EXERCISE1.3                                                                            PAGE:14

 

Question 1. Write the following in decimal form and say what kind of decimal expansion each has: NCERT solution for class 9 Maths Chapter-1 Number Systems/image001.png

(i) NCERT solution for class 9 Maths Chapter-1 Number Systems/image002.png

(ii) NCERT solution for class 9 Maths Chapter-1 Number Systems/image003.png

(iii) NCERT solution for class 9 Maths Chapter-1 Number Systems/image004.png

(iv) NCERT solution for class 9 Maths Chapter-1 Number Systems/image005.png

(v) NCERT solution for class 9 Maths Chapter-1 Number Systems/image006.png

(vi) NCERT solution for class 9 Maths Chapter-1 Number Systems/image007.png

 

Solution :
(i) NCERT solution for class 9 Maths Chapter-1 Number Systems/image002.png

On dividing 36 by 100, we get

NCERT solution for class 9 Maths Chapter-1 Number Systems/image008.png

Therefore, we conclude thatNCERT solution for class 9 Maths Chapter-1 Number Systems/image009.png, which is a terminating decimal.

(ii) NCERT solution for class 9 Maths Chapter-1 Number Systems/image003.png

On dividing 1 by 11, we get

NCERT solution for class 9 Maths Chapter-1 Number Systems/image010.png

We can observe that while dividing 1 by 11, we got the remainder as 1, which will continue to be 1.

Therefore, we conclude thatNCERT solution for class 9 Maths Chapter-1 Number Systems/image014.png, which is a non-terminating decimal and recurring decimal.

(iii) NCERT solution for class 9 Maths Chapter-1 Number Systems/image004.pngNCERT solution for class 9 Maths Chapter-1 Number Systems/image012.png

On dividing 33 by 8, we get

NCERT solution for class 9 Maths Chapter-1 Number Systems/image013.png

We can observe that while dividing 33 by 8, we got the remainder as 0.

Therefore, we conclude thatNCERT solution for class 9 Maths Chapter-1 Number Systems/image014.png, which is a terminating decimal.

(iv) NCERT solution for class 9 Maths Chapter-1 Number Systems/image005.png

On dividing 3 by 13, we get

NCERT solution for class 9 Maths Chapter-1 Number Systems/image015.png

We can observe that while dividing 3 by 13 we got the remainder as 3, which will continue to be 3 after carrying out 6 continuous divisions.

Therefore, we conclude thatNCERT solution for class 9 Maths Chapter-1 Number Systems/image016.png, which is a non-terminating decimal and recurring decimal.

(v) NCERT solution for class 9 Maths Chapter-1 Number Systems/image006.png

On dividing 2 by 11, we get

NCERT solution for class 9 Maths Chapter-1 Number Systems/image017.png

We can observe that while dividing 2 by 11, first we got the remainder as 2 and then 9, which will continue to be 2 and 9 alternately.

Therefore, we conclude thatNCERT solution for class 9 Maths Chapter-1 Number Systems/image018.png, which is a non-terminating decimal and recurring decimal.

(vi) NCERT solution for class 9 Maths Chapter-1 Number Systems/image007.png

On dividing 329 by 400, we get

NCERT solution for class 9 Maths Chapter-1 Number Systems/image019.png

We can observe that while dividing 329 by 400, we got the remainder as 0.

Therefore, we conclude that NCERT solution for class 9 Maths Chapter-1 Number Systems/image020.png, which is a terminating decimal.

 

Question 2.
You know that NCERT solution for class 9 Maths Chapter-1 Number Systems/image021.png. Can you predict what the decimal expansions ofNCERT solution for class 9 Maths Chapter-1 Number Systems/image022.pngare, without actually doing the long division? If so, how?

[Hint: Study the remainders while finding the value of NCERT solution for class 9 Maths Chapter-1 Number Systems/image023.pngcarefully.]

Solution :
We are given thatNCERT solution for class 9 Maths Chapter-1 Number Systems/image024.png.

We need to find the values ofNCERT solution for class 9 Maths Chapter-1 Number Systems/image025.png, without performing long division.

We know that, NCERT solution for class 9 Maths Chapter-1 Number Systems/image025.pngcan be rewritten as

NCERT solution for class 9 Maths Chapter-1 Number Systems/image026.png.

On substituting value of NCERT solution for class 9 Maths Chapter-1 Number Systems/image023.pngas NCERT solution for class 9 Maths Chapter-1 Number Systems/image027.png, we get

NCERT solution for class 9 Maths Chapter-1 Number Systems/image034.pngNCERT solution for class 9 Maths Chapter-1 Number Systems/image029.png

NCERT solution for class 9 Maths Chapter-1 Number Systems/image030.pngNCERT solution for class 9 Maths Chapter-1 Number Systems/image031.png

NCERT solution for class 9 Maths Chapter-1 Number Systems/image032.pngNCERT solution for class 9 Maths Chapter-1 Number Systems/image033.png

NCERT solution for class 9 Maths Chapter-1 Number Systems/image034.pngNCERT solution for class 9 Maths Chapter-1 Number Systems/image035.png

NCERT solution for class 9 Maths Chapter-1 Number Systems/image036.pngNCERT solution for class 9 Maths Chapter-1 Number Systems/image037.png

Therefore, we conclude that, we can predict the values ofNCERT solution for class 9 Maths Chapter-1 Number Systems/image025.png, without performing long division, to get

NCERT solution for class 9 Maths Chapter-1 Number Systems/image038.pngNCERT solution for class 9 Maths Chapter-1 Number Systems/image039.pngNCERT solution for class 9 Maths Chapter-1 Number Systems/image040.png

 

Question 3. Express the following in the form NCERT solution for class 9 Maths Chapter-1 Number Systems/image041.png, where p and q are integers and qNCERT solution for class 9 Maths Chapter-1 Number Systems/image042.png0.

