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    NCERT Solutions for Class 9 Maths Chapter 1 Number Systems

    Edited By Ramraj Saini | Updated on Oct 06, 2023 08:02 AM IST

    NCERT Solutions For Class 9 Maths Chapter 1 Number Systems

    Number Systems Class 9 Questions And Answers are provided here. These NCERT solutions are prepared by expert team at Careers360 according to latest CBSE syllabus 2023-24. Apart from school exams, the solutions are also beneficial if you are aiming for exams like- National Talent Search Examination (NTSE), Indian National Olympiad (INO), SSC, CAT, etc. The Chapter starts with an introduction which includes rational numbers, whole numbers, and integers followed by important topics like irrational numbers, real numbers, how to represent real numbers on the number line, operations on real numbers, and many more.

    With the understanding of rational numbers, we will also study how to represent square roots of 2, 3, 5, and other non-rational numbers. NCERT solutions for Class 9 Maths chapter 1 Number Systems are written keeping important basics in mind so that a student can get 100% out of it. students can find all the NCERT solutions for class 9 here. practice these to command the concepts.

    Number Systems Class 9 Questions And Answers PDF Free Download

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    Number Systems Class 9 Solutions - Important Formulae

    Any unique real number can be represented on a number line.

    1696559183963

    If r = rational number and s = irrational number

    Then (r + s), (r – s), (r × s), and (r ⁄ s) all are irrational.

    Rules for positive real numbers:

    • √ab = √a × √b

    • √(a/b) = √a/√b

    • (√a + √b) × (√a – √b) = a−b

    • (a + √b) × (a − √b) = a2 − b

    • (√a+√b)2 = a2 + 2√ab + b

    To rationalise the denominator of 1/√(a + b), one must multiply it by √(a – b) / √(a – b), where a and b are both integers.

    Suppose a = real number (greater than 0) and p and q are rational numbers:

    • ap × bp = (ab)(p)

    • (ap)q = apq

    • ap / aq = (a)(p-q)

    • ap / bp = (a/b)p

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    Free download NCERT Solutions for Class 9 Maths Chapter 1 Number Systems for CBSE Exam.

    Number Systems Class 9 NCERT Solutions (Intext Questions and Exercise)

    Number systems class 9 questions and answers - Exercise 1.1

    Q1 Is zero a rational number? Can you write it in the form \frac{p}{q} , where p and q are integers and q ≠ 0?

    Answer:

    Any number that can represent in the form of \frac{p}{q} (where \ q \neq 0) is a rational number

    Now, we can write 0 in the form of \frac{p}{q} for eg. \frac{0}{1},\frac{0}{2},\frac{0}{-1} etc.

    Therefore, 0 is a rational number.

    Q2 Find six rational numbers between 3 and 4.

    Answer:

    There are an infinite number of rational numbers between 3 and 4. one way to take them is
    \Rightarrow 3 = \frac{21}{7} \ and \ 4 = \frac{28}{7}
    Therefore, six rational numbers between 3 and 4 are \frac{22}{7}, \frac{23}{7},\frac{24}{7},\frac{25}{7},\frac{26}{7},\frac{27}{7}

    Q3 Find five rational numbers between \frac{3}{5} and \frac{4}{5} .

    Answer:

    We can write
    \Rightarrow \frac{3}{5}= \frac{3\times 6}{5\times 6} = \frac{18}{30}
    And
    \Rightarrow \frac{4}{5}= \frac{4\times 6}{5\times 6} = \frac{24}{30}
    Therefore, five rational numbers between \frac{3}{5} and \frac{4}{5} . are \frac{19}{30},\frac{20}{30},\frac{21}{30},\frac{22}{30},\frac{23}{30},

    Q4 (i) State whether the following statements are true or false. Give reasons for your answers. (i)Every natural number is a whole number.

    Answer:

    (i) TRUE
    The number that is starting from 1, i.e 1, 2, 3, 4, 5, 6, .................. are natural numbers
    The number that is starting from 0. i.e, 0, 1, 2, 3, 4, 5.............are whole numbers
    Therefore, we can clearly see that the collection of whole numbers contains all natural numbers.

    Q4 (ii) State whether the following statements are true or false. Give reasons for your answers. (ii) Every integer is a whole number.

    Answer:

    (ii) FALSE
    Because integers may be negative or positive but whole numbers are always positive. for eg. -1 is an integer but not a whole number.

    Q4 (iii) State whether the following statements are true or false. Give reasons for your answers.(iii) Every rational number is a whole number.

    Answer:

    (iii) FALSE
    Numbers that can be represented in the form of \frac{p}{q} \ ( where \ q \neq 0) are a rational number.
    And numbers that are starting from 0 i.e. 0,1,2,3,4,......... are whole numbers
    Therefore, we can clearly see that every rational number is not a whole number for eg. \frac{3}{4} is a rational number but not a whole number


    Class 9 maths chapter 1 question answer - Exercise 1.2

    Q1 (i) State whether the following statements are true or false. Justify your answers. (i) Every irrational number is a real number.

