Real Number System Class 9th Mathematics Term 1 Tamilnadu Board Solution

Class 9^th Mathematics Term 1 Tamilnadu Board Solution

Exercise 2.1

1. State whether the following statements are true or false. i. Every natural…
2. Is zero a rational number? Give reasons for your answer.
3. Find any two rational numbers between - 5/7 and - 2/7

Exercise 2.2

1. Convert the following rational numbers into decimals and state the kind of…
2. Without actual division, find which of the following rational numbers have…
3. Express the following decimal expansions into rational numbers. i. 0. bar 18 ii.…
4. Explain 1/13 in decimal form. Find the number of digits in the repeating block.…
5. Find the decimal expansions of 1/7 and 2/7 by division method. Without using the…

Exercise 2.3

1. Locate √5 on the number line.
2. Find any three irrational numbers between √3 and √5.
3. Find any two irrational numbers between 3 and 3.5.
4. Find any two irrational numbers between 0.15 and 0.16.
5. Insert any two irrational numbers between 4/7 and 5/7 .
6. Find any two irrational numbers between √3 and 2.
7. Find a rational number and also an irrational number between 1.1011001110001...…
8. Find any two rational numbers between 0.12122122212222... and 0.2122122212222...…

Exercise 2.5

1. A number having non-terminating and recurring decimal expansion is -A. an…
2. If a number has a non-terminating non-recurring decimal expansion then the…
3. Decimal form of - 3/4 isA. −0.75 B. −0.50 C. −0.25 D. −0.125
4. The p/q form of 0. bar 3 isA. 1/7 B. 2/7 C. 1/3 D. 2/3
5. Which one of the following is not true?A. Every natural number is a rational…
6. Which one of the following has a terminating decimal expansion?A. 5/32 B. 7/9 C.…
7. Which one of the following is an irrational number?A. π B. root 9 C. 1/4 D. 1/5…
8. Which of the following are irrational numbers? i. root 2 + root 3 ii. root 4 +…

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Exercise 2.1

Question 1.

State whether the following statements are true or false.

i. Every natural number is a whole number.

ii. Every whole number is a natural number.

iii. Every integer is a rational number.

iv. Every rational number is a whole number.

v. Every rational number is an integer.

vi. Every integer is a whole number.

Answer:

(i) Yes, because natural numbers are 1, 2, 3, …, and so on

And whole numbers are 0, 1, 2, 3, …,

Therefore, Every natural number is a whole number.

(ii) No,

0 is a whole number, but not a natural number.

(iii) Yes, Any integer 'p' can be represented in the form  and 1≠0,

Therefore, every integar is a rational number.

(iv) No,  is a rational number, but not a whole number.

(v) No,  is a rational number, but not an integer.

(vi) No, negative integers are not whole numbers,

For example, -1 is an integer but not a whole number.

Question 2.

Is zero a rational number? Give reasons for your answer.

Answer:

Yes, 0 can be re-written as , where q is any non-negative integer and therefore 0 is a rational number.

Question 3.

Find any two rational numbers between  and 

Answer:

As,

-5 < -4 < -3 < -2

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Exercise 2.2

Question 1.

Convert the following rational numbers into decimals and state the kind of decimal expansion.

i.  ii. 

iii.  iv. 

v.  vi. 

vii.  viii. 

Answer:

i.  , the decimal expansion is terminating.

ii. , the decimal expansion is non-terminating and recurring.

iii. , the decimal expansion is non-terminating and recurring.

iv. , the decimal expansion is terminating.

v. , the decimal expansion is non-terminating and recurring.

the decimal expansion is non-terminating and recurring.

vi.  the decimal expansion is non-terminating and recurring.

vii. , the decimal expansion is non-terminating and recurring.

viii. , the decimal expansion is terminating.

Question 2.

Without actual division, find which of the following rational numbers have terminating decimal expansion.

i.  ii. 

iii.  iv. 

Answer:

i.  , As the rational number can be represented in form , therefore its terminating.

ii. , As the rational number can't be represented in form , therefore its non-terminating and recurring.

iii.  , As the rational number can be represented in form , therefore its terminating.

iv. , As the rational number can't be represented in form , therefore its non-terminating and recurring.

Question 3.

Express the following decimal expansions into rational numbers.

i.  ii. 

iii.  iv. 

v.  vi. 

