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RD Sharma Solutions Class 9 Maths Chapter 1 Number System
RD Sharma Class 9 Maths Chapter 1 Number System: Exercise-wise Solutions
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Question 1: Is zero a rational number? Can you write it in the form p/q, where p and q are integers and q ≠ 0?
Solution:
Yes, zero is a rational number.
It can be written in p/q form, provided that q ≠ 0.
For example, 0/1 or 0/3 or 0/4 etc.
Question 2: Find five rational numbers between 1 and 2.
Solution:
We know that one rational number between two numbers m and n = (m+n)/2
To find: 5 rational numbers between 1 and 2
Step 1: Rational number between 1 and 2
= (1+2)/2
= 3/2
Step 2: Rational number between 1 and 3/2
= (1+3/2)/2
= 5/4
Step 3: Rational number between 1 and 5/4
= (1+5/4)/2
= 9/8
Step 4: Rational number between 3/2 and 2
= 1/2 [(3/2) + 2)]
= 7/4
Step 5: Rational number between 7/4 and 2
= 1/2 [7/4 + 2]
= 15/8
Arrange all the results: 1 < 9/8 < 5/4 < 3/2 < 7/4 < 15/8 < 2
Therefore required integers are, 9/8, 5/4, 3/2, 7/4, 15/8
Question 3: Find six rational numbers between 3 and 4.
Solution:
Steps to find n rational numbers between any two numbers:
Step 1: Multiply and divide both the numbers by n+1.
In this example, we have to find 6 rational numbers between 3 and 4. Here n = 6
Multiply 3 and 4 by 7
3 x 7/7 = 21/7 and
4 x 7/7 = 28/7
Step 2: Choose 6 numbers between 21/7 and 28/7
3 = 21/7 < 22/7 < 23/7 < 24/7 < 25/7 < 26/7 < 27/7 < 28/7 = 4
Therefore, 6 rational numbers between 3 and 4 are
22/7, 23/7, 24/7, 25/7, 26/7, 27/7
Question 4: Find five rational numbers between 3/5 and 4/5.
Solution:
Steps to find n rational numbers between any two numbers:
Step 1: Multiply and divide both the numbers by n+1.
In this example, we have to find 5 rational numbers between 3/5 and 4/5. Here n = 5
Multiply 3/5 and 4/5 by 6
3/5 x 6/6 = 18/30 and
4/5 x 6/6 = 24/30
Step 2: Choose 5 numbers between 18/30 and 24/30
3/5 = 18/30 < 19/30 < 20/30 < 21/30 < 22/30 < 23/30 < 24/30 = 4/5
Therefore, 5 rational numbers between 3/5 and 4/5 are
19/30, 20/30, 21/30, 22/30, 23/30
Question 5: Are the following statements true or false? Give reasons for your answer.
(i) Every whole number is a natural number.
(ii) Every integer is a rational number.
(iii) Every rational number is an integer.
(iv) Every natural number is a whole number,
(v) Every integer is a whole number.
(vi) Every rational number is a whole number.
Solution:
(i) False.
Reason: As 0 is not a natural number.
(ii) True.
(iii) False.
Reason: Numbers such as 1/2, 3/2, and 5/3 are rational numbers but not integers.
(iv) True.
(v) False.
Reason: Negative numbers are not whole numbers.
(vi) False.
Reason: Proper fractions are not whole numbers.
Exercise 1.2
Question 1: Express the following rational numbers as decimals.
(i) 42/100 (ii) 327/500 (iii) 15/4
Solution:
Question 2: Express the following rational numbers as decimals.
(i) 2/3 (ii) -4/9 (iii) -2/15 (iv) -22/13 (v) 437/999 (vi) 33/26
Solution:
(i) Divide 2/3 using long division:
(ii) Divide using long division: -4/9
(iii) Divide using long division: -2/15
(iv) Divide using long division: -22/13
(v) Divide using long division: 437/999
(vi) Divide using long division: 33/26
Question 3: 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. Can you guess what property q must satisfy?
Solution:
The decimal representation will be terminating if the denominators have factors 2 or 5, or both. Therefore, p/q is a terminating decimal when the prime factorization of q must have only powers of 2 or 5 or both.
