PSEB Solutions for Class 9 Maths Chapter 1 Number Systems

PSEB 9th Class Maths Solutions Chapter 1 Number Systems

Ex 1.1

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?
Answer:
Yes. Zero is a rational number. Zero is a whole number as well as an Integer. As we know, all integers are rational numbers, zero is also a rational number.

Question 2.
Find six rational numbers between 3 and 4.
Answer:
Since six rational numbers are to be found. between rationals 3 and 4, we express both of them with denominator 7(6 + 1).

Question 4.
State whether the following statements are true or false. Give reasons for your answers :
(i) Every natural number is a whole number.
Answer:
The given statement is true as the collection of whole numbers contain all the natural numbers.

(ii) Every integer is a whole number.
Answer:
The given statement is false as any negative integer like – 2, – 3, – 5, etc. Is not a whole number. The collection of whole numbers contains 0 and all natural numbers, but not the opposites of the natural numbers.

(iii) Every rational number is a whole number.
Answer:
The given statement is false as any rational number lying between two consecutive whole numbers is not a whole number.

e.g., 5/2 is a rational number lying between whole numbers 2 and 3, but it is not a whole number.
Skill Testing Exercise

Ex 1.2

Question 1.
Are the following statements are true or false. Justify your answers:
(i) Every irrational number is a real number.
Answer:
This statement is true. The collection of real numbers contain all the rational numbers and all the irrational numbers.

(ii) Every point on the number line is of the form √m, where m is a natural number.
Answer:
This statement is false. The negative numbers represented on the number line are never of the form √m, where m is a natural number. For natural number m, √m is always a positive number.

(iii) Every real number is an irrational number.
Answer:
This statement is false. The collection of real numbers is made-up of rational numbers and irrational numbers. So, every real number is either a rational number or an irrational number.

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:
No, The square roots of all positive integers are not irrational. The square root of any perfect square number is always a rational number.
e.g., √4 = 2, √9 = 3, √16 = 4, ……

Question 3.
Show how √5 can be represented on the number line.
Answer:

Start by taking point O on the number line which represents O. Choosing proper unit, locate point A on the number line to represent
1. Draw seg AB of unit length perpendicular to seg OA. Construct seg OB. Then, by Pythagoras’ theorem, OB = √2. Draw seg BC of unit length perpendicular to seg OB. Construct seg OC.
Then, OC = √3. Draw seg CD of unit length perpendicular to seg OC. Construct seg OD.
Then, OD = √4. Draw seg DE of unit length perpendicular to seg OD. Construct seg 0E.
Then, 0E = √5. Draw an arc with centre O and radius OE to Intersect the number line at P Then, point P on the number line represents √5.

Alternate Method:
This can be solved In fewer steps as shown in the following manner:
On the number line l, take a point O corresponding to 0. Choosing the proper unit, take a point A on l such that OA = 2 units. Construct right angled Δ OAP such that ΔOAP = 90° and AP = 1 unit.
PSEB 9th Class Maths Solutions Chapter 1 Number Systems Ex 1.2 2
Then, by Pythagoras theorem,
OP2 = OA2 + AP2
= (2)2 + (1)2 = 4 + 1 = 5
∴ OP = √5
Draw an arc with centre O and radius OP to intersect l at B. B is the point corresponding to Draw an arc with centre O and radius OP to intersect I at B. B is the point corresponding to √5.

Question 4.
Classroom activity (Constructing the ‘square root spiral’): Take a large sheet of paper and construct the ‘square root spiral’ in the following fashion. Start with a point O and draw a line segment OP1 of unit length. Draw a line segment P1P2 perpendicular to OP1 of unit length (see Fig. 1.9). Now draw a line segment P2P3 perpendicular to OP2. Then draw a line segment P3P4 perpendicular to OP3. Continuing in this manner, you can get the line segment Pn-1Pn by drawing a line segment of unit length perpendicular to OPn-1. In this manner, you will have created the points P2, P3, …………, Pn, ………, and joined them to create a beautiful spiral depicting √2, √3, √4, ………..

Answer:
Take a large sheet of paper and construct the ‘square root spiral’ in the following fashion:

Start with a point O and draw a line segment OP1 of unit length. Draw a line segment P1P2 perpendicular to OP1 of unit length (see Fig. 1.9). Now draw a line segment P2P3 perpendicular to OP2. Then draw a line segment P3P4 perpendicular to OP3. Continuing in this manner, you can get the line segment Pn-1Pn by drawing a line segment of unit length perpendicular to OPn-1. In this manner, you will have created the points P2, P3, …………, Pn, ………, and joined them to create a beautiful spiral depicting √2, √3, √4, ………..

Ex 1.3

Question 1.
Write the following in decimal form and say
what kind of decimal expansion each has:

(i) 36/100
Answer:
The decimal expansion of 36/100 is terminating.

Question 5.
What can the maximum number of digits be in the repeating block of digits in the decimal expansion of 1/17? Perform the division to check your answer.
Answer:
The maximum number of digits in the repeating block of digits in the decimal expansion of 1/17 can be 16(17 – 1).

Question 7.
Write three numbers whose decimal expansions are non-terminating non-recurring.
Answer:
We know that the decimal expansion of an irrational number is non-terminating non recurring. There are infinitely many irrational numbers. We can state few of them as below:
o.o1001000100001 …,

Question 9.
Classify the following numbers as rational or irrational:
(i) √23
Answer:
√23 is an irrational number.

(ii) √225
Answer:
√225 = 15 is a rational number

(iii) 0.3796
Answer:
0.3796 is a rational number.

(iv) 7.478478……..
Answer:
7.478478…… = 7.478 is a rational number.

(v) 1.101001000100001…….
Answer:
1.101001000100001……. is an irrational number.

Ex 1.4

Question 1.
Visualise 3.765 on the number line, using successive magnification.
Answer:

Point P in fig. 4 represents 3.765 on the number line.

Ex 1.5

Question 1.
Classify the following numbers as rational or irrational:
(i) 2 – √5
Answer:
2 – √5 is an irrational number as it is the difference of rational number (2) and irrational number (√5).

(ii) (3 + √23) – √23
Answer:
Thus, (3 + √23) – √23 is a rational number even if it is the difference of two irrational numbers (3 + √23) and √23.

(v) 2π
Answer:
2π is an irrational number as it is the product of rational number 2 and irrational number π.

Question 2.
Simplify each of the following expressions:
(i) (3 + √3) (2 + √2)
Answer:
(3 + √3) (2 + √2) = 6 + 3√2 + 2√3 = √6

(ii) (3 + √3) (3 – √3)
Answer:
(3 + √3) (3 – √3) = (3)2 – (√3)2 = 9 – 3 = 6

(iii) (√5 + √2)2
Answer:
(√5)2 + 2(√5) (√2) + (√2)2
= 5 + 2√10 + 2
= 7 + 2√10

(iv) (√5 – √2) (√5 + √2)
Answer:
(√5 – √2) (√5 + √2) = (√5)2 – (√2)2
= 5 – 2 = 3

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:
π is defined as the ratio of circumference (C) of a circle to its diameter (d). As seen in the process of successive magnification used to represent real numbers on the number line, we see that more and more accuracy can be obtained by successive magnification. But, since the real numbers exhibit gaps. we can never measure the exact length of the circumference and the diameter. Any one or both may be having length represented by an irrational number. Hence, there is no contradiction that π being the ratio of c and d is still an irrational number.

Note: in the study of mathematics at higher level, you may study that π is a transcendental number and also the proof of π being an irrational number.

Question 4.
Represent √9.3 on the number line.
Answer: