ii) (3 + √23) - √23
iii) \\frac{2√7}{7√7})
iv) \\frac{1}{√2})
v) 2π
i) 2 - √5 is irrational as it contains the irrational number √5 in it. √5 cannot be expressed in the form \\frac{p}{q}) where p and q are integers and q ≠ 0.
ii) (3 + √23) - √23 = 3 + √23 - √23 = 3 is a natural number.
All natural numbers are rational numbers. 3 can be expressed in the form \\frac{p}{q}).
iii) \\frac{2√7}{7√7}) = \\frac{2}{7}) which is rational as it can be expressed in the form \\frac{p}{q}).
Property used for (iv) and (v): The product or quotient of a rational number and an irrational number is an irrational number.
iv) \\frac{1}{√2}) contains the irrational number √2. So, \\frac{1}{√2}) is irrational. It cannot be expressed in the form \\frac{p}{q}) where p and q are integers and q ≠ 0.
v) 2π is irrational because π is irrational.
Question 2
Simplify each of the following expressions:
(i) (3 + √3) (2 + √2)
(ii) (3 + √3) (3 - √3)
(iii) (√5 + √2)^{2}
(iv) (√5 - √2)(√5 + √2)
Question 3
Recall, π is defined as the ratio of the circumference (c) of a circle to its diameter (d). That is, π = \\frac{c}{d}). This seems to contradict the fact that π is irrational. How can you resolve this contradiction?
There is no contradiction. The measure of c will be irrational if d is rational or the measure of d will be irrational if c is rational or both c and d will be irrational. Essentially, at least one of c or d will be irrational. Because at least one of c or d is not an integer, though it is of the form \\frac{c}{d}), π is not a rational number.
Question 5
Rationalise the denominators of the following:
(i) \\frac{1}{√7})
(ii) \\frac{1}{√7 - √6})
(iii) \\frac{1}{√5 + √2})
(iv) \\frac{1}{√7 - 2})
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