Question 1

Question 1Write the following in decimal form and say what kind of decimal expansion each has :(i) 36100\dfrac{36}{100}10036​(ii) 111\dfrac{1}{11}111​(iii) 4184\dfrac{1}{8}481​(iv) 313\dfrac{3}{13}133​(v) 211\dfrac{2}{11}112​(vi) 329400\dfrac{329}{400}400329​

Accepted Answer

(i) 36100\dfrac{36}{100}10036​ = 0.36, terminating decimal expansion(ii) 111\dfrac{1}{11}111​11)0.090911)100‾11)22−99‾11x3−210011)x322−99‾11)x32222111)x3221111‾\begin{array}{l} \phantom{11)}{0.0909} \ 11\overline{\smash{\big)}\quad100\quad} \ \phantom{11)}\phantom{22}\underline{-99} \ \phantom{{11}x^3-2}100 \ \phantom{{11)}x^322}\underline{-99} \ \phantom{{11)}{x^32222}}1 \ \phantom{{11)}{x^322}}\overline{\phantom{1111}} \end{array}11)0.090911)10011)22−99​11x3−210011)x322−99​11)x32222111)x3221111​The remainder 1 keeps repeating.Hence, 111\dfrac{1}{11}111​ = 0.09‾0.\overline{09}0.09. This is a non-terminating repeating decimal expansion(iii) 418=3384\dfrac{1}{8} = \dfrac{33}{8}481​=833​8)4.1258)33‾8)−32 ‾8)−108−−8‾8)−8208−−16‾8)−82408−8−40‾8−82208−811111‾\begin{array}{l} \phantom{8)}{\enspace 4.125} \ 8\overline{\smash{\big)}\enspace 33\quad} \ \phantom{8)}\underline{-32\space} \ \phantom{8)-}10 \ \phantom{8-}\underline{-8} \ \phantom{8)-8}20 \ \phantom{8-}\underline{-16} \ \phantom{8)-82}40 \ \phantom{8-8}\underline{-40\enspace} \ \phantom{8-822}0 \ \phantom{8-8}\overline{\phantom{11111}} \end{array}8)4.1258)338)−32 ​8)−108−−8​8)−8208−−16​8)−82408−8−40​8−82208−811111​Hence, 4184\dfrac{1}{8}481​ = 4.125. This is a terminating decimal expansion.(iv) 313\dfrac{3}{13}133​8)0.23076923076913)30‾13)−26 ‾13)−408−−39‾8)−810013303−91‾13303333901330331−78‾13303333312013303333−117‾1330333333330133033333)−26‾1330333333340133033333311111‾\begin{array}{l} \phantom{8)}{\enspace 0.230769230769} \ 13\overline{\smash{\big)}\enspace 30\qquad\qquad\quad} \ \phantom{13)}\underline{-26\space} \ \phantom{13)-}40 \ \phantom{8-}\underline{-39} \ \phantom{8)-8}100 \ \phantom{13303}\underline{-91\enspace} \ \phantom{13303333}90 \ \phantom{1330331}\underline{-78\enspace} \ \phantom{133033333}120 \ \phantom{13303333}\underline{-117\enspace} \ \phantom{13303333333}30 \ \phantom{133033333)}\underline{-26\enspace} \ \phantom{13303333333}40 \ \phantom{1330333333}\overline{\phantom{11111}} \end{array}8)0.23076923076913)3013)−26 ​13)−408−−39​8)−810013303−91​13303333901330331−78​13303333312013303333−117​1330333333330133033333)−26​1330333333340133033333311111​Hence, 313\dfrac{3}{13}133​ = 0.230769‾0.\overline{230769}0.230769. This is a non-terminating repeating decimal expansion.(v) 211\dfrac{2}{11}112​8)0.181811)20‾11)−11 ‾11)−908−−88‾8)−8)201330)−11‾1330333)90133033−88‾13303333213303311111‾\begin{array}{l} \phantom{8)}{\enspace 0.1818} \ 11\overline{\smash{\big)}\enspace 20\qquad} \ \phantom{11)}\underline{-11\space} \ \phantom{11)-}90 \ \phantom{8-}\underline{-88} \ \phantom{8)-8)}20 \ \phantom{1330)}\underline{-11\enspace} \ \phantom{1330333)}90 \ \phantom{133033}\underline{-88\enspace} \ \phantom{13303333}2 \ \phantom{133033}\overline{\phantom{11111}} \end{array}8)0.181811)2011)−11 ​11)−908−−88​8)−8)201330)−11​1330333)90133033−88​13303333213303311111​Hence, 211=0.18‾\dfrac{2}{11} = 0.\overline{18}112​=0.18. This is a non-terminating repeating decimal expansion.(vi) 329400\dfrac{329}{400}400329​ = 0.8225, terminating decimal expansion400)0.8225400)3290‾400)−3200 ‾400)−900400)2−800‾400)2181000400)21−800‾400)212)2000400)21−2000‾400)211110400)211111111‾\begin{array}{l} \phantom{400)}{0.8225} \ 400\overline{\smash{\big)}\enspace 3290\quad} \ \phantom{400)}\underline{-3200\space} \ \phantom{400)-}900 \ \phantom{400)2}\underline{-800} \ \phantom{400)218}1000 \ \phantom{400)21}\underline{-800} \ \phantom{400)212)}2000 \ \phantom{400)21}\underline{-2000\enspace} \ \phantom{400)21111}0 \ \phantom{400)21}\overline{\phantom{1111111}} \end{array}400)0.8225400)3290400)−3200 ​400)−900400)2−800​400)2181000400)21−800​400)212)2000400)21−2000​400)211110400)211111111​Hence, 329400\dfrac{329}{400}400329​ = 0.8225. This is a terminating decimal expansion.