Exercise 1.3
Question 1: Express each of the following decimals in the form p/q:
(i) 0.39
(ii) 0.750
(iii) 2.15
(iv) 7.010
(v) 9.90
(vi) 1.0001
Solution:
(i)
0.39 = 39/100
(ii)
0.750 = 750/1000 = 3/4
(iii)
2.15 = 215/100 = 43/20
(iv)
7.010 = 7010/1000 = 701/100
(v)
9.90 = 990/100 = 99/10
(vi)
1.0001 = 10001/10000
Question 2: Express each of the following decimals in the form p/q:
Solution:
(i) Let x = 0.4̅
or x = 0.4̅ = 0.444 …. (1)
Multiplying both sides by 10
10x = 4.444 …..(2)
Subtract (1) by (2), and we get
10x – x = 4.444… – 0.444…
9x = 4
x = 4/9
=> 0.4̅ = 4.9
(ii) Let x = 0.3737.. …. (1)
Multiplying both sides by 100
100x = 37.37… …..(2)
Subtract (1) from (2), and we get
100x – x = 37.37… – 0.3737…
100x – x = 37
99x = 37
x = 37/99
(iii) Let x = 0.5454… (1)
Multiplying both sides by 100
100x = 54.5454…. (2)
Subtract (1) from (2), and we get
100x – x = 54.5454…. – 0.5454….
99x = 54
x = 54/99
(iv) Let x = 0.621621… (1)
Multiplying both sides by 1000
1000x = 621.621621…. (2)
Subtract (1) from (2), and we get
1000x – x = 621.621621…. – 0.621621….
999x = 621
x = 621/999
or x = 23/37
(v) Let x = 125.3333…. (1)
Multiplying both sides by 10
10x = 1253.3333…. (2)
Subtract (1) from (2), and we get
10x – x = 1253.3333…. – 125.3333….
9x = 1128
or x = 1128/9
or x = 376/3
(vi) Let x = 4.7777…. (1)
Multiplying both sides by 10
10x = 47.7777…. (2)
Subtract (1) from (2), and we get
10x – x = 47.7777…. – 4.7777….
9x = 43
x = 43/9
(vii) Let x = 0.47777….
Multiplying both sides by 10
10x = 4.7777…. …(1)
Multiplying both sides by 100
100x = 47.7777…. (2)
Subtract (1) from (2), and we get
100x – 10x = 47.7777…. – 4.7777…
90x = 43
x = 43/90
Exercise 1.4
Question 1: Define an irrational number.
Solution:
A number which cannot be expressed in the form of p/q, where p and q are integers and q ≠ 0. It is a non-terminating or non-repeating decimal.
Question 2: Explain how irrational numbers differ from rational numbers.
Solution:
An irrational number is a real number which can be written as a decimal but not as a fraction i.e. it cannot be expressed as a ratio of integers.
It cannot be expressed as terminating or repeating decimals.
For example, √2 is an irrational number
A rational number is a real number which can be written as a fraction, and as a decimal i.e. it can be expressed as a ratio of integers.
It can be expressed as a terminating or repeating decimal.
For example, 0.10 and 5/3 are rational numbers
Question 3: Examine whether the following numbers are rational or irrational:
Solution:
(i) √7
Not a perfect square root, so it is an irrational number.
(ii) √4
A perfect square root of 2.
We can express 2 in the form of 2/1, so it is a rational number.
(iii) 2 + √3
Here, 2 is a rational number, but √3 is an irrational number.
Therefore, the sum of a rational and irrational number is an irrational number.
(iv) √3 + √2
√3 is not a perfect square, thus an irrational number.
√2 is not a perfect square, thus an irrational number.
Therefore, the sum of √2 and √3 gives an irrational number.
(v) √3 + √5
√3 is not a perfect square, and hence, it is an irrational number
Similarly, √5 is not a perfect square, and it is an irrational number.
Since the sum of two irrational numbers is an irrational number, √3 + √5 is an irrational number.
(vi) (√2 – 2)2
(√2 – 2)2 = 2 + 4 – 4 √2
= 6 – 4 √2
Here, 6 is a rational number but 4√2 is an irrational number.
Since the sum of a rational and an irrational number is an irrational number, (√2 – 2)2 is an irrational number.
