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Number System

Class 9th Mathematics CBSE Solution
Exercise 1.1
  1. Is zero a rational number? Can you write it in the form p/q, where p and q are…
  2. Find six rational numbers between 3 and 4.
  3. Find five rational numbers between 3/5 and 4/5
  4. State whether the following statements are true or false. Give reasons for your…
Exercise 1.2
  1. State whether the following statements are true or false. Justify your answers.…
  2. Are the square roots of all positive integers irrational? If not, give an…
  3. Show how can be represented on the number line.
  4. Classroom activity constructing the square root spiral. Take a large sheet of paper and…
Exercise 1.3
  1. Write the following in decimal form and say what kind of decimal expansion each…
  2. You know that Can you predict what the decimal expansions of are, without…
  3. Express the following in the form p/q, where p and q are integers and (i) (ii)…
  4. Express 0.99999 .. in the form p/q. Are you surprised by your answer? With your…
  5. What can the maximum number of digits be in the repeating block of digits in the…
  6. Look at several examples of rational numbers in the form where and are integers…
  7. Write three numbers whose decimal expansions are non-terminating non-recurring.…
  8. Find three different irrational numbers between the rational numbers 5/7 and…
  9. Classify the following numbers as rational or irrational: (i) (ii) (iii) 0.3796…
Exercise 1.4
  1. Visualize 3.765 on the number line, using successive magnification.…
  2. Visualizeon the number line, up to 4 decimal places.
Exercise 1.5
  1. Classify the following numbers as rational or irrational: (i) (ii) (iii) (iv)…
  2. Simplify each of the following expressions: (i) (ii) (iii) (iv)
  3. Recall, is defined as the ratio of the circumference (say c) of a circle to its…
  4. Represent on the number line.
  5. Rationalize the denominators of the following: (i) (ii) (iii) (iv)…
Exercise 1.6
  1. (i) (ii) (iii) Find:
  2. (i) 9^3/2 (ii) 32^2/5 (iii) 16^3/4 (iv) 125^-1/3 Find:
  3. Simplify: (i) 2^2/3 2^1/5 (ii) (1/3^3)^7 (iii) 11^1/2/11^1/4 (iv) 7^1/2 , 8^1/2…

Exercise 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 and it can be represented in the form of as
or where z is an integer and z ≠0.


Question 2.

Find six rational numbers between 3 and 4.


Answer:

There are infinite rational numbers between 3 and 4.

So, we can represent 3 and 4 in terms of fractions as an example:

(Multiplying & dividing by 8)

NOTE: We can multiply & divide by other numbers as well.
We are multiplying and dividing at the same time because it is easier to write other fractions in between.
The decimal numbers between 3 & 4 can also be written for e.g. 3.1, 3.2, 3.3, etc.

Now, we can write the six rational numbers between 3 and 4 as:


Now let us try dividing with some other number.

(Multiplying and dividing by 10)


And now we can write the rational numbers between 3 and 4 as



So, you can take any number.

Question 3.

Find five rational numbers between 3/5 and 4/5


Answer:

Between and there exist infinite rational numbers & we know that rational numbers are fractions, whole numbers and decimals.

Trick: To find "n" rational numbers between any two numbers, multiply & divide both by "n+1".

Example: If we need to find 5 rational numbers between & , we multiply and divide both the numbers by 6 to obtain the set of rational numbers.

Hence, five rational numbers between and are:


Now its not necessary to only multiply and divide by n+1, you can do by any number greater than n.
For example if you multiply and divide by 10

So the rational numbers between 3 and 4 now will be


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.


Answer:

(i) True, since the collection of whole numbers contains all natural numbers.

Natural numbers are numbers starting from 1, i.e 1, 2, 3, 4, 5, 6, ..................
And whole numbers are numbers starting from 0. i.e, 0, 1, 2, 3, 4, 5.............

And you can see that all natural numbers are within whole numbers.

(ii) False, as integers may be negative or positive but whole numbers are always positive.

Now integers are numbers that are both negative and positive and include zero also, i.e, ..........-3, -2, -1, 0, 1, 2, 3, 4, .......
And whole numbers are numbers starting from 0. i.e, 0, 1, 2, 3, 4, 5............

And clearly, negative numbers are missing from whole numbers.

(iii) False, as rational numbers may be fractional but whole numbers are not fractional.