Number Systems - Interactive Study Notes | Bright Tutorials

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BRIGHT TUTORIALS

BRIGHT TUTORIALSCBSE Class IX | Academic Year 2026-2027
    
    9403781999Excellence in Education
  
  Mathematics | Number SystemsWeb Content &bull; Interactive Notes
  
    Number Systems &mdash; 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 &ldquo;dense&rdquo; with both types!

Identifying Rational vs Irrational
      
        NumberTypeReason
        
          &radic;4Rational&radic;4 = 2 (perfect square)
          &radic;7Irrational7 is not a perfect square
          0.333...RationalRecurring decimal = 1/3
          0.10100100010...IrrationalNon-terminating, non-recurring
          &pi;IrrationalNon-terminating, non-recurring
          22/7RationalIt is p/q form (just an approximation of &pi;)

Rationalisation &mdash; Quick Method
      To rationalise a denominator with surds, multiply top and bottom by the conjugate:
      
        Conjugate of (a + &radic;b) is (a &minus; &radic;b)
        Conjugate of (&radic;a &minus; &radic;b) is (&radic;a + &radic;b)
      
      The denominator becomes rational because (a+b)(a&minus;b) = a&sup2; &minus; b&sup2;.

Quick Self-Check
      
        Is &radic;(16/9) rational or irrational? (Rational: = 4/3)
        Simplify: &radic;50 + &radic;18 (= 5&radic;2 + 3&radic;2 = 8&radic;2)
        Rationalise: 1/(&radic;3 + 1) (= (&radic;3 &minus; 1)/2)
        Find: 81/3 (= 2)

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Question 2

Question 2You know that 17\dfrac{1}{7}71​ = 0.142857‾0.\overline{142857}0.142857. Can you predict what the decimal expansions of 27\dfrac{2}{7}72​, 37\dfrac{3}{7}73​, 47\dfrac{4}{7}74​, 57\dfrac{5}{7}75​, 67\dfrac{6}{7}76​ are, without actually doing the long division? If so, how?[Hint : Study the remainders while finding the value of 17\dfrac{1}{7}71​ carefully.]

Accepted Answer

Given,17\dfrac{1}{7}71​ = 0.142857‾0.\overline{142857}0.142857(i) 27\dfrac{2}{7}72​ = 2×172 	imes \dfrac{1}{7}2×71​= 2×0.142857‾2 	imes 0.\overline{142857}2×0.142857= 0.285714‾0.\overline{285714}0.285714(ii) 37\dfrac{3}{7}73​ = 3×173 	imes \dfrac{1}{7}3×71​= 3×0.142857‾3 	imes 0.\overline{142857}3×0.142857= 0.428571‾0.\overline{428571}0.428571(iii) 47\dfrac{4}{7}74​ = 4×174 	imes \dfrac{1}{7}4×71​= 4×0.142857‾4 	imes 0.\overline{142857}4×0.142857= 0.571428‾0.\overline{571428}0.571428(iv) 57\dfrac{5}{7}75​ = 5×175 	imes \dfrac{1}{7}5×71​= 5×0.142857‾5 	imes 0.\overline{142857}5×0.142857= 0.714285‾0.\overline{714285}0.714285(v)67\dfrac{6}{7}76​ = 6×176 	imes \dfrac{1}{7}6×71​= 6×0.142857‾6 	imes 0.\overline{142857}6×0.142857= 0.857142‾0.\overline{857142}0.857142

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BRIGHT TUTORIALS

BRIGHT TUTORIALSCBSE Class IX | Academic Year 2026-2027
    
    9403781999Excellence in Education
  
  Mathematics | Number SystemsWeb Content &bull; Interactive Notes
  
    Number Systems &mdash; 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 &ldquo;dense&rdquo; with both types!

