Number System: 9 Math

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.

(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.

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

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

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.

(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.

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

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

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

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