    Answer:

    (i) TRUE
    Since the real numbers are the collection of all rational and irrational numbers.

    Q1 (ii) State whether the following statements are true or false. Justify your answers. (ii) Every point on the number line is of the form \sqrt{m} , where m is a natural number.

    Answer:

    (ii) FALSE
    Because negative numbers are also present on the number line and no negative number can be the square root of any natural number

    Q1 (iii) State whether the following statements are true or false. Justify your answers. (iii) Every real number is an irrational number.

    Answer:

    (iii) FALSE
    As real numbers include both rational and irrational numbers. Therefore, every real number cannot be an irrational number.

    For eg. 4 is a real number but not an irrational number

    Q2 Are the square roots of all positive integers irrational? If not, give an example of the square root of a number that is a rational number.

    Answer:

    NO, Square root of all positive integers is not irrational. For the eg square of 4 is 2 which is a rational number.

    Q3 Show how \sqrt{5} can be represented on the number line.

    Answer:

    We know that
    \sqrt{5} = \sqrt{(2)^2+(1)^2}
    Now,
    1639723473448 Let OA be a line of length 2 unit on the number line. Now, construct AB of unit length perpendicular to OA. and join OB.
    Now, in right angle triangle OAB, by Pythagoras theorem
    OB = \sqrt{(2)^2+(1)^2}=\sqrt{5}
    Now, take O as centre and OB as radius, draw an arc intersecting number line at C. Point C represent \sqrt{5} on a number line.

    Class 9 maths chapter 1 question answer - Exercise: 1.3

    Q1 (i) Write the following in decimal form and say what kind of decimal expansion each has : (i) \frac{36}{100}

    Answer:

    We can write \frac{36}{100} as
    \Rightarrow \frac{36}{100}= 0.36
    Since the decimal expansion ends after a finite number of steps. Hence, it is terminating

    Q1 (ii) Write the following in decimal form and say what kind of decimal expansion each has : (ii) \frac{1}{11}

    Answer:

    We can rewrite \frac{1}{11} as

    \Rightarrow \frac{1}{11}= 0.09090909..... = 0.\overline{09}
    Since decimal expansion repeats itself so it is a non-terminating recurring decimal expansion.

    Q1 (iii) Write the following in decimal form and say what kind of decimal expansion each has : (iii) 4\frac{1}{8}

    Answer:

    We can rewrite 4\frac{1}{8} as

    \Rightarrow 4\frac{1}{8} = \frac{33}{8}= 4.125
    Since the decimal expansion ends after a finite number. Therefore, it is terminating

    Q1 (iv) Write the following in decimal form and say what kind of decimal expansion each has : (iv) \frac{3}{13}

    Answer:

    We can rewrite \frac{3}{13} as

    \Rightarrow \frac{3}{13} = 0.230769230769 = 0.\overline{230769}
    Since decimal expansion repeats itself so it is a non-terminating recurring decimal expansion.

    Q1 (v) Write the following in decimal form and say what kind of decimal expansion each has: (v) \frac{2}{11}

    Answer:

    We can rewrite \frac{2}{11} as

    \Rightarrow \frac{2}{11} = 0.181818......= 0.\overline{18}
    Since decimal expansion repeats itself so it is a non-terminating recurring decimal expansion.

    Q1 (vi) Write the following in decimal form and say what kind of decimal expansion each has : (vi) \frac{329}{400}

    Answer:

    We can rewrite \frac{329}{400} as

    \Rightarrow \frac{329}{400}= 0.8225
    Since decimal expansion ends after finite no. of figures. Hence, it is terminating.

    Q3 (i) Express the following in the form \frac{p}{q} , where p and q are integers and q ≠ 0. (i) 0.\bar{6}

    Answer:

    Let x = 0.\overline6= 0.6666.... -(i)

    Now, multiply by 10 on both sides

    10x= 6.6666...

    \Rightarrow 10x = 6 + x \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \(using \ (i))

    \Rightarrow 9x = 6

    \Rightarrow x = \frac{6}{9} = \frac{2}{3}

    Therefore, \frac{p}{q} form of 0.\bar{6} is \frac{2}{3}

    Q3 (ii) Express the following in the form \frac{p}{q} , where p and q are integers and q ≠ 0. (ii) 0.4\bar{7}

    Answer:

    We can write 0.4\overline7 as

    \Rightarrow 0.4\overline7 = \frac{4}{10}+ \frac{0.777..}{10} -(i)

    Now,

    Let x = 0.\overline7= 0.7777.... -(ii)

    Now, multiply by 10 on both sides

    10x= 7.7777...