Identifying Rational vs Irrational
      
        NumberTypeReason
        
          &radic;4Rational&radic;4 = 2 (perfect square)
          &radic;7Irrational7 is not a perfect square
          0.333...RationalRecurring decimal = 1/3
          0.10100100010...IrrationalNon-terminating, non-recurring
          &pi;IrrationalNon-terminating, non-recurring
          22/7RationalIt is p/q form (just an approximation of &pi;)

Rationalisation &mdash; Quick Method
      To rationalise a denominator with surds, multiply top and bottom by the conjugate:
      
        Conjugate of (a + &radic;b) is (a &minus; &radic;b)
        Conjugate of (&radic;a &minus; &radic;b) is (&radic;a + &radic;b)
      
      The denominator becomes rational because (a+b)(a&minus;b) = a&sup2; &minus; b&sup2;.

Quick Self-Check
      
        Is &radic;(16/9) rational or irrational? (Rational: = 4/3)
        Simplify: &radic;50 + &radic;18 (= 5&radic;2 + 3&radic;2 = 8&radic;2)
        Rationalise: 1/(&radic;3 + 1) (= (&radic;3 &minus; 1)/2)
        Find: 81/3 (= 2)

Bright Tutorials | Hariom Nagar, Nashik Road | 9403781999 | brighttutorials.in

Question 3

Question 5What can the maximum number of digits be in the repeating block of digits in the decimal expansion of 117\dfrac{1}{17}171​ ? Perform the division to check your answer.

Accepted Answer

117\dfrac{1}{17}171​ = 0.0588235294117647‾0.\overline{0588235294117647}0.0588235294117647sixteen digit are of repeating blocks.17)0.0588235294117647‾17)100‾17))−85 ‾17)−15017))−136‾17))−14017)))−136‾13303333401330331−34‾1330333336013303333−51‾1330333333390133033333)−85‾133033333333)5013303333333−34‾1330333333333)160133033333333−153‾13303333333333337013303333333333)−68‾13303333333333333)201330333333333333)−17‾1330333333333333333)3013303333333333333))−17‾1330333333333333333)))1301330333333333333333)−119‾13303333333333333333)))11013303333333333333333)−102‾133033333333333333333333)8013303333333333333333333−68‾133033333333333333333333)12013303333333333333333333−119‾1330333333333333333333333)113303333333333333333333111111‾\begin{array}{l} \phantom{17)}{0.\overline{0588235294117647}} \ 17\overline{\smash{\big)}\enspace 100\qquad\qquad\qquad} \ \phantom{17))}\underline{-85\space} \ \phantom{17)-}150 \ \phantom{17))}\underline{-136} \ \phantom{17))-}140 \ \phantom{17)))}\underline{-136\enspace} \ \phantom{13303333}40 \ \phantom{1330331}\underline{-34\enspace} \ \phantom{133033333}60 \ \phantom{13303333}\underline{-51\enspace} \ \phantom{13303333333}90 \ \phantom{133033333)}\underline{-85\enspace} \ \phantom{133033333333)}50 \ \phantom{13303333333}\underline{-34\enspace} \ \phantom{1330333333333)}160 \ \phantom{133033333333}\underline{-153\enspace} \ \phantom{1330333333333333}70 \ \phantom{13303333333333)}\underline{-68\enspace} \ \phantom{13303333333333333)}20 \ \phantom{1330333333333333)}\underline{-17\enspace} \ \phantom{1330333333333333333)}30 \ \phantom{13303333333333333))}\underline{-17\enspace} \ \phantom{1330333333333333333)))}130 \ \phantom{1330333333333333333)}\underline{-119\enspace} \ \phantom{13303333333333333333)))}110 \ \phantom{13303333333333333333)}\underline{-102\enspace} \ \phantom{133033333333333333333333)}80 \ \phantom{13303333333333333333333}\underline{-68\enspace} \ \phantom{133033333333333333333333)}120 \ \phantom{13303333333333333333333}\underline{-119\enspace} \ \phantom{1330333333333333333333333)}1 \ \phantom{13303333333333333333333}\overline{\phantom{111111}} \end{array}17)0.058823529411764717)10017))−85 ​17)−15017))−136​17))−14017)))−136​13303333401330331−34​1330333336013303333−51​1330333333390133033333)−85​133033333333)5013303333333−34​1330333333333)160133033333333−153​13303333333333337013303333333333)−68​13303333333333333)201330333333333333)−17​1330333333333333333)3013303333333333333))−17​1330333333333333333)))1301330333333333333333)−119​13303333333333333333)))11013303333333333333333)−102​133033333333333333333333)8013303333333333333333333−68​133033333333333333333333)12013303333333333333333333−119​1330333333333333333333333)113303333333333333333333111111​