Now, Rational numbers are numbers that can be expressed in p/q form where q not equal to zero. So they include fractions on the other hand whole numbers are numbers starting from 0. i.e, 0, 1, 2, 3, 4, 5............

Therefore, Whole numbers does not contain all rational numbers.


Exercise 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 where m is a natural number

(iii) Every real number is an irrational number.


Answer:

(i) True, since the real numbers are the collection of rational and irrational numbers.

For example √2 = 1.414... which can be represented in a number line.

All the numbers which can be represented in a number line are real.

(ii) False, since negative numbers cannot be expressed as the square of any other real number m.

for example: 4 = √2 but there is no number whose square root is -4.


(iii) False, as real numbers consist of both rational and irrational numbers. Therefore, every real number may not be an irrational number.

For example: √2 is irrational and 1/2 is rational and both are real numbers.


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. For example = 5.



Question 3.

Show how √5 can be represented on the number line.


Answer:

Steps of construction:

Step 1: Let AB be a line of length 2 unit on the number line.

Step 2: At B, draw a perpendicular line BC of length 1 unit and then join CA.


Step 3: Now, ABC is a right angled triangle.


Using Pythagoras theorem,

AB2 + BC2 = CA2

22 + 12 = CA2

CA2 = 5

CA = √5


Thus, CA is a line of length √5 units.


Step 4: Taking CA as a radius and A as the center, construct an arc touching the number line.


The number line gets intersected by the arc at a point which is at √5 distance from 0, as it is a radius of the circle with center A


Thus, √5 is represented on the number line as shown in the figure below.



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 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


Answer:

Activity

For the square root spiral follow the given steps:-

1. Draw a line AB of length 1 unit.
2. Draw another line BC of length 1 unit perpendicular to AB.
3. Now, Join point A and point C forming a line AC.
Here, AC represents a line of length √2 units. (This can be easily found using Pythagoras Theorem in right ΔABC)

4. Now, Draw a perpendicular CD of length 1 unit at point C and join points A and D.
AD here represents length √3.
5. Similarly proceeding further we get Square Root Spiral.


Exercise 1.3
Question 1.

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

(i) (ii) 1/11

(iii) (iv) 3/13

(v) 2/11 (vi)


Answer:

Decimal Expansion which ends after the division is called terminating expansion. And when a fraction is divided such that the value repeats itself, it is a recurring expansion. When the expansion never ends it is called non terminating expansion.

For example: 0.2342516782943257 is neither repeating nor terminating expansion.


(i) = 0.36

(Since, decimal expansion has finite no. of figures. Hence, it is terminating).

(ii) = 0.09090909….


=

(Since decimal expansion repeats itself so it is a non-terminating recurring decimal expansion).


(iii)


= 4.125

(Since, decimal expansion has finite no. of figures. Hence, it is terminating).


(iv) = 0.230769230769


=

(Since decimal expansion repeats itself so it is a non-terminating recurring decimal expansion).

(v) = 0.1818181818….


=
(Since decimal expansion repeats itself so it is a non-terminating recurring decimal expansion)


(vi) = 0.8225


(Since, decimal expansion has finite no. of figures. Hence, it is terminating.)


Question 2.

You know that Can you predict what the decimal expansions of are, without actually doing the long division? If so how?

[Hint: Study the remainders while finding the value of 1/7 carefully.]


Answer:

Yes, we can do this by the following method:

2/7 = 2 x 1/7 = 2 x 0.͞1͞42͞8͞5͞7 = 0.2͞8͞5͞71͞4


3/7 = 3 x 1/7 = 3 x 0.1͞42͞8͞5͞7 = 0.4͞28͞5͞7͞1


4/7 = 4 x 1/7 = 4 x 0.1͞42͞8͞5͞7 = 0.͞57͞1͞4͞2͞8


5/7 = 5 x 1/7 = 5 x 0.1͞42͞8͞5͞7 = 0.7͞1͞4͞2͞8͞5


6/7 = 6 x 1/7 = 6 x 0.1͞42͞8͞5͞7 = 0.͞8͞5͞7͞1͞4͞2


Question 3.