    \Rightarrow 10x = 7 + x \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \(using \ (ii))

    \Rightarrow 9x = 7

    \Rightarrow x = \frac{7}{9}
    Now, put the value of x in equation (i). we will get

    \Rightarrow 0.4\overline7 = \frac{4}{10}+ \frac{7}{10\times 9}= \frac{4}{10}+ \frac{7}{90} = \frac{36+7}{90} = \frac{43}{90}

    Therefore, \frac{p}{q} form of 0.4\overline7 is \frac{43}{90}

    Q3 (iii) Express the following in the form \frac{p}{q} , where p and q are integers and q ≠ 0. (iii) 0.\overline{001}

    Answer:

    Let x = 0.\overline{001}= 0.001001.... -(i)

    Now, multiply by 1000 on both sides

    1000x= 1.001001...

    \Rightarrow 1000x = 1 + x \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \(using \ (i))

    \Rightarrow 999x = 1

    \Rightarrow x = \frac{1}{999}

    Therefore, \frac{p}{q} form of 0.\overline{001} is \frac{1}{999}

    Q4 Express 0.99999 .... in the form \frac{p}{q} . Are you surprised by your answer?

    Answer:

    Let x = 0.\overline{9}= 0.9999.... -(i)

    Now, multiply by 10 on both sides

    10x= 9.999....

    \Rightarrow 10x = 9 + x \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \(using \ (i))

    \Rightarrow 9x = 9

    \Rightarrow x = \frac{9}{9} = 1

    Therefore, \frac{p}{q} form of 0.999.... is 1

    The difference between 1 and 0.999999 is o.000001 which is almost negligible.

    Therefore, 0.999 is too much closer to 1. Hence, we can write 0.999999.... as 1

    Q5 What can the maximum number of digits be in the repeating block of digits in the decimal expansion of \frac{1}{17} ? Perform the division to check your answer.

    Answer:

    We can rewrite \frac{1}{17} as

    \Rightarrow \frac{1}{17} = 0.05882352941176470588235294117647= 0.\overline{0588235294117647}
    Therefore, there are total 16 number of digits be in the repeating block of digits in the decimal expansion of \frac{1}{17}

    Q6 Look at several examples of rational numbers in the form \frac{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:

    We can observe that when q is 2, 4, 5, 8, 10… then the decimal expansion is terminating. For example:

    \frac{3}{2}= 1.5 , denominator q = 2^1

    \frac{8}{5}= 1.6 , denominator q = 5^1

    \frac{15}{10} = 1.5 , denominator q =10=2\times 5= 2^1 , 5^1

    Therefore,

    It can be observed that the terminating decimal can be obtained in a condition where prime factorization of the denominator of the given fractions has the power of 2 only or 5 only or both.

    Q7 Write three numbers whose decimal expansions are non-terminating non-recurring.

    Answer:

    Three numbers whose decimal expansions are non-terminating non-recurring are
    1) 0.02002000200002......
    2) 0.15115111511115.......
    3) 0.27227222722227.......

    Q8 Find three different irrational numbers between the rational numbers \frac{5}{7} and \frac{9}{11} .

    Answer:

    We can write \frac{5}{7} as

    \Rightarrow \frac{5}{7} = 0.714285714285.... = 0.\overline{714285}

    And \frac{9}{11} as

    \Rightarrow \frac{9}{11} = 0.818181.... = 0.\overline{81}
    Therefore, three different irrational numbers between the rational numbers \frac{5}{7} and \frac{9}{11} are

    1) 0.72737475....
    2) 0.750760770780...
    3) 0.790780770760....

    Q9 (i) Classify the following numbers as rational or irrational : \sqrt{23}

    Answer:

    We can rewrite \sqrt{23} in decimal form as

    \Rightarrow \sqrt{23} = 4.7958152....

    Now, as the decimal expansion of this number is non-terminating non-recurring.

    Therefore, it is an irrational number.

    Q9 (ii) Classify the following numbers as rational or irrational : \sqrt{225}

    Answer:

    We can rewrite \sqrt{225} as

    \Rightarrow \sqrt{225} = 15
    We can clearly see that it is a rational number because we can represent it in \frac{p}{q} form

    Q9 (iii) Classify the following numbers as rational or irrational : 0.3796

    Answer:

    We can rewrite 0.3796 as

    \Rightarrow 0.3796 = \frac{3796}{10000}
    Now, we can clearly see that it is a rational number as the decimal expansion of this number is terminating and we can also write it in \frac{p}{q} form.

    Q9 (iv) Classify the following numbers as rational or irrational : 7.478478....

    Answer:

    We can rewrite 7.478478.... as

    \Rightarrow 7.478478.... = 7.\overline{478}
    Now, as the decimal expansion of this number is non-terminating recurring. Therefore, it is a rational number.

    Q9 (v) Classify the following numbers as rational or irrational : 1.101001000100001...

    Answer:

    In the case of number 1.101001000100001...
    As the decimal expansion of this number is non-terminating non-repeating. Therefore, it is an irrational number.

    Class 9 maths chapter 1 NCERT solutions - Exercise: 1.4

    Q1 Visualise 3.765 on the number line, using successive magnification.