Number Systems - Interactive Study Notes | Bright Tutorials

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BRIGHT TUTORIALS

BRIGHT TUTORIALSCBSE Class IX | Academic Year 2026-2027
    
    9403781999Excellence in Education
  
  Mathematics | Number SystemsWeb Content &bull; Interactive Notes
  
    Number Systems &mdash; 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 &ldquo;dense&rdquo; with both types!

Identifying Rational vs Irrational
      
        NumberTypeReason
        
          &radic;4Rational&radic;4 = 2 (perfect square)
          &radic;7Irrational7 is not a perfect square
          0.333...RationalRecurring decimal = 1/3
          0.10100100010...IrrationalNon-terminating, non-recurring
          &pi;IrrationalNon-terminating, non-recurring
          22/7RationalIt is p/q form (just an approximation of &pi;)

Rationalisation &mdash; Quick Method
      To rationalise a denominator with surds, multiply top and bottom by the conjugate:
      
        Conjugate of (a + &radic;b) is (a &minus; &radic;b)
        Conjugate of (&radic;a &minus; &radic;b) is (&radic;a + &radic;b)
      
      The denominator becomes rational because (a+b)(a&minus;b) = a&sup2; &minus; b&sup2;.

Quick Self-Check
      
        Is &radic;(16/9) rational or irrational? (Rational: = 4/3)
        Simplify: &radic;50 + &radic;18 (= 5&radic;2 + 3&radic;2 = 8&radic;2)
        Rationalise: 1/(&radic;3 + 1) (= (&radic;3 &minus; 1)/2)
        Find: 81/3 (= 2)

Bright Tutorials | Hariom Nagar, Nashik Road | 9403781999 | brighttutorials.in

Question 4

Question 6Look at several examples of rational numbers in the form pq\dfrac{p}{q}qp​ (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?

Accepted Answer

Suppose, pq\dfrac{p}{q}qp​ = 25\dfrac{2}{5}52​ = 0.4or910\dfrac{9}{10}109​ = 0.9or32\dfrac{3}{2}23​ = 1.5q must contain factors of 2 or 5 or both.For example,In 78\dfrac{7}{8}87​, 8 = 2 x 2 x 2andin 710\dfrac{7}{10}107​, 10 = 2 x 5

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BRIGHT TUTORIALS

BRIGHT TUTORIALSCBSE Class IX | Academic Year 2026-2027
    
    9403781999Excellence in Education
  
  Mathematics | Number SystemsWeb Content &bull; Interactive Notes
  
    Number Systems &mdash; 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 &ldquo;dense&rdquo; with both types!

Identifying Rational vs Irrational
      
        NumberTypeReason
        
          &radic;4Rational&radic;4 = 2 (perfect square)
          &radic;7Irrational7 is not a perfect square
          0.333...RationalRecurring decimal = 1/3
          0.10100100010...IrrationalNon-terminating, non-recurring
          &pi;IrrationalNon-terminating, non-recurring
          22/7RationalIt is p/q form (just an approximation of &pi;)

Rationalisation &mdash; Quick Method
      To rationalise a denominator with surds, multiply top and bottom by the conjugate:
      
        Conjugate of (a + &radic;b) is (a &minus; &radic;b)
        Conjugate of (&radic;a &minus; &radic;b) is (&radic;a + &radic;b)
      
      The denominator becomes rational because (a+b)(a&minus;b) = a&sup2; &minus; b&sup2;.

Quick Self-Check
      
        Is &radic;(16/9) rational or irrational? (Rational: = 4/3)
        Simplify: &radic;50 + &radic;18 (= 5&radic;2 + 3&radic;2 = 8&radic;2)
        Rationalise: 1/(&radic;3 + 1) (= (&radic;3 &minus; 1)/2)
        Find: 81/3 (= 2)

Bright Tutorials | Hariom Nagar, Nashik Road | 9403781999 | brighttutorials.in

Question 5

Question 9Classify the following numbers as rational or irrational :(i) 23\sqrt{23}23​(ii) 225\sqrt{225}225​(iii) 0.3796(iv) 7.478478...(v) 1.101001000100001...