Express the following in the form p/q, where p and q are integers and
(i) (ii)

(iii)


Answer:

(i) = 0.666…

Let x = 0.666666…


Multiplying both sides by 10


10 x = 6.666= 6 + 0.666


10 x = 6 + x


9 x = 6



(ii)


Let x = 0.4777777
Now multiplying both sides by 10,
10 x = 4.77777777........ eq(1)
Now multiplying eq(1) by 100 we get
100 x = 47.777777........ eq(2)
Now eq(2) - eq(1)
100 x - 10 x = 47 - 4
90 x = 43
x = 43/90

(iii)


Let x = 0.001001…eq(1)


Multiplying both sides by 1000


1000x = 1.001001… = 1 + 0.001001... eq(2)
we can see that,

1000x = 1 + x

999x = 1


Question 4.

Express 0.99999 ….. in the form p/q. Are you surprised by your answer? With your teacher and classmates discuss why the answer makes sense.


Answer:

Let x = 0.99999......... ......(i)

multiply both sides by 10

10x = 9.9999...... ....(2)

subtract (1) from (2).

9 x = 9

As x = 9/9 or x = 1.

Therefore, on converting 0.99999.... in the p/q form, we get the answer as1. .

The difference between 1 and 0.999999 is o.000001 which is negligible.


Hence, 0.999 is too much near to 1. Therefore, 1 as an answer can be justified


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:

1/17 = 0.0͞5͞8͞8͞2͞3͞5͞2͞9͞4͞1͞1͞7͞6͞4͞7

There are 16 digits in the repeating numbers of the decimal expansion of 1/17.


Division check:




After 0.0588235294117647 digits will start repeating itself.


Question 6.

Look at several examples of rational numbers in the form where and are integers with no common factors other than 1 and having terminating decimal representations (expansions). Can you guess what property q must satisfy?


Answer:

We observe that when q is 2, 4, 5, 8, 10… then the decimal expansion is terminating. For example:

, denominator q = 21


, denominator q = 23


, denominator q = 51


It can be observed that the terminating decimal can be obtained in a condition where prime factorization of the denominator of the given fractions has the power of 2 only or 5 only or both.


Question 7.

Write three numbers whose decimal expansions are non-terminating non-recurring.


Answer:

Numbers with non-terminating decimal expression means that they never completely divided.

For example, when you divide 10 by 3, the answer you get keeps on dividing and no single value can be obtained but this number is recurring because 10/3 = 2.666666666 and in this value 6 is repeating or recurring.

So we have to write a number which is non terminating and non-repeating. Write any number which does not end and does not repeat.

(0.123123456345876....... or 2.123231245627549.... there can be infinite such numbers)

Three numbers whose decimal expansions are non-terminating non-recurring are:

0.304004000400003…


0.80506000900005…


0.7205200820008200008200000…

OR

All irrational numbers are non-terminating and non-repeating.

Example: √2 ,√3 and √5


Question 8.

Find three different irrational numbers between the rational numbers 5/7 and 9/11


Answer:

Irrational Numbers: Numbers that cannot be expressed in p/q form are called irrational number.
Any number that is non-repeating and non terminating is an irrational number.
To find irrational numbers first we find a decimal expansion of given fractions.


Now, for irrational numbers, we write any non-terminating and non-repeating decimal number between given numbers.


Three different irrational numbers can be:

0.74074007400074000074…

0.75075007500075000075…

0.78078007800078000078…


Question 9.

Classify the following numbers as rational or irrational:

(i)

(ii)

(iii) 0.3796

(iv) 7.478478…

(v) 1.101001000100001….


Answer:

Rational NUmber: A number that can be expressed in the form p/q, where q ≠ 0.

Irrational Number: A number which can not be expressed in the form p/q. The numbers that are non-repeating and non-terminating are called irrational.

(i)
= 4.79583152331…

Since the decimal expansion is non-terminating and non-recurring hence, it is an irrational number


(ii) = 15 = 15/1


It is a rational number as it can be represented in the form of p/q


(iii) 0.3796


Since the decimal expansion is terminating number.


Therefore, it is a rational number


(iv) 7.478478 = 7.4͞7͞8


Since the decimal expansion is a non-terminating recurring number.


Therefore, it is a rational number


(v) 1.101001000100001…


Since the number has decimal expansion as non-terminating and non-repeating.


Hence, it is an irrational number



Exercise 1.4
Question 1.

Visualize 3.765 on the number line, using successive magnification.


Answer:

First, we will divide the number line between points 3 and 4. And then we will divide the points between 3.7 and 3.8 as the number is between both of them.
The number to visualize is 3.765 so now visualize the points between 3.76 and 3.77.




Question 2.

Visualizeon the number line, up to 4 decimal places.


