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CBSE Class 9 Mathematics Question 4 of 9

Number Systems — Question 6

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Question 6

Look at several examples of rational numbers in the form pq\dfrac{p}{q} (q ≠ 0), where p and q are integers with no common factors other than 1 and having terminating decimal representations (expansions). Can you guess what property q must satisfy?

Answer

Suppose, pq\dfrac{p}{q} = 25\dfrac{2}{5} = 0.4

or

910\dfrac{9}{10} = 0.9

or

32\dfrac{3}{2} = 1.5

q must contain factors of 2 or 5 or both.

For example,

In 78\dfrac{7}{8}, 8 = 2 x 2 x 2

and

in 710\dfrac{7}{10}, 10 = 2 x 5

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Mathematics | Number SystemsWeb Content • Interactive Notes

Number Systems — Interactive Study Guide

Master the real number system, irrational numbers, surds, rationalisation, and exponent laws.

The Number Hierarchy

Think of numbers as nested boxes: Natural numbers are inside Whole numbers, which are inside Integers, which are inside Rational numbers, which are inside Real numbers.

Key Insight: Between any two rational numbers, there are infinitely many irrational numbers, and vice versa. The number line is “dense” with both types!

Identifying Rational vs Irrational

NumberTypeReason
√4Rational√4 = 2 (perfect square)
√7Irrational7 is not a perfect square
0.333...RationalRecurring decimal = 1/3
0.10100100010...IrrationalNon-terminating, non-recurring
πIrrationalNon-terminating, non-recurring
22/7RationalIt is p/q form (just an approximation of π)

Rationalisation — Quick Method

To rationalise a denominator with surds, multiply top and bottom by the conjugate:

  • Conjugate of (a + √b) is (a − √b)
  • Conjugate of (√a − √b) is (√a + √b)

The denominator becomes rational because (a+b)(a−b) = a² − b².

Quick Self-Check

  1. Is √(16/9) rational or irrational? (Rational: = 4/3)
  2. Simplify: √50 + √18 (= 5√2 + 3√2 = 8√2)
  3. Rationalise: 1/(√3 + 1) (= (√3 − 1)/2)
  4. Find: 81/3 (= 2)

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