Gujarat Board Textbook Solutions Class 9 Maths Chapter 1 Number Systems

GSEB Solutions Class 9 Maths Chapter 1 Number Systems

Ex 1.1

Question 1.
Is zero a rational number? Can you write it in the form of p/q, where p and q are integers and q≠0?
Solution:
Yes, zero is a rational number, because 0 can be written in the form of p/q, where p and q are integers and q≠0.
We can write

Question 2.
Find six rational numbers between 3 and 4.
Solution:
Infinitely many rational numbers can exist between 3 and 4.
Rational number between 3 and 4

Question 4.
State whether the following statements are true or false. Give reasons for your answers.
(i) Every natural number is a whole number
(ii) Every integer is a whole number.
(iii) Every rational number is a whole number.
Solution:
(i) True, because the collection (set) of whole numbers contains all the natural numbers

(ii) False, – 1 is an integer but it is not a whole number.

(iii) False, 2/3 is a rational number but it is not a whole number.

Ex 1.2

Question 1.
State whether the following statements are true or false. Justify your answers.
(i) Every irrational number is a real number.
(ii) Every point on the number line is of the form √m, where m is a natural number.
(iii) Every real number is an irrational number.
Solution:
(i) True, because collection (set) of a real number consist rational and irrational number.

(iii) False, because a real number can be either a rational number or irrational number. For example, 3 is a real number but not 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.
Solution:
No. For example, √4 = 2 and √9 = 3 are rational number, but not irrational.

Question 3.
Show how √5 can be represented on the number line.
Solution:
Consider a unit square OABC with each side unit in length, then by Pythagoras theorem
OB2 = OA2 + AB2
OB2 = 12 + 12 = 1 + 1
OB = √2 units
Then construct BD = 1 unit perpendicular to OB.
Then again by Pythagoras theorem in ΔODE

With centre O and radius equal to OF draw an arc which intersects the number line at the point P, then

Question 4.
(Classroom activity) (Constructing the square root spiral)
Solution:
Let us take a large sheet of paper and construct the square root spiral in the following position. 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. Now draw a line segment P2P3 perpendicular to OP2. Then draw a line segment P3P4 perpendicular to OP3. Continuing in this manner, we can get the line segment Pn-1 Pn by drawing a line segment of unit length perpendicular to OPn-1 In this manner, we 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
Solution:

The decimal expansion is non-terminating repeating.

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.
Solution:
Long division method

We observe that by long division method maximum number of digits in repeating block in the decimal expansion of 1/17 is 16, thus answer is verified.

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:

We observe that the denominator of all the above rational numbers are of the form 2m x 5n i.e., the prime factorization of denominators has only powers of 2 or powers of 5 or both.

Question 7.
Write three numbers whose decimal expansions are non-terminating non-recurring.
Solution:
(i) 0.012012001200012…
(ii) 0.21021002100021000021…
(iii) 0.32032003200032000032…

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

(iii) 0.3796
The decimal expression is terminating.
Hence 0.3796 is a rational number.

(iv) 7.478478…
∴ 7.478478… = 7.748
The decimal expansion is non-terminating recurring.
∴ 7.478478… is a rational number.

(v) 1.101001000100001…
∵ The decimal expansion is non – terminating non-recurring.
∴ 1.101001000100001… is an irrational number.

Ex 1.4

Question 1.
Visualise 3.765 on the number line using successive magnification.
Solution:
Step 1.
We locate 3.765 on the number line between 3 and 4. To do this we magnify the segment between 3 and 4 and divide this segment in 10 equal parts which shown in figure 1(b).

Step 2.
We locate 3.765 on the number line between 3.7 and 3.8 so we magnify the line segment between 3.7 and 3.8 and divide this segment into 10 equal parts which shown in figure 1(c).

Step 3.
Now we observe that 3.765 is between 3.76 and 3.77. Hence again magnify this portion of line segment and divide it into 10 equal parts. The portion of the line segment 3.765 is clearly located on the number line as shown in figure 1(d).

Question 2.
Visualise 4.26 on the number line, up to 4 places.
Solution:
Step 1.
We locate 4.26 up to 4 decimal places i.e., 4.2626 on the number line. We see that 4.2626 is located between 4 and 5 on the number line which is shown in figure 2(a). Divide the segment between 4 and 5 into 10 equal parts and mark each point of division as shown in figure 2(b).

Step 2.
We locate 4.2626 between somewhere of 4.2 and 4.3 and divide this segment into 10 equal parts which is shown in figure 2(c).

Step 3.
Now we magnify the portion between 4.2 and 4.3 by magnifying glass and locate the point 4.2626 in between the point 4.26 and 4.27 which is shown in figure 2(d).

Step 4.
Now locate the point 4.2626 magnifying the line segment between 4.262 and 4.263 by magnifying glass and divide the segment into 10 equal parts and mark the point 4.2626 which is shown in figure 2(e).

Ex 1.5

Question 1.
Classify the following numbers as rational or irrational.
Solution:

(v) 2π
Here, 2 is a rational number (2 ≠ 0)
and π is an irrational number.
∴ π is an irrational number.
(Because the product of a non-zero rational number with an irrational number is an irrational number).

Question 2.
Simplify each of the following expressions:

Question 3.
Recall, n 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 n is irrational. How will you resolve this contradiction?
Solution:
π = c/d
There is no contradiction as either c or d is irrational and hence n is an irrational.

Question 4.