Answer:

4.2͞6 = 4.2626




Exercise 1.5
Question 1.

Classify the following numbers as rational or irrational:

(i)

(ii)

(iii)

(iv)

(v)


Answer:

(i) 2 - = 2 – 2.2360679…

= -0.2360679…


Since the number is in non-terminating non-recurring, therefore, it is an irrational number


(ii) (3 + ) - = 3 + -


= 3


= 3/1


A number is a rational number as it can be represented in the form of p/q.


(iii) = 2/7


The number is a rational number as it can be represented in the form of p/q.


(iv) =


= 0.7071067811.....


Since the number is a non-terminating and non-recurring number.


Hence, it is an irrational number


(v) 2π = 2 * 3.1415…


= 6.2830......


Since the number is a non-terminating and non-recurring number.


Hence, it is an irrational number


Question 2.

Simplify each of the following expressions:

(i)

(ii)

(iii)

(iv)


Answer:

(i) (3 + √3) (2 + √2)

= 3 (2 + √2) + √3 (2 + √2)

= 3 × 2 + 3√2 + 2√3 + √2 x √3


= 6 + 3√2 + 2√3 + √6


(ii) (3 + √3) (3 - √3) [Since, (a + b) (a – b) = a2 – b2]


= 32 – (√3)2


= 9 – 3


= 6


(iii) (√5 + √2)2 [Since, (a + b)2 = a2 + b2 + 2ab]


= (√5)2 + (√2)2 + 2 × √5×√2


= 5 + 2 + 2 ×


= 7 + 2√10


(iv) (√5 - √2) (√5 + √2) [Since, (a + b) (a – b) = a2 – b2]


= (√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, This seems to contradict the fact that π is irrational. How will you resolve this contradiction?


Answer:

There is no contradiction in the statement.

When we measure any value with a scale, we only obtain an approximate value.


We never obtain an accurate value.


Hence, we cannot say that either c or d is irrational.


The value of π is almost equal to 22/7 or 3.142857…


Question 4.

Represent on the number line.


Answer:

Step 1: Draw a line segment AB of 9.3 unit. Then, extend it to C so that BC = 1 unit.

Step 2: Now, AC = 10.3 units. Find the center of AC and mark it as O


Step 3: Draw a semi-circle with radius OC and center O.


Step 4: Draw a perpendicular line BD to AC at point B intersecting the semi-circle at D. And then, join OD


Step 5: Now, OBD is a right angled triangle


Here, OD (Radius of semi-circle)


OC


BC = 1


Then, OB = OC – BC


Using Pythagoras theorem,


OD2 = BD2 + OB2


()2 = BD2 + ()2


BD2 = ()2 - ()2


BD2 = () ( + )


BD2 = 9.3


BD=


Thus, the length of BD is


Step 6: Taking BD as radius and B as the center, construct an arc which touches the line segment.


Now, the point where it touches the line segment is at a distance of from O as shown in the figure below



Question 5.

Rationalize the denominators of the following:

(i) (ii)

(iii) (iv)


Answer:

Rationalizing the denominator means that we have to remove the irrational component from the denominator of the fraction.



(ii)


(iii) =


=


=


=


(iv) =


=


=


=



Exercise 1.6
Question 1.

Find:

(i) (ii)

(iii)


Answer:

(i) (64)1/2

Writing Prime factors of 64 we get that : 64 = 8 x 8

So, (64)1/2 = (26)1/2

(64 )1/2 = 23
= 8


(ii) (32)1/5

Writing Prime Factors of 32 we get that: 32 = 2 x 2 x 2 x 2 x 2

So, 32 = 25

(32)1/5 = (25)1/5

(32)1/5 = 2

(iii)(125)1/3

Writing Prime Factors of 125 we get that: 125 = 5 x 5 x 5

So, 125 = 5 3


(125)1/3 = (53)1/3

(125) 1/3 = 5


= 5


Question 2.

Find:

(i) (ii)

(iii) (iv)


Answer:

(i) 93/2 = (32)3/2

= 32*3/2


= 33


= 27


(ii) 322/5 = (25)2/5


= (2)5*2/5


= 22


= 4


(iii) (16)3/4 = (24)3/4


= 23


= 8


(iv) (125)-1/3 =


=


Question 3.

Simplify:

(i)

(ii)

(iii) (iv)


Answer:

(i) am . an = a(m + n)


(ii) (am)n = amn




(iii)





(iv) am . bm = (a b)m