Accepted Answer

(i) 23\sqrt{23}23​ is a prime number. It is irrational because 23\sqrt{23}23​ does not have a complete root.(ii) 225\sqrt{225}225​ is not a prime number. 15 is root of 225\sqrt{225}225​ so it is rational.(iii) 0.3796 is terminating decimal expansion. Hence, it is rational.(iv) 7.478478... = 7.478‾7.\overline{478}7.478, it is non-terminating but repeating decimal expansion. Hence, it is rational.(v) 1.101001000100001... is non-terminating and non-repeating decimal expansion. Hence, it is irrational.

Number Systems - Interactive Study Notes | Bright Tutorials

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BRIGHT TUTORIALS

BRIGHT TUTORIALSCBSE Class IX | Academic Year 2026-2027
    
    9403781999Excellence in Education
  
  Mathematics | Number SystemsWeb Content &bull; Interactive Notes
  
    Number Systems &mdash; 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 &ldquo;dense&rdquo; with both types!

Identifying Rational vs Irrational
      
        NumberTypeReason
        
          &radic;4Rational&radic;4 = 2 (perfect square)
          &radic;7Irrational7 is not a perfect square
          0.333...RationalRecurring decimal = 1/3
          0.10100100010...IrrationalNon-terminating, non-recurring
          &pi;IrrationalNon-terminating, non-recurring
          22/7RationalIt is p/q form (just an approximation of &pi;)

Rationalisation &mdash; Quick Method
      To rationalise a denominator with surds, multiply top and bottom by the conjugate:
      
        Conjugate of (a + &radic;b) is (a &minus; &radic;b)
        Conjugate of (&radic;a &minus; &radic;b) is (&radic;a + &radic;b)
      
      The denominator becomes rational because (a+b)(a&minus;b) = a&sup2; &minus; b&sup2;.

Quick Self-Check
      
        Is &radic;(16/9) rational or irrational? (Rational: = 4/3)
        Simplify: &radic;50 + &radic;18 (= 5&radic;2 + 3&radic;2 = 8&radic;2)
        Rationalise: 1/(&radic;3 + 1) (= (&radic;3 &minus; 1)/2)
        Find: 81/3 (= 2)

Bright Tutorials | Hariom Nagar, Nashik Road | 9403781999 | brighttutorials.in

Question 6

Question 2(iii)Simplify the following expression:(5+2)(\sqrt{5} + \sqrt{2})(5​+2​) 2

Accepted Answer

(5+2)(\sqrt{5} + \sqrt{2})(5​+2​) 2= (a + b)2= a2 + 2ab + b2= (5)(\sqrt{5})(5​) 2 + 2 x (5)(\sqrt{5})(5​) x (2)(\sqrt{2})(2​) + (2)(\sqrt{2})(2​) 2 [∵ (a + b)2 = a2 + 2ab + b2]= 5 + 2 (10)(\sqrt{10})(10​) + 2= 7 + 2 (10)(\sqrt{10})(10​)Hence,(5+2)(\sqrt{5} + \sqrt{2})(5​+2​) 2 = 7 + 2 (10)(\sqrt{10})(10​)

Number Systems - Interactive Study Notes | Bright Tutorials

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BRIGHT TUTORIALS

BRIGHT TUTORIALSCBSE Class IX | Academic Year 2026-2027
    
    9403781999Excellence in Education
  
  Mathematics | Number SystemsWeb Content &bull; Interactive Notes
  
    Number Systems &mdash; 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 &ldquo;dense&rdquo; with both types!

Identifying Rational vs Irrational
      
        NumberTypeReason
        
          &radic;4Rational&radic;4 = 2 (perfect square)
          &radic;7Irrational7 is not a perfect square
          0.333...RationalRecurring decimal = 1/3
          0.10100100010...IrrationalNon-terminating, non-recurring
          &pi;IrrationalNon-terminating, non-recurring
          22/7RationalIt is p/q form (just an approximation of &pi;)

Rationalisation &mdash; Quick Method
      To rationalise a denominator with surds, multiply top and bottom by the conjugate:
      
        Conjugate of (a + &radic;b) is (a &minus; &radic;b)
        Conjugate of (&radic;a &minus; &radic;b) is (&radic;a + &radic;b)
      
      The denominator becomes rational because (a+b)(a&minus;b) = a&sup2; &minus; b&sup2;.