Answer:

i. let 

⇒ 100x = 18.181818 … = 18 + 0.181818…

⇒ 100x = 18 + x

⇒ 99x = 18

ii. let 

⇒ 1000x = 427.427427 … = 18 + 0.427427427…

⇒ 1000x = 427 + x

⇒ 999x = 427

iii. let 

⇒ 10000x = 1.00010001 … = 1 + 0.00010001…

⇒ 10000x = 1 + x

⇒ 9999x = 1

iv. As, 

Let x = 0.454545…

⇒ 1 + x = 1 + 0.454545..

⇒ 1+ x = 1.45454545

⇒ 100 + 100x = 145.454545… = 145 + 0.454545…

⇒ 100 + 100x = 145 + x

⇒ 99x = 45

And

v. As, 

Let x = 0.3333…

⇒ 7 + x = 7 + 0.3333…

⇒ 7 + x = 7.3333…

⇒ 70 + 10x = 73.3333… = 73 + 0.3333…

⇒ 70 + 10x = 73 + x

⇒ 9x = 3

And

vi. let 

⇒ 1000x = 416.416416 … = 416 + 0.416416…

⇒ 1000x = 416 + x

⇒ 999x = 416

Question 4.

Explain  in decimal form. Find the number of digits in the repeating block.

Answer:

No of divisions in repeating block = 6

Question 5.

Find the decimal expansions of  and  by division method. Without using the long division method, deduce the decimal expressions of  from the decimal expansion of .

Answer:

Also,

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Exercise 2.3

Question 1.

Locate √5 on the number line.

Answer:

Step 1:

Draw a number line. Mark points O and A such that O represents the number zero and

A represents the number 2. i.e., OA = 2 unit

Step 2:

Draw AB ⊥ OA such that AB = 1unit.

Step 3:

Join OB

In right triangle OAB, by Pythagorean theorem,

OB² = OA² + AB²

OB² = 2² + 1²

OB² = 4 + 1

OB = √5

Step 4:

With O as centre and radius OB, draw an arc to intersect the number line at C on the right side of O. Clearly OC = OB = √5 . Thus, C corresponds to √5 on the number line.

Question 2.

Find any three irrational numbers between √3 and √5.

Answer:

As, √3 = 1.73205…

And √5 = 2.23606…

We have to find three non-terminating and non-recurring numbers between 1.73205… and 2.23606…

Three possible answers are

1.740400400040000…

1.750500500050000…

1.760600600060000…

Question 3.

Find any two irrational numbers between 3 and 3.5.

Answer:

We have to find two non-terminating and non-recurring numbers between 3 and 3.5

Two possible answers are

3.10100100010000…

3.20200200020000…

Question 4.

Find any two irrational numbers between 0.15 and 0.16.

Answer:

We have to find two non-terminating and non-recurring numbers between 0.15 and 0.16

Two possible answers are

0.15100100010000…

0.15200200020000…

Question 5.

Insert any two irrational numbers between  and .

Answer:

Clearly, 

and 

We have to find two non-terminating and non-recurring numbers between 0.571428571428… and 0.714285714285…

Two possible answers are

0.580800800080000…

0.590900900090000…

Question 6.

Find any two irrational numbers between √3 and 2.

Answer:

Clearly,

As, √3 = 1.73205…

We have to find two non-terminating and non-recurring numbers between 1.73205… and 2

Two possible answers are

1.740400400040000…

1.750500500050000…

Question 7.

Find a rational number and also an irrational number between 1.1011001110001... and 2.1011001110001...

Answer:

Clearly,

1.1011001110001… < 2 < 2.1011001110001…

As, 2 is a rational number.

Therefore, 2 is a required rational number.

Now, We have to find a non-terminating and non-recurring number between 1.1011001110001… and 2.1011001110001…

And one of the possible answer is

1.20200200020000…

Question 8.

Find any two rational numbers between 0.12122122212222... and 0.2122122212222...

Answer:

Clearly,

0.12122122212222…<0.13<0.14<0.2122122122212222…

And 0.13, 0.14 are terminating, therefore rational numbers.

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Exercise 2.5

Question 1.

A number having non-terminating and recurring decimal expansion is –
A. an integer

B. a rational number

C. an irrational number

D. a whole number

Answer:

By definition, a rational number is one which follows either of the following 2 conditions:

1. It can be expressed inform where p and q are integers (q≠0).

2. Its decimal expansion is either terminating or non-terminating recurring.

Ex. – 0.35 (terminating decimal expansion) → Rational number

0.7777…. =  (non-terminating but recurring i.e. the number does not terminate but its digits follow a pattern.) → Rational number

π = 3.14…… (Non-terminating non recurring) → Irrational number

Hence, the answer.

Question 2.

