# NCERT Solutions for Class 9 Maths Chapter 1 - Number Systems

Practise with TopperLearning’s NCERT Solutions for CBSE Class 9 Mathematics Chapter 1 Number Systems to understand the real number system. Revise concepts such as irrational numbers, whole numbers, natural numbers, integers and more. Go through Maths answers by experts to figure out whether the given decimal is non-terminating repeating or terminating.

Our study resources will strengthen your knowledge of the foundational concepts taught in class according to the latest CBSE Class 9 Maths syllabus. Check our concept insights to grasp the problem-solving techniques shared by experts in our NCERT solutions. These solutions and our other Maths resources will help you revise the number systems thoroughly.

## Chapter 1 - Number Systems Exercise Ex. 1.1

**Concept Insight:**Key idea to answer this question is "every integer is a rational number and zero is a non negative integer". Also 0 can be expressed in form in various ways as 0 divided by any number is 0. simplest is .

3 and 4 can be represented as respectively.

**Concept Insight:**Since there are infinite number of rational numbers between any two numbers so the answer is not unique here. The trick is to convert the number to equivalent form by multiplying and dividing by the number atleast 1 more than the rational numbers to be inserted.

**Concept Insight**: Since there are infinite number of rational numbers between any two numbers so the answer is not unique here. The trick is to convert the number to equivalent form by multiplying and dividing by the number at least 1 more than the rational numbers required.

Alternatively for any two rational numbers a and b, is also a rational number which lies between a and b.

(ii) False, integers include negative of natural numbers as well, which are clearly not whole numbers. For example -1 is an integer but not a whole number.

(iii) False, rational numbers includes fractions and integers as well. For example is a rational number but not whole number.

**Concept Insight:**Key concept involved in this question is the hierarchy of number systems

Since Mathematics is an exact science every fact has a proof but in order to negate a statement even one counter example is sufficient.

## Chapter 1 - Number Systems Exercise Ex. 1.2

(ii) False, Since negative integers cannot be expressed as the square root of any natural number.

(iii) False, real number includes both rational and irrational numbers. So every real number can not be an irrational number.

**Concept Insight:**Mentioning the reasons is important in this problem. Real Numbers consists of rational and irrational numbers and not vice versa. Every real number corresponds to a point on number line and vice versa.

Recall real number includes negative numbers also. Square root of negative numbers is not defined.

For example

Thus the square roots of all positive integers are not irrational

**Concept Insight:**In general only the square root of a prime number is irrational.

Therefore square root of perfect square numbers are rational.

^{2}+1

^{2}

Taking positive square root we get

2. Now construct AB of unit length perpendicular to OA. Join OB

3. Now taking O as centre and OB as radius draw an arc, intersecting number line at point C.

4. Point C represents on number line. [length (OB) = length (OC)]

**Concept Insight:**For a positive integer n, can be located on number line , if is located using Pythagoras Theorem . If is a perfect square then this method is useful.

## Chapter 1 - Number Systems Exercise Ex. 1.3

(ii)

(iii)

(iv)

(v)

(vi)

**Concept Insight:**The decimal expansion of a rational number is either terminating or non terminating recurring.

Decimal expansion terminates in case the prime factors of denominator includes 2 or 5 only.

**Concept Insight:**Multiples of the given decimal expansion can be obtained by simple multiplication with the given constant. Cross check the answer by performing long division.

Multiplying by 10 we get

10x = 6.666 ... (ii)

(ii) - (i) gives

9x = 6

Or x =

(ii)

Let x = 0.4777 ... (i)

10x = 4.777 ...

100x = 47.777 ... (ii)

(ii) - (i) gives

99 x = 43

(iii)

Let x = 0.001001 ...(i)

1000x = 1.001001 ...(ii)

(ii) - (i) gives

x =

**Concept Insight:**The key idea to express a recurring decimal in the p/q form is to multiply the number by the 10n where n = number of digits repeating.

This is done to make the repeating block a whole number part of the decimal. By subtracting the two expressions x can be expressed in the P/q form

10x = 9.9999 ... ...(ii)

(ii) - (i) gives

9x = 9

x = 1

**Concept Insight:**.9999999 ..... is nothing but 1 when expressed in p/q form.

There are 16 digits in repeating block of decimal expansion of .

**Concept Insight:**Maximum number of digits that can repeat will be 1 less than the prime number in denominator.

Terminating decimal may be obtained in the situation where prime factorisation of the denominator of the given fractions are having power of 2 only or 5 only or both.

**Concept Insight:**A rational number in its simplest form will terminate only when prime factors of its denominator consists of 2 or 5 only.

0.505005000051509 ... ... ...

0.72012009200011500007200000 ... ... ...

7.03124509761202 ... ... ... ... ... ...

**Concept Insight:**Recall that a non terminating non recurring decimal is an irrational number. Answer to such questions is not unique.

0.73073007300073000073 ... ... ...

0.79079007900079000079 ... ... ...

**Concept Insight: **There is infinite number of rational and irrational numbers between any two rational numbers. Convert the number into its decimal form to find irrationals between them.

Alternatively following result can be used to answer

As decimal expansion of this number is terminating, so it is a rational number.

**Concept Insight:**A number is rational if its decimal expansion is either terminating or non terminating but recurring. A number which cannot be expressed in p/q form is irrational. Square root of prime numbers is always irrational.

## Chapter 1 - Number Systems Exercise Ex. 1.4

**Concept Insight:**Divide the number line between the number to be represented in 10 parts starting the whole number part.

**Concept Insight:**Divide the number line between the number to be represented in 10 parts starting the whole number part.

## Chapter 1 - Number Systems Exercise Ex. 1.5

**Concept Insight:**Do the simplifications as indicated and see whether the number is terminating, non terminating recurring or neither terminating nor repeating. Remember Sum/difference/Product of a rational and irrational number may or may not be irrational.

**Concept Insight:**Apply the algebraic identities (a+b)

^{2}, (a-b)

^{2},(a+b)(a-b) etc to simplify the given expressions.

Equivalent Identities used here are

**Concept Insight:**A rational number is the number of the form where p and q are

(v) Taking B as centre and BE as radius draw an arc intersecting number line at F. BF is i.e point F represents on number line

ED

^{2}=EB

^{2}+DB

^{2}Using Pythagoras theorem

**Concept Insight:**This method based on the application of Pythagoras theorem can be used to represent root of any rational number on the number line.

In ODB

**Concept Insight:**Rationalisation of denominator means converting the irrational denominator to rational i.e . removing the radical sign from denominator.A number of the form can be converted to rational form by multiplying with its conjugate. Remember the algebraic identities

## Chapter 1 - Number Systems Exercise Ex. 1.6

**Concept Insight:**Express the number in exponent notation and use the rule Exponent m must be such that it is divisible by n.

**Concept Insight:**Express the number in exponent notation and use the rule of exponents.

**Concept Insight:**Use the rule of exponents

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