    Answer:

    3.765 can be visualised as in the following steps.
    First, we draw a number line and mark points on it after that we will divide the number line between points 3 and 4. And then we will divide the points between 3.7 and 3.8 as the number is between both of them.
    1639723522139

    Q2 Visualise 4.\overline{26} on the number line, up to 4 decimal places.

    Answer:

    We can rewrite 4.\overline{26} as \Rightarrow 4.\overline{26}= 4.262626...
    Now, 4.2626 can be visualised as in the following steps.

    1639723563861

    Class 9 maths chapter 1 NCERT solutions - Exercise: 1.5

    Q1 (i) Classify the following numbers as rational or irrational: 2-\sqrt{5}

    Answer:

    Value of \sqrt{5} is 2.23606798....
    Now,
    \Rightarrow 2 - \sqrt{5} = 2 - 2.23606798... = -0.23606798...
    Now,
    Since the number is in non-terminating non-recurring. Therefore, it is an irrational number.

    Q1 (ii) Classify the following numbers as rational or irrational: \left ( 3+\sqrt{23} \right )-\sqrt{23}

    Answer:

    Given number is \left ( 3+\sqrt{23} \right )-\sqrt{23}

    \Rightarrow \left ( 3+\sqrt{23} \right )-\sqrt{23} = 3+\sqrt{23}-\sqrt{23} = 3
    Now, it is clearly a rational number because we can represent it in the form of \frac{p}{q}

    Q1 (iii) Classify the following numbers as rational or irrational: \frac{2\sqrt{7}}{7\sqrt{7}}

    Answer:

    Given number is \frac{2\sqrt{7}}{7\sqrt{7}}

    \Rightarrow \frac{2\sqrt{7}}{7\sqrt{7}} = \frac{2}{7}
    As we can clearly see that it can be represented in \frac{p}{q} form. Therefore, it is a rational number.

    Q1 (iv) Classify the following numbers as rational or irrational: \frac{1}{\sqrt{2}}

    Answer:

    Given number is \frac{1}{\sqrt{2}}

    \Rightarrow \frac{1}{\sqrt{2}}= \frac{\sqrt{2}}{2}=\frac{1.41421356..}{2} =0.7071068...
    Now,
    Clearly, as the decimal expansion of this expression is non-terminating and non-recurring. Therefore, it is an irrational number.

    Q1 (v) Classify the following numbers as rational or irrational: 2\pi

    Answer:

    Given number is 2\pi
    We know that the value of \pi = 3.14159265...
    Now,
    \Rightarrow 2\pi = 2\times 3.14159265... = 6.2831853...
    Now,
    Clearly, as the decimal expansion of this expression is non-terminating and non-recurring. Therefore, it is an irrational number.

    Q2 (i) Simplify each of the following expressions: (3+\sqrt{3})(2+\sqrt{2})

    Answer:

    Given number is (3+\sqrt{3})(2+\sqrt{2})
    Now, we will reduce it into
    \Rightarrow (3+\sqrt{3})(2+\sqrt{2})= 3.2+3.\sqrt{2}+\sqrt{3}.2+\sqrt{3}.\sqrt{2}
    = 6+3\sqrt{2}+2\sqrt{3}+\sqrt{6}

    Therefore, answer is 6+3\sqrt{2}+2\sqrt{3}+\sqrt{6}

    Q2 (ii) Simplify each of the following expressions: \left (3+\sqrt{3} \right )\left ( 3-\sqrt{3} \right )

    Answer:

    Given number is

    \left (3+\sqrt{3} \right )\left ( 3-\sqrt{3} \right )
    Now, we will reduce it into
    \Rightarrow (3+\sqrt{3})(3-\sqrt{3})= \left ( (3)^2-(\sqrt{3})^2 \right ) \left ( using \ (a+b)(a-b)=a^2-b^2 \right )
    =9 - 3 = 6

    Therefore, answer is 6

    Q2 (iii) Simplify each of the following expressions: \left (\sqrt{5}+\sqrt{2} \right )^{2}

    Answer:

    Given number is \left (\sqrt{5}+\sqrt{2} \right )^{2}

    Now, we will reduce it into
    \Rightarrow \left (\sqrt{5}+\sqrt{2} \right )^{2} = \left ( (\sqrt{5})^2+(\sqrt{2})^2+2.\sqrt{5}.\sqrt{2} \right ) \left ( using \ (a+b)^2=a^2+b^2 +2ab\right )
    =5+2+2\sqrt{10}
    =7+2\sqrt{10}


    Therefore, the answer is 7+2\sqrt{10}

    Q2 (iv) Simplify each of the following expressions: \left (\sqrt{5}-\sqrt{2} \right )\left ( \sqrt{5}+\sqrt{2} \right )

    Answer:

    Given number is \left (\sqrt{5}-\sqrt{2} \right )\left ( \sqrt{5}+\sqrt{2} \right )

    Now, we will reduce it into
    \Rightarrow \left (\sqrt{5}-\sqrt{2} \right )\left ( \sqrt{5}+\sqrt{2} \right )= \left ( (\sqrt{5})^2-(\sqrt{2})^2 \right ) \left ( using \ (a+b)(a-b)=a^2-b^2\right )
    =5-2
    =3

    Therefore, the answer is 3 .