Quick Self-Check
      
        Is &radic;(16/9) rational or irrational? (Rational: = 4/3)
        Simplify: &radic;50 + &radic;18 (= 5&radic;2 + 3&radic;2 = 8&radic;2)
        Rationalise: 1/(&radic;3 + 1) (= (&radic;3 &minus; 1)/2)
        Find: 81/3 (= 2)

Bright Tutorials | Hariom Nagar, Nashik Road | 9403781999 | brighttutorials.in

Question 7

Question 4Represent 9.3\sqrt{9.3}9.3​ on the number line.

Accepted Answer

Steps:Draw a line and take AB = 9.3 units on it.From B, measure a distance of 1 unit and mark C on the number line. Mark the midpoint of AC as O.With 'O' as center and OC as radius, draw a semicircle.At B, draw a perpendicular to cut the semicircle at D.With B as center and BD as radius draw an arc to cut the number line at E. Thus, taking B as origin the distance BE = 9.3\sqrt{9.3}9.3​Hence, point E represents 9.3\sqrt{9.3}9.3​ on the number line.

Number Systems - Interactive Study Notes | Bright Tutorials

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BRIGHT TUTORIALS

BRIGHT TUTORIALSCBSE Class IX | Academic Year 2026-2027
    
    9403781999Excellence in Education
  
  Mathematics | Number SystemsWeb Content &bull; Interactive Notes
  
    Number Systems &mdash; 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 &ldquo;dense&rdquo; with both types!

Identifying Rational vs Irrational
      
        NumberTypeReason
        
          &radic;4Rational&radic;4 = 2 (perfect square)
          &radic;7Irrational7 is not a perfect square
          0.333...RationalRecurring decimal = 1/3
          0.10100100010...IrrationalNon-terminating, non-recurring
          &pi;IrrationalNon-terminating, non-recurring
          22/7RationalIt is p/q form (just an approximation of &pi;)

Rationalisation &mdash; Quick Method
      To rationalise a denominator with surds, multiply top and bottom by the conjugate:
      
        Conjugate of (a + &radic;b) is (a &minus; &radic;b)
        Conjugate of (&radic;a &minus; &radic;b) is (&radic;a + &radic;b)
      
      The denominator becomes rational because (a+b)(a&minus;b) = a&sup2; &minus; b&sup2;.

Quick Self-Check
      
        Is &radic;(16/9) rational or irrational? (Rational: = 4/3)
        Simplify: &radic;50 + &radic;18 (= 5&radic;2 + 3&radic;2 = 8&radic;2)
        Rationalise: 1/(&radic;3 + 1) (= (&radic;3 &minus; 1)/2)
        Find: 81/3 (= 2)

Bright Tutorials | Hariom Nagar, Nashik Road | 9403781999 | brighttutorials.in

Question 8

Question 5Rationalize the denominators of the following:(i) 17\dfrac{1}{\sqrt{7}}7​1​(ii) 17−6\dfrac{1}{\sqrt{7} - \sqrt{6} }7​−6​1​(iii)15+2\dfrac{1}{\sqrt{5} + \sqrt{2} }5​+2​1​(iv)17−2\dfrac{1}{\sqrt{7} - 2 }7​−21​

