Number System: 9 Math


mathematics Class Nine

NCERT Exercise 1.3(part-2) and NCERT Exercise 1.4: 9th math

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 ?

Solution

Numbers which have either 2 or 5 or both as factor of denominator have terminating decimal expansions. Thus, q must have either 2 or 5 or both as factor.

Examples:

`1/2 = 0.5` terminating decimal

`1/3=0.bar(3)` non-terminating repeating.

`1/4 = 0.25` terminating

`1/5 = 0.2` terminating

`1/6=0.bar(16)` non terminating repeating.

`1/7=0.bar(142857)` non-terminating repeating

`1/8=0.125` terminating

`1/9=0.bar(1)` non-terminating repeating.

Question (7) Write three numbers whose decimal expansions are non-terminating non-recurring.

Solution

Decimal expansion of Irrational numbers is non-terminating non-repeating.

`sqrt2, sqrt3`, and `sqrt5` are three numbers whose decimal expansions are non-terminating non-repeating. Because these are irrational numbers.

Question (8) Find three different irrational numbers between the rational numbers `5/7` and `9/11`.

Solution

`5/7` = 0.7142857142857 . . . And

`9/11` = 8181818 . . . .

Thus, possible irrational numbers between `5/7` and `9/11` can be

0.740740074000 . . . .

0.750750075000750000 . .

0.760760076000 . . . .

Question (9) Classify the following numbers as rational or irrational.

(i) `sqrt(23)`

Solution

`sqrt(23)` = 4.79583152. . . . .

Since, decimal expansion of given number is non-repeating non-recurring, thus it is an irrational number.

Thus, Answer = Irrational

(ii) `sqrt(225)`

Solution

`sqrt(225)=15`

Thus, Answer: Rational

(iii) 0.3796

Solution

`0.3796 =3796/10000`

Since, given number can be represent in the form of `p/q` where `q!=0` and p and q are integers, thus given number is rational.

Thus, Answer = Rational

(iv) 7.478478 . . . .

Solution

Given, 7.478478 . . . .

Since, given number is non-repeating recurring, thus it is an Rational Number.

Thus, Answer = Rational

(v) 1.101001000100001 . . . .

Solution

Given, 1.101001000100001 . . . .

Since, given number is non-repeating non-recurring, thus it is an irrational number.

Thus, Answer = Irrational

Solution of NCERT Exercise 1.4

Question (1) Visualize 3.765 on the number line, using successive magnification.

Solution

(a) First point and take span between 3 and 4 on the number line and distribute the span between 10 equal parts.

9 math rational number ncert exercise 1.4 ques1a

(b) Second, now magnify and take span between 3.7 and 3.8 on the number line and distribute the span between 10 equal parts.

9 math rational number ncert exercise 1.4 ques1b

(c) Third, magnify and take span between 3.76 and 3.77 on the number line and distribute the span between 10 equal parts.

9 math rational number ncert exercise 1.4 ques1c

(d) Fourth, magnify and take 3.765 on the number line from the magnification between 3.76 and 3.77

9 math rational number ncert exercise 1.4 ques1d

Question (2) Visualize `4.bar(26)` on the number line, upto 4 decimal places.

Solution

(a) First point and take span between 4 and 5 on the number line and distribute the span between 10 equal parts.

9 math rational number ncert exercise 1.4 ques2a

(b) Second, now magnify and take span between 4.2 and 4.3 on the number line and distribute the span between 10 equal parts.

9 math rational number ncert exercise 1.4 ques2b

(c) Third, magnify and take span between 4.26 and 4.27 on the number line and distribute the span between 10 equal parts.

9 math rational number ncert exercise 1.4 ques2c

(d) Third, magnify and take span between 4.262 and 4.263 on the number line and distribute the span between 10 equal parts.

9 math rational number ncert exercise 1.4 ques2d

(e) Fourth, magnify and take 4.2626 on the number line from the magnification between 4.262 and 4.2623

9 math rational number ncert exercise 1.4 ques2e

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