    Q3 Recall, \pi is defined as the ratio of the circumference (say c) of a circle to its diameter (say d). That is, \pi=\frac{c}{d} ⋅ This seems to contradict the fact that \pi is irrational. How will you resolve this contradiction?

    Answer:

    There is no contradiction.
    When we measure a length with scale or any other instrument, we only obtain an approximate rational value. We never obtain an exact value.
    For this reason, we cannot say that either c or d is irrational.
    Therefore, the fraction \frac{c}{d} is irrational. Hence, the value of \pi is approximately equal to \frac{22}{7} = 3.142857....

    Therefore, \pi is irrational.

    Q4 Represent \sqrt{9.3} on the number line.

    Answer:

    1639723609742 Draw a line segment OB of 9.3 unit. Then, extend it to C so that BC = 1 unit. Find the mid-point D of OC and draw a semi-circle on OC while taking D as its centre and OD as the radius. Now, Draw a perpendicular to line OC passing through point B and intersecting the semi-circle at E. Now, Take B as the centre and BE as radius, draw an arc intersecting the number line at F. the length BF is 9\sqrt{3} units.

    Q5 (i) Rationalise the denominators of the following: \frac{1}{\sqrt{7}}

    Answer:

    Given number is \frac{1}{\sqrt{7}}

    Now, on rationalisation, we will get

    \Rightarrow \frac{1}{\sqrt 7} = \frac{1}{\sqrt 7}\times \frac{\sqrt 7}{\sqrt 7 } = \frac{\sqrt7}{7}

    Therefore, the answer is \frac{\sqrt7}{7}

    Q5 (ii) Rationalise the denominators of the following: \frac{1}{\sqrt{7}-\sqrt{6}}

    Answer:

    Given number is \frac{1}{\sqrt{7}-\sqrt{6}}

    Now, on rationalisation, we will get

    \Rightarrow \frac{1}{\sqrt 7-\sqrt6} = \frac{1}{\sqrt 7-\sqrt 6}\times \frac{\sqrt 7+\sqrt 6 }{\sqrt 7+\sqrt 6 } = \frac{\sqrt7+\sqrt 6}{(\sqrt7)^2-(\sqrt6)^2} = \frac{\sqrt7+\sqrt6}{7-6} = \sqrt7+\sqrt6


    Therefore, the answer is \sqrt7+\sqrt6

    Q5 (iii) Rationalise the denominators of the following: \frac{1}{\sqrt{5}+\sqrt{2}}

    Answer:

    Given number is \frac{1}{\sqrt{5}+\sqrt{2}}

    Now, on rationalisation, we will get

    \\= \frac{1}{\sqrt 5+\sqrt2} \\\\= \frac{1}{\sqrt 5+\sqrt 2}\times \frac{\sqrt 5-\sqrt 2 }{\sqrt 5-\sqrt 2 }\\\\ = \frac{\sqrt5-\sqrt 2}{(\sqrt5)^2-(\sqrt2)^2} \\\\= \frac{\sqrt5-\sqrt2}{5-2}
    = \frac{\sqrt5-\sqrt2}{3}

    Therefore, the answer is \frac{\sqrt5-\sqrt2}{3}

    Q5 (iv) Rationalise the denominators of the following: \frac{1}{\sqrt{7}-2}

    Answer:

    Given number is \frac{1}{\sqrt{7}-2}

    Now, on rationalisation, we will get

    \Rightarrow \frac{1}{\sqrt 7-2} = \frac{1}{\sqrt 7- 2}\times \frac{\sqrt 7+ 2 }{\sqrt 7+ 2 } = \frac{\sqrt7+ 2}{(\sqrt7)^2-(2)^2} = \frac{\sqrt7+2}{7-4}
    = \frac{\sqrt7+2}{3}

    Therefore, answer is \frac{\sqrt7+2}{3}

    Class 9 number systems ncert solutions - Exercise: 1.6

    Q1 (i) Find : 64^{\frac{1}{2}}

    Answer:

    Given number is 64^{\frac{1}{2}}

    Now, on simplifying it we will get

    \Rightarrow 64^{\frac{1}{2}} = (8^2)^\frac{1}{2} = 8

    Therefore, answer is 8

    Q1 (ii) Find : 32^{\frac{1}{5}}

    Answer:

    Given number is 32^{\frac{1}{5}}

    Now, on simplifying it we will get

    \Rightarrow 32^{\frac{1}{5}} = (2^5)^\frac{1}{5} = 2

    Therefore, the answer is 2

    Q1 (iii) Find : 125^{\frac{1}{3}}

    Answer:

