NCERT

NCERT Solution Class IX Mathematics Number System Question 4 (Ex 1.5)

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Question 4:

Represent √93 on the number line.

Answer:

Mark a line segment OB = 9.3 on number line. Further, take BC of 1 unit. Find the

mid-point D of OC and draw a semi-circle on OC while taking D as its centre. Draw a

perpendicular to line OC passing through point B. Let it intersect the semi-circle at E.

Taking B as centre and BE as radius, draw an arc intersecting number line at F. BF

is √93.

NCERT Solution Class IX Mathematics Number System Question 3 (Ex 1.5)

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Question 3:

Recall, π is defined as the ratio of the circumference (say c) of a circle to its diameter (say d). That is, π=c/d. This seems to contradict the fact that π 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 may not realise that either c or d is irrational. Therefore, the fraction c/d is irrational. Hence, π is irrational.

NCERT Solution Class IX Mathematics Number System Question 2 (Ex 1.5)

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Question 2:

Simplify each of the following expressions:

(i) (3 + √3) (2+√2)        (ii) (3+√3) (3−√3)

(iii) (√5 +√2)2                    (iv) (√5−√2) (√5+√2)

Answer:

(i) (3+√3) (2+√2) = 3(2+√2) +√3(2+√2)

=6+3√2 + 2√3 + √6

(ii) (3+√3)(3−√3)=(3)2 – (√3)2

= 9 – 3 =6

(iii) (√5+√2)2 = (√5)2 + (√2)2 + 2(√5) (√2)

=5+2+2√10 = 7+2√10

(iv) (√5 – √2) (√5+√2) =(√5)2 – (√2)2

5 – 2 = 3 

NCERT Solution Class IX Mathematics Number System Question 1 (Ex 1.5)

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Question 1:

Classify the following numbers as rational or irrational:

 

Answer:

(i) √2 – 5 = 2 − 2.2360679…

= − 0.2360679…
As the decimal expansion of this expression is non-terminating non-recurring, therefore, it is an irrational number.

As it can be represented in p/q form, therefore, it is a rational number.

As it can be represented in p/q form, therefore, it is a rational number.

As the decimal expansion of this expression is non-terminating non-recurring,
therefore, it is an irrational number.

(v) 2π = 2(3.1415 …)
= 6.2830 …

As the decimal expansion of this expression is non-terminating non-recurring,
therefore, it is an irrational number.

NCERT Solution Class IX Mathematics Number System Question 9 (Ex 1.3)

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Question 9:

Classify the following numbers as rational or irrational:

(i)  √23         (ii) √225        (iv) 0.3796

(iv) 7.478478       (v) 1.101001000100001…

Answer:

(i) √23 = 4.79583152331…  

As the decimal expansion of this number is non-terminating non-recurring, therefore, it is an irrational number.

It is a rational number as it can be represented in p/q form.

(iii) 0.3796

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

As the decimal expansion of this number is non-terminating recurring, therefore, it is a rational number.

(v) 1.10100100010000 …

As the decimal expansion of this number is non-terminating non-repeating, therefore, it is an irrational number.

NCERT Solution Class IX Mathematics Number System Question 6 (Ex 1.3)

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Question 6:

Look at several examples of rational numbers in the form 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:

Terminating decimal expansion will occur when denominator q of rational number p/q is either of 2, 4, 5, 8, 10, and so on…

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

NCERT Solution Class IX Mathematics Number System Question 2 (Ex 1.2)

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Question 2:

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:

If numbers such as √4 = 2, √9 = 3 are considered,

Then here, 2 and 3 are rational numbers. Thus, the square roots of all positive
integers are not irrational.

NCERT Solution Class IX Mathematics Number System Question 1 (Ex 1.2)

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Question 1:

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

(ii) Every point on the number line is of the form √m , where m is a natural number.

(iii) Every real number is an irrational number.

Answer:

(i) True; since the collection of real numbers is made up of rational and irrational
numbers.
(ii) False; as negative numbers cannot be expressed as the square root of any other
number.
(iii) False; as real numbers include both rational and irrational numbers. Therefore,
every real number cannot be an irrational number.

NCERT Solution Class IX Mathematics Number System Question 4 (Ex 1.1)

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Question 4:

State whether the following statements are true or false. Give reasons for your answers.

(i) Every natural number is a whole number.
(ii) Every integer is a whole number.
(iii) Every rational number is a whole number.

Answer:

(i) True; since the collection of whole numbers contains all natural numbers.
(ii) False; as integers may be negative but whole numbers are positive. For example:
 −3 is an integer but not a whole number.
(iii) False; as rational numbers may be fractional but whole numbers may not be.          For example: 1/5 is a rational number but not a whole number.

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