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CBSE Class 9 Mathematics Question 2 of 9

Number Systems — Question 2

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Question 2

You know that 17\dfrac{1}{7} = 0.142857‾0.\overline{142857}. Can you predict what the decimal expansions of 27\dfrac{2}{7}, 37\dfrac{3}{7}, 47\dfrac{4}{7}, 57\dfrac{5}{7}, 67\dfrac{6}{7} are, without actually doing the long division? If so, how?

[Hint : Study the remainders while finding the value of 17\dfrac{1}{7} carefully.]

Answer

Given,

17\dfrac{1}{7} = 0.142857‾0.\overline{142857}

(i) 27\dfrac{2}{7} = 2×172 \times \dfrac{1}{7}

= 2×0.142857‾2 \times 0.\overline{142857}

= 0.285714‾0.\overline{285714}

(ii) 37\dfrac{3}{7} = 3×173 \times \dfrac{1}{7}

= 3×0.142857‾3 \times 0.\overline{142857}

= 0.428571‾0.\overline{428571}

(iii) 47\dfrac{4}{7} = 4×174 \times \dfrac{1}{7}

= 4×0.142857‾4 \times 0.\overline{142857}

= 0.571428‾0.\overline{571428}

(iv) 57\dfrac{5}{7} = 5×175 \times \dfrac{1}{7}

= 5×0.142857‾5 \times 0.\overline{142857}

= 0.714285‾0.\overline{714285}

(v)67\dfrac{6}{7} = 6×176 \times \dfrac{1}{7}

= 6×0.142857‾6 \times 0.\overline{142857}

= 0.857142‾0.\overline{857142}

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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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