Accepted Answer

(i)17\dfrac{1}{\sqrt{7}}7​1​Multiply by (7)(\sqrt{7})(7​) in numerator and denominator= 17\dfrac{1}{\sqrt{7}}7​1​ x 77\dfrac{\sqrt{7}}{\sqrt{7}}7​7​​= 77\dfrac{\sqrt{7}}{7}77​​Hence, 17\dfrac{1}{\sqrt{7}}7​1​ = 77\dfrac{\sqrt{7}}{7}77​​(ii) 17−6\dfrac{1}{\sqrt{7} - \sqrt{6}}7​−6​1​Multiply by 17+6\dfrac{1}{\sqrt{7} + \sqrt{6}}7​+6​1​ in numerator and denominator= 17−6\dfrac{1}{\sqrt{7} - \sqrt{6} }7​−6​1​ x 7+67+6\dfrac{\sqrt{7}+ \sqrt{6}}{\sqrt{7} + \sqrt{6}}7​+6​7​+6​​= 7+6(7)2−(6)2\dfrac{\sqrt{7} + \sqrt{6}}{(\sqrt{7})^2 - (6)^2 }(7​)2−(6)27​+6​​= 7+67−6\dfrac{\sqrt{7} + \sqrt{6} }{7 - 6}7−67​+6​​= 7+61\dfrac{\sqrt{7} + \sqrt{6} }{1}17​+6​​= (7+6)(\sqrt{7} + \sqrt{6})(7​+6​)=Hence, 17−6\dfrac{1}{\sqrt{7} - \sqrt{6}}7​−6​1​ = (7+6)(\sqrt{7} + \sqrt{6})(7​+6​)(iii)15+2\dfrac{1}{\sqrt{5} + \sqrt{2}}5​+2​1​Multiply by 5+2{\sqrt{5} + \sqrt{2}}5​+2​ in numerator and denominator= 15+2\dfrac{1}{\sqrt{5} + \sqrt{2}}5​+2​1​ x 5−25−2\dfrac{\sqrt{5} - \sqrt{2}}{\sqrt{5} - \sqrt{2}}5​−2​5​−2​​= 5−2(5)2−(2)2\dfrac{\sqrt{5} - \sqrt{2}} {(\sqrt{5})^2 - (\sqrt{2})^2 }(5​)2−(2​)25​−2​​= 5−25−2\dfrac{\sqrt{5} - \sqrt{2}} {5 - 2}5−25​−2​​= 5−23\dfrac{\sqrt{5} - \sqrt{2}} {3}35​−2​​=Hence, 15+2\dfrac{1}{\sqrt{5} + \sqrt{2}}5​+2​1​ = 5−23\dfrac{\sqrt{5} - \sqrt{2}} {3}35​−2​​(iv)17−2\dfrac{1}{\sqrt{7} - 2}7​−21​Multiply by 7+2{\sqrt{7} + {2}}7​+2 in numerator and denominator= 17−2\dfrac{1}{\sqrt{7} - 2}7​−21​ X 7+27+2\dfrac{\sqrt{7} + 2 }{\sqrt{7} + 2 }7​+27​+2​= 7+2(7)2−(2)2\dfrac{\sqrt{7} + 2} {(\sqrt{7})^2 - (2)^2 }(7​)2−(2)27​+2​= 7+27−4\dfrac{\sqrt{7} + 2}{7 - 4}7−47​+2​= 7+23\dfrac{\sqrt{7} + 2}{3}37​+2​= Hence, 17−2\dfrac{1}{\sqrt{7} - 2 }7​−21​ = 7+23\dfrac{\sqrt{7} + 2}{3}37​+2​

Number Systems - Interactive Study Notes | Bright Tutorials

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BRIGHT TUTORIALS

BRIGHT TUTORIALSCBSE Class IX | Academic Year 2026-2027
    
    9403781999Excellence in Education
  
  Mathematics | Number SystemsWeb Content &bull; Interactive Notes
  
    Number Systems &mdash; 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 &ldquo;dense&rdquo; with both types!

Identifying Rational vs Irrational
      
        NumberTypeReason
        
          &radic;4Rational&radic;4 = 2 (perfect square)
          &radic;7Irrational7 is not a perfect square
          0.333...RationalRecurring decimal = 1/3
          0.10100100010...IrrationalNon-terminating, non-recurring
          &pi;IrrationalNon-terminating, non-recurring
          22/7RationalIt is p/q form (just an approximation of &pi;)

Rationalisation &mdash; Quick Method
      To rationalise a denominator with surds, multiply top and bottom by the conjugate:
      
        Conjugate of (a + &radic;b) is (a &minus; &radic;b)
        Conjugate of (&radic;a &minus; &radic;b) is (&radic;a + &radic;b)
      
      The denominator becomes rational because (a+b)(a&minus;b) = a&sup2; &minus; b&sup2;.

Quick Self-Check
      
        Is &radic;(16/9) rational or irrational? (Rational: = 4/3)
        Simplify: &radic;50 + &radic;18 (= 5&radic;2 + 3&radic;2 = 8&radic;2)
        Rationalise: 1/(&radic;3 + 1) (= (&radic;3 &minus; 1)/2)
        Find: 81/3 (= 2)

Bright Tutorials | Hariom Nagar, Nashik Road | 9403781999 | brighttutorials.in

Question 9

Question 3Simplify:(i) (223)\left(2^{\smash{\frac{2}{3}}} \right)(232​). (215)\left(2^{\smash{\frac{1}{5}}} \right)(251​)(ii) (133)7\Big(\dfrac{1}{3^3}\Big)^7(331​)7(iii) 11121114\dfrac{11^\frac{1}{2}}{11^\frac{1}{4}}1141​1121​​(iv) (712)\left(7^{\smash{\frac{1}{2}}} \right)(721​). (812)\left(8^{\smash{\frac{1}{2}}} \right)(821​)