    Given number is 125^{\frac{1}{3}}

    Now, on simplifying it we will get

    \Rightarrow 125^{\frac{1}{3}} = (5^3)^\frac{1}{3} = 5

    Therefore, the answer is 5

    Q2 (i) Find : 9^{\frac{3}{2}}

    Answer:

    Given number is 9^{\frac{3}{2}}

    Now, on simplifying it we will get

    \Rightarrow 9^{\frac{3}{2}} = (3^2)^\frac{3}{2} = 3^3 = 27

    Therefore, the answer is 27

    Q2 (ii) Find : 32^{\frac{2}{5}}

    Answer:

    Given number is 32^{\frac{2}{5}}

    Now, on simplifying it we will get

    \Rightarrow 32^{\frac{2}{5}} = (2^5)^\frac{2}{5} = 2^2 = 4

    Therefore, the answer is 4

    Q2 (iii) Find : 16^{\frac{3}{4}}

    Answer:

    Given number is 16^{\frac{3}{4}}

    Now, on simplifying it we will get

    \Rightarrow 16^{\frac{3}{4}} = (2^4)^\frac{3}{4} = 2^3 = 8

    Therefore, the answer is 8

    Q2 (iv) Find : 125^{\frac{-1}{3}}

    Answer:

    Given number is 125^{\frac{-1}{3}}

    Now, on simplifying it we will get

    \Rightarrow 125^{\frac{-1}{3}} = (5^3)^\frac{-1}{3} = 5^{-1} = \frac{1}{5}
    Therefore, the answer is \frac{1}{5}

    Q3 (i) Simplify : 2^{\frac{2}{3}}.2^{\frac{1}{5}}

    Answer:

    Given number is 2^{\frac{2}{3}}.2^{\frac{1}{5}}

    Now, on simplifying it we will get

    \Rightarrow 2^{\frac{2}{3}}.2^{\frac{1}{5}} = 2^{\frac{2}{3}+\frac{1}{5}} = 2^{\frac{10+3}{15}} = 2^\frac{13}{15} \left ( \because a^n.a^m = a^{n+m} \right )

    Therefore, the answer is 2^{\frac{13}{15}}

    Q3 (ii) Simplify : \left (\frac{1}{3^{3}} \right )^{7}

    Answer:

    Given number is \left (\frac{1}{3^{3}} \right )^{7}

    Now, on simplifying it we will get

    \Rightarrow \left ( \frac{1}{3^3} \right )^7= \frac{1^7}{3^{3\times7}} = \frac{1}{3^{21}} = 3^{-21} \left ( \because (a^n)^m = a^{n.m} \ and \ \frac{1}{a^m}= a^{-m}\right )

    Therefore, the answer is 3^{-21}

    Q3 (iii) Simplify : \frac{11^{\frac{1}{2}}}{11^{\frac{1}{4}}}

    Answer:

    Given number is \frac{11^{\frac{1}{2}}}{11^{\frac{1}{4}}}

    Now, on simplifying it we will get

    \Rightarrow \frac{11^{\frac{1}{2}}}{11^{\frac{1}{4}}} = 11^{\frac{1}{2}-\frac{1}{4}}= 11^{\frac{2-1}{4}} = 11^\frac{1}{4} \left ( \because \frac{a^n}{a^m}= a^{n-m}\right )
    Therefore, the answer is 11^{\frac{1}{4}}

    Q3 (iv) Simplify : 7^{\frac{1}{2}}.8^{\frac{1}{2}}

    Answer:

    Given number is 7^{\frac{1}{2}}.8^{\frac{1}{2}}

    Now, on simplifying it we will get

    \Rightarrow 7^{\frac{1}{2}}.8^{\frac{1}{2}}= (7\times8)^{\frac{1}{2}} = 56^{\frac{1}{2}} \left ( \because a^n.b^n=(a.b)^n\right )
    Therefore, the answer is 56^{\frac{1}{2}}

    Number systems class 9 solutions - Topics

    • Irrational Numbers
    • Real Numbers and their Decimal Expansions
    • Representing Real Numbers on the Number Line
    • Operations on Real Numbers
    • Laws of Exponents for Real Numbers

    Also Read :

    Key Features of NCERT Solutions for Class 9 Maths Chapter 1 Number Systems

    • In-depth Knowledge: This chapter provides in-depth knowledge of numbers and their applications.
    • Conceptual Clarity: By studying this chapter, you will gain a profound conceptual understanding of the number system.
    • Assistance: NCERT solutions for Class 9 Maths Chapter 1 are available to assist you when you encounter difficulties with problems related to this chapter.
    • Total Exercises: The chapter consists of a total of 6 exercises.
    • Number of Questions: These exercises encompass 27 questions in total.
    • Weightage: Chapter 1 holds a weightage of 8 marks in the CBSE Class 9 final examination.
    • Comprehensive Solutions: NCERT maths chapter 1 class 9 cover the solutions to each and every question present in the practice exercises of the NCERT syllabus.

    Interested students can practice class 9 maths ch 1 question answer using the following exercises.