If a number has a non-terminating non-recurring decimal expansion then the number is –
A. a rational number

B. a natural number

C. an irrational number

D. an integer.

Answer:

As explained in the above question, for a number to be rational its decimal expansion has to be either terminating or non-terminating and recurring.

For example –

π = 3.14…… (Non-terminating non recurring) → Irrational number

Hence, the answer.

Question 3.

Decimal form of  is
A. −0.75

B. −0.50

C. −0.25

D. −0.125

Answer:

To convert into its equivalent decimal expansion, we can either proceed by long division method or multiply the denominator and (hence) numerator by a number such that the denominator turns out to be 100.

Now, let 4×a = 100

Solving, we get a = 25.

Hence multiplying numerator and denominator by 25.

∴ (-3/4) × (25/25) = (-75/100) = -0.75 (Placing decimal point before 2 digits).

Hence, the answer.

Question 4.

The  form of  is
A. 

B. 

C. 

D. 

Answer:

Let x = 0. = 0.3333... → (eq)1

Since there is only one repeating digit after decimal point, multiply equation 1 by 10.

10x = 3.3333… = 3 + 0.3333… = 3 + x → (eq)2

(eq)2 –(eq)1

10x-x = 3.3333… - 0.3333… = 3

∴ 9x = 3

∴ x = 1/3

Hence, the answer.

Question 5.

Which one of the following is not true?
A. Every natural number is a rational number

B. Every real number is a rational number

C. Every whole number is a rational number

D. Every integer is a rational number.

Answer:

Real numbers is the set which includes both rational and irrational numbers.

∴ Rational numbers and Irrational numbers are 2 independent subsets of Real numbers. Hence every real number is not a rational number (since real numbers also consists of irrational numbers.)

Option (A) is correct because every natural number can be expressed in  form (q≠0 i.e a rational number).

Option (C) is correct because every whole number can be expressed in  form (q≠0 i.e a rational number).

Option (D) is correct because every integer can be expressed in  form (q≠0 i.e a rational number).

Natural numbers ⊂ Whole numbers ⊂ Integers ⊂ Rational Numbers ⊂ Real numbers

Real numbers ⊃ (Rational numbers + Irrational numbers).

Hence, only option B is incorrect.

Question 6.

Which one of the following has a terminating decimal expansion?
A. 

B. 

C.

D. 

Answer:

If a rational number  (q≠0) can be expressed in the form  , where p ∈ Z, and m,n ∈ W, then the rational number will have a terminating decimal expansion.

Otherwise, the rational number will have a non-terminating and recurring decimal

expansion.

Now, 32 = 2⁵ × 5⁰ Hence, it has terminating decimal expansion.Hence, option A is correct.

Option B is incorrect ∵ 9 = 3² and it cannot be expressed in the form 2^m × 5ⁿ.

Option C is incorrect ∵ 15 = 3 × 5 and it cannot be expressed in the form 2^m × 5ⁿ.

Option D is incorrect ∵ 12 = 3 × 2² and it cannot be expressed in the form 2^m × 5ⁿ.

Hence, only option A is correct.

Question 7.

Which one of the following is an irrational number?
A. π

B. 

C.

D. 

Answer:

Option A is correct since π = 3.141592… i.e. the decimal expansion is non-terminating and non-recurring.

Option B is incorrect because  = ±3 and it is an integer (and hence a rational number, Integers ⊂ Rational Numbers).

Option C is incorrect as it is expressed in the form  (p = 1 and q = 4) and is a rational number (by definition, q≠0).

Option D is incorrect as it is expressed in the form  (p = 1 and q = 5) and is a rational number (by definition, q≠0).

Hence, option A is correct.

Question 8.

Which of the following are irrational numbers?

i.  ii. 

iii.  iv. 
A. (ii), (iii) and (iv)

B. (i), (ii) and (iv)

C. (i), (ii) and (iii)

D. (i), (iii) and (iv)

Answer:

i.√(2 + √3 ) is irrational ∵ √3 is irrational and 2 is rational and sum of a rational and irrational number is always irrational. Hence 2 + √3 is irrational. Hence, its square root is also irrational.

ii.√(4 + √25) is rational ∵ √25( = 5) is rational and 4 is also rational.

Hence, √(4 + 5) = √9 = 3 is rational.

iii. ∛(5 + √7) is irrational ∵ √7 is irrational and 5 is rational and sum of a rational and irrational number is always irrational. Hence 5 + √7 is irrational. Hence, its cube root is also irrational.

iv.√(8-∛8 ) is irrational. Here both  ( = 2) and 8 are rational. Hence

8 – 8^1/3 = 8 – 2 = 6 is rational. But √6 is irrational.

Hence, the answer.