Accepted Answer

(i) (223)\left(2^{\smash{\frac{2}{3}}} \right)(232​). (215)\left(2^{\smash{\frac{1}{5}}} \right)(251​)= (223+15)\left(2^{\smash{\frac{2}{3} + \frac{1}{5}}} \right)(232​+51​)= ((2)10+315)\left((2)^{\smash{\frac{10+3}{15}}} \right)((2)1510+3​)= ((2)1315)\left((2)^{\smash{\frac{13}{15}}} \right)((2)1513​)Hence, (223)\left(2^{\smash{\frac{2}{3}}} \right)(232​). (215)\left(2^{\smash{\frac{1}{5}}} \right)(251​) = ((2)1315)\left((2)^{\smash{\frac{13}{15}}} \right)((2)1513​)(ii) (133)7\Big(\dfrac{1}{3^3}\Big)^7(331​)7=(133)7=(1733×7)=(1321)= \Big(\dfrac{1}{3^3}\Big)^7 \[1em] = \Big(\dfrac{1^7}{3^{3 \times 7}}\Big) \[1em] = \Big(\dfrac{1}{3^{21}}\Big)=(331​)7=(33×717​)=(3211​)Hence, (133)7=(1321)\Big(\dfrac{1}{3^3}\Big)^7 = \Big(\dfrac{1}{3^{21}}\Big)(331​)7=(3211​)(iii) 11121114\dfrac{11^\frac{1}{2}}{11^\frac{1}{4}}1141​1121​​= (1112−14)\left(11^{\smash{\frac{1}{2} - \frac{1}{4}}} \right)(1121​−41​)= (112−14)\left(11^{\smash{\frac{2-1}{4}}} \right)(1142−1​)= (1114)\left(11^{\smash{\frac{1}{4}}} \right)(1141​)Hence, 11121114\dfrac{11^\frac{1}{2}}{11^\frac{1}{4}}1141​1121​​ = (1114)\left(11^{\smash{\frac{1}{4}}} \right)(1141​)(iv) (712)\left(7^{\smash{\frac{1}{2}}} \right)(721​). (812)\left(8^{\smash{\frac{1}{2}}} \right)(821​)=(7×8)12[∵am×bm=(ab)m]=5612= (7 \times 8)^{\dfrac{1}{2}}\quad[\because \text{a}^m \times \text{b}^m = (\text{ab})^m] \[1em] = 56^{\frac{1}{2}}=(7×8)21​[∵am×bm=(ab)m]=5621​Hence, (712).(812)=5612\left(7^{\smash{\frac{1}{2}}} \right).\left(8^{\smash{\frac{1}{2}}} \right) = 56^{\frac{1}{2}}(721​).(821​)=5621​

Number Systems - Interactive Study Notes | Bright Tutorials

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BRIGHT TUTORIALS

BRIGHT TUTORIALSCBSE Class IX | Academic Year 2026-2027
    
    9403781999Excellence in Education
  
  Mathematics | Number SystemsWeb Content &bull; Interactive Notes
  
    Number Systems &mdash; 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 &ldquo;dense&rdquo; with both types!

Identifying Rational vs Irrational
      
        NumberTypeReason
        
          &radic;4Rational&radic;4 = 2 (perfect square)
          &radic;7Irrational7 is not a perfect square
          0.333...RationalRecurring decimal = 1/3
          0.10100100010...IrrationalNon-terminating, non-recurring
          &pi;IrrationalNon-terminating, non-recurring
          22/7RationalIt is p/q form (just an approximation of &pi;)

Rationalisation &mdash; Quick Method
      To rationalise a denominator with surds, multiply top and bottom by the conjugate:
      
        Conjugate of (a + &radic;b) is (a &minus; &radic;b)
        Conjugate of (&radic;a &minus; &radic;b) is (&radic;a + &radic;b)
      
      The denominator becomes rational because (a+b)(a&minus;b) = a&sup2; &minus; b&sup2;.

Quick Self-Check
      
        Is &radic;(16/9) rational or irrational? (Rational: = 4/3)
        Simplify: &radic;50 + &radic;18 (= 5&radic;2 + 3&radic;2 = 8&radic;2)
        Rationalise: 1/(&radic;3 + 1) (= (&radic;3 &minus; 1)/2)
        Find: 81/3 (= 2)

Bright Tutorials | Hariom Nagar, Nashik Road | 9403781999 | brighttutorials.in

Number Systems

Exercise 13

Exercise 14

Exercise 15