    NCERT Solutions for Class 9 - Chapter Wise

    Chapter No.
    Chapter Name
    Chapter 1
    Number Systems
    Chapter 2
    Chapter 3
    Chapter 4
    Chapter 5
    Chapter 6
    Chapter 7
    Chapter 8
    Chapter 9
    Chapter 10
    Chapter 11
    Chapter 12
    Chapter 13
    Chapter 14
    Chapter 15

    NCERT Solitions for Class 9 - Subject Wise

    How to Use NCERT Solutions for Class 9 Maths Chapter 1 Number Systems?

    • First of all, go through the conceptual text given in the book before the exercises.
    • After covering the conceptual theory, you must go through some examples to understand the application of that particular concept.
    • After observing the application, come to the practice exercises available in the textbook
    • While solving the practice exercises, you can take the help of NCERT solutions for Class 9 Maths chapter 1 Number Systems to boost your preparation.

    NCERT Books And NCERT Syllabus

    Keep working hard & happy learning!

    Frequently Asked Question (FAQs)

    1. What are the important topics in NCERT class 9 maths chapter 1 Number Systems ?

    The basic concept of the number system, rational numbers, whole numbers, and integers, irrational numbers, real numbers, representation of real numbers on the number line, operations on real numbers are the important topics of this chapter. Students can practice math 9th class chapter 1 solutions to get in-depth understanding of concepts.

    2. Which are the most difficult chapters of NCERT Class 9 Maths syllabus?

    Most of the students consider geometry especially triangles as the most difficult chapter in the CBSE Class 9 Maths. To solve more problems students can also refer to NCERT exemplar questions.

    3. Does CBSE provides the solutions of NCERT for Class 9 Maths ?

    No, CBSE doesn’t provide NCERT solutions for any class or subject. To get a good score in CBSE exams students can follow NCERT syllabus and book.

    4. Where can I find the complete solutions of NCERT for Class 9 Maths ?

    Here, students can get detailed NCERT solutions for Class 9 Maths  by clicking on the link. practicing these real numbers class 9 solutions are important as these provide confidence to students which ultimately lead to high score.

    5. How does the NCERT solutions are helpful ?

    NCERT solutions are helpful for the students if they stuck while solving NCERT problems. Also, these solutions are provided in a very detailed manner which will give them conceptual clarity.

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    Get answers from students and experts

    A block of mass 0.50 kg is moving with a speed of 2.00 ms-1 on a smooth surface. It strikes another mass of 1.00 kg and then they move together as a single body. The energy loss during the collision is

    Option 1)

    0.34\; J

    Option 2)

    0.16\; J

    Option 3)

    1.00\; J

    Option 4)

    0.67\; J

    A person trying to lose weight by burning fat lifts a mass of 10 kg upto a height of 1 m 1000 times.  Assume that the potential energy lost each time he lowers the mass is dissipated.  How much fat will he use up considering the work done only when the weight is lifted up ?  Fat supplies 3.8×107 J of energy per kg which is converted to mechanical energy with a 20% efficiency rate.  Take g = 9.8 ms−2 :

    Option 1)

    2.45×10−3 kg

    Option 2)

     6.45×10−3 kg

    Option 3)

     9.89×10−3 kg

    Option 4)

    12.89×10−3 kg

     

    An athlete in the olympic games covers a distance of 100 m in 10 s. His kinetic energy can be estimated to be in the range

    Option 1)

    2,000 \; J - 5,000\; J

    Option 2)

    200 \, \, J - 500 \, \, J

    Option 3)

    2\times 10^{5}J-3\times 10^{5}J

    Option 4)

    20,000 \, \, J - 50,000 \, \, J

    A particle is projected at 600   to the horizontal with a kinetic energy K. The kinetic energy at the highest point

    Option 1)

    K/2\,

    Option 2)

    \; K\;

    Option 3)

    zero\;

    Option 4)

    K/4

    In the reaction,

    2Al_{(s)}+6HCL_{(aq)}\rightarrow 2Al^{3+}\, _{(aq)}+6Cl^{-}\, _{(aq)}+3H_{2(g)}

    Option 1)

    11.2\, L\, H_{2(g)}  at STP  is produced for every mole HCL_{(aq)}  consumed

    Option 2)

    6L\, HCl_{(aq)}  is consumed for ever 3L\, H_{2(g)}      produced

    Option 3)

    33.6 L\, H_{2(g)} is produced regardless of temperature and pressure for every mole Al that reacts

    Option 4)

    67.2\, L\, H_{2(g)} at STP is produced for every mole Al that reacts .

    How many moles of magnesium phosphate, Mg_{3}(PO_{4})_{2} will contain 0.25 mole of oxygen atoms?