(vii) (2 – √2)(2 + √2)
We can write the given expression as;
(2 – √2)(2 + √2) = ((2)2 − (√2)2)
[Since, (a + b)(a – b) = a2 – b2]
= 4 – 2 = 2 or 2/1
Since 2 is a rational number, (2 – √2)(2 + √2) is a rational number.
(viii) (√3 + √2)2
We can write the given expression as;
(√3 + √2)2 = (√3)2 + (√2)2 + 2√3 x √2
= 3 + 2 + 2√6
= 5 + 2√6
[using identity, (a+b)2 = a2 + 2ab + b2]
Since the sum of a rational number and an irrational number is an irrational number, (√3 + √2)2 is an irrational number.
(ix) √5 – 2
√5 is an irrational number, whereas 2 is a rational number.
The difference of an irrational number and a rational number is an irrational number.
Therefore, √5 – 2 is an irrational number.
(x) √23
Since, √23 = 4.795831352331…
As the decimal expansion of this number is non-terminating and non-recurring, it is an irrational number.
(xi) √225
√225 = 15 or 15/1
√225 is a rational number as it can be represented in the form of p/q, and q is not equal to zero.
(xii) 0.3796
As the decimal expansion of the given number is terminating, it is a rational number.
(xiii) 7.478478……
As the decimal expansion of this number is a non-terminating recurring decimal, it is a rational number.
(xiv) 1.101001000100001……
As the decimal expansion of the given number is non-terminating and non-recurring, it is an irrational number.
Question 4: Identify the following as rational or irrational numbers. Give the decimal representation of rational numbers:
Solution:
(i) √4
√4 = 2, which can be written in the form of a/b. Therefore, it is a rational number.
Its decimal representation is 2.0.
(ii) 3√18
3√18 = 9√2
Since the product of a rational and an irrational number is an irrational number.
Therefore, 3√18 is an irrational number.
Or 3 × √18 is an irrational number.
(iii) √1.44
√1.44 = 1.2
Since every terminating decimal is a rational number, √1.44 is a rational number.
And its decimal representation is 1.2.
(iv) √9/27
√9/27 = 1/√3
Since the quotient of a rational and an irrational number is irrational numbers, √9/27 is an irrational number.
(v) – √64
– √64 = – 8 or – 8/1
Therefore, – √64 is a rational number.
Its decimal representation is –8.0.
(vi) √100
√100 = 10
Since 10 can be expressed in the form of a/b, such as 10/1, √100 is a rational number.
And its decimal representation is 10.0.
Question 5: In the following equation, find which variables x, y, z etc. represent rational or irrational numbers:
Solution:
(i) x2 = 5
Taking square root on both sides,
x = √5
√5 is not a perfect square root, so it is an irrational number.
(ii) y2 = 9
y2 = 9
or y = 3
3 can be expressed in the form of a/b, such as 3/1, so it is a rational number.
(iii) z2 = 0.04
z2 = 0.04
Taking square root on both sides, we get
z = 0.2
0.2 can be expressed in the form of a/b, such as 2/10, so it is a rational number.
(iv) u2 = 17/4
Taking square root on both sides, we get
u = √17/2
Since the quotient of an irrational and a rational number is irrational, u is an Irrational number.
(v) v2 = 3
Taking square root on both sides, we get
v = √3
Since √3 is not a perfect square root, so v is an irrational number.
(vi) w2 = 27
Taking square root on both sides, we get
w = 3√3
Since the product of a rational and irrational is an irrational number, w is an irrational number.
(vii) t2 = 0.4
Taking square root on both sides, we get
t = √(4/10)
t = 2/√10
Since the quotient of a rational and an irrational number is an irrational number, t is an irrational number.
Important Topics: RD Sharma Solutions Class 9 Maths Chapter 1
- Number System Introduction
- Review of Numbers
- The decimal representation of rational numbers
- Conversion of decimal numbers into rational numbers
- Conversion of decimal numbers into rational numbers
- Irrational Numbers
- Some useful results on Irrational Numbers
- Representing Irrational Numbers on the Number Line
- Real Numbers and real number line
- Existence of the square root of a positive real number
- Visualization of representation of real numbers
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