    Option 1)

    0.02

    Option 2)

    3.125 × 10-2

    Option 3)

    1.25 × 10-2

    Option 4)

    2.5 × 10-2

    If we consider that 1/6, in place of 1/12, mass of carbon atom is taken to be the relative atomic mass unit, the mass of one mole of a substance will

    Option 1)

    decrease twice

    Option 2)

    increase two fold

    Option 3)

    remain unchanged

    Option 4)

    be a function of the molecular mass of the substance.

    With increase of temperature, which of these changes?

    Option 1)

    Molality

    Option 2)

    Weight fraction of solute

    Option 3)

    Fraction of solute present in water

    Option 4)

    Mole fraction.

    Number of atoms in 558.5 gram Fe (at. wt.of Fe = 55.85 g mol-1) is

    Option 1)

    twice that in 60 g carbon

    Option 2)

    6.023 × 1022

    Option 3)

    half that in 8 g He

    Option 4)

    558.5 × 6.023 × 1023

    A pulley of radius 2 m is rotated about its axis by a force F = (20t - 5t2) newton (where t is measured in seconds) applied tangentially. If the moment of inertia of the pulley about its axis of rotation is 10 kg m2 , the number of rotations made by the pulley before its direction of motion if reversed, is

    Option 1)

    less than 3

    Option 2)

    more than 3 but less than 6

    Option 3)

    more than 6 but less than 9

    Option 4)

    more than 9

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    A career as a merchandiser requires one to promote specific products and services of one or different brands, to increase the in-house sales of the store. Merchandising job focuses on enticing the customers to enter the store and hence increasing their chances of buying a product. Although the buyer is the one who selects the lines, it all depends on the merchandiser on how much money a buyer will spend, how many lines will be purchased, and what will be the quantity of those lines. In a career as merchandiser, one is required to closely work with the display staff in order to decide in what way a product would be displayed so that sales can be maximised. In small brands or local retail stores, a merchandiser is responsible for both merchandising and buying. 

    2 Jobs Available
    Procurement Manager

    The procurement Manager is also known as  Purchasing Manager. The role of the Procurement Manager is to source products and services for a company. A Procurement Manager is involved in developing a purchasing strategy, including the company's budget and the supplies as well as the vendors who can provide goods and services to the company. His or her ultimate goal is to bring the right products or services at the right time with cost-effectiveness. 

    2 Jobs Available
    Production Planner

    Individuals who opt for a career as a production planner are professionals who are responsible for ensuring goods manufactured by the employing company are cost-effective and meets quality specifications including ensuring the availability of ready to distribute stock in a timely fashion manner. 

    2 Jobs Available
    Product Manager

    A Product Manager is a professional responsible for product planning and marketing. He or she manages the product throughout the Product Life Cycle, gathering and prioritising the product. A product manager job description includes defining the product vision and working closely with team members of other departments to deliver winning products.  

    3 Jobs Available
    ITSM Manager

    ITSM Manager is a professional responsible for heading the ITSM (Information Technology Service Management) or (Information Technology Infrastructure Library) processes. He or she ensures that operation management provides appropriate resource levels for problem resolutions. The ITSM Manager oversees the level of prioritisation for the problems, critical incidents, planned as well as proactive tasks. 

    3 Jobs Available
    Information Security Manager

    Individuals in the information security manager career path involves in overseeing and controlling all aspects of computer security. The IT security manager job description includes planning and carrying out security measures to protect the business data and information from corruption, theft, unauthorised access, and deliberate attack 

    3 Jobs Available
    Computer Programmer

    Careers in computer programming primarily refer to the systematic act of writing code and moreover include wider computer science areas. The word 'programmer' or 'coder' has entered into practice with the growing number of newly self-taught tech enthusiasts. Computer programming careers involve the use of designs created by software developers and engineers and transforming them into commands that can be implemented by computers. These commands result in regular usage of social media sites, word-processing applications and browsers.

    3 Jobs Available
    Computer System Analyst

    Individuals in the computer systems analyst career path study the hardware and applications that are part of an organization's computer systems, as well as how they are used. They collaborate closely with managers and end-users to identify system specifications and business priorities, as well as to assess the efficiency of computer systems and create techniques to boost IT efficiency. Individuals who opt for a career as a computer system analyst support the implementation, modification, and debugging of new systems after they have been installed.

    2 Jobs Available
    Test Manager

    A Test Manager is a professional responsible for planning, coordinating and controlling test activities. He or she develops test processes and strategies to analyse and determine test methods and tools for test activities. The test manager jobs involve documenting tests that have been carried out, analysing and evaluating software quality to determine further recommended procedures. 

    2 Jobs Available
    Azure Developer

    A career as Azure Developer comes with the responsibility of designing and developing cloud-based applications and maintaining software components. He or she possesses an in-depth knowledge of cloud computing and Azure app service. 

    2 Jobs Available
    Deep Learning Engineer

    A Deep Learning Engineer is an IT professional who is responsible for developing and managing data pipelines. He or she is knowledgeable about analyzing and storing data collected from various sources.  A Career as a Deep Learning Engineer needs to help the  data scientists and analysts to create effective data sets.

    2 Jobs Available
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