(i) NCERT solution for class 9 Maths Chapter-1 Number Systems/image043.png

(ii) NCERT solution for class 9 Maths Chapter-1 Number Systems/image044.png

(iii) NCERT solution for class 9 Maths Chapter-1 Number Systems/image045.png

Solution :
(i) NCERT solution for class 9 Maths Chapter-1 Number Systems/image046.pngNCERT solution for class 9 Maths Chapter-1 Number Systems/image047.png

We need to multiply both sides by 10 to get

NCERT solution for class 9 Maths Chapter-1 Number Systems/image048.png

We need to subtract (a)from (b), to get

NCERT solution for class 9 Maths Chapter-1 Number Systems/image049.png

We can also writeNCERT solution for class 9 Maths Chapter-1 Number Systems/image050.pngas NCERT solution for class 9 Maths Chapter-1 Number Systems/image051.pngorNCERT solution for class 9 Maths Chapter-1 Number Systems/image052.png.

Therefore, on converting NCERT solution for class 9 Maths Chapter-1 Number Systems/image043.png in theNCERT solution for class 9 Maths Chapter-1 Number Systems/image041.pngform, we get the answer a NCERT solution for class 9 Maths Chapter-1 Number Systems/image053.png.

(ii) NCERT solution for class 9 Maths Chapter-1 Number Systems/image054.pngNCERT solution for class 9 Maths Chapter-1 Number Systems/image055.png

We need to multiply both sides by 10 to get

NCERT solution for class 9 Maths Chapter-1 Number Systems/image056.png

We need to subtract (a)from (b), to get

NCERT solution for class 9 Maths Chapter-1 Number Systems/image057.png

We can also write NCERT solution for class 9 Maths Chapter-1 Number Systems/image058.pngasNCERT solution for class 9 Maths Chapter-1 Number Systems/image059.png orNCERT solution for class 9 Maths Chapter-1 Number Systems/image060.png.

Therefore, on convertingNCERT solution for class 9 Maths Chapter-1 Number Systems/image044.png in theNCERT solution for class 9 Maths Chapter-1 Number Systems/image041.pngform, we get the answer as NCERT solution for class 9 Maths Chapter-1 Number Systems/image061.png.

(iii) NCERT solution for class 9 Maths Chapter-1 Number Systems/image062.pngNCERT solution for class 9 Maths Chapter-1 Number Systems/image063.png

We need to multiply both sides by 1000 to get

NCERT solution for class 9 Maths Chapter-1 Number Systems/image064.png

We need to subtract (a)from (b), to get

NCERT solution for class 9 Maths Chapter-1 Number Systems/image065.png

We can also write NCERT solution for class 9 Maths Chapter-1 Number Systems/image066.png as NCERT solution for class 9 Maths Chapter-1 Number Systems/image067.png

Therefore, on convertingNCERT solution for class 9 Maths Chapter-1 Number Systems/image045.png in theNCERT solution for class 9 Maths Chapter-1 Number Systems/image041.pngform, we get the answer asNCERT solution for class 9 Maths Chapter-1 Number Systems/image068.png4

 

Question 4. Express NCERT solution for class 9 Maths Chapter-1 Number Systems/image069.png in the formNCERT solution for class 9 Maths Chapter-1 Number Systems/image041.png. Are you surprised by your answer? Discuss why the answer makes sense with your teacher and classmates.
Solution :
NCERT solution for class 9 Maths Chapter-1 Number Systems/image070.png

We need to multiply both sides by 10 to get

NCERT solution for class 9 Maths Chapter-1 Number Systems/image071.png

We need to subtract (a)from (b), to get

NCERT solution for class 9 Maths Chapter-1 Number Systems/image072.png

We can also writeNCERT solution for class 9 Maths Chapter-1 Number Systems/image073.pngasNCERT solution for class 9 Maths Chapter-1 Number Systems/image074.png.

Therefore, on convertingNCERT solution for class 9 Maths Chapter-1 Number Systems/image075.png in theNCERT solution for class 9 Maths Chapter-1 Number Systems/image041.pngform, we get the answer asNCERT solution for class 9 Maths Chapter-1 Number Systems/image076.png.

Yes, at a glance we are surprised at our answer.

But the answer makes sense when we observe that 0.9999……… goes on forever. SO there is not gap between 1 and 0.9999……. and hence they are equal.

 

Question 5. What can the maximum number of digits be in the recurring block of digits in the decimal expansion ofNCERT solution for class 9 Maths Chapter-1 Number Systems/image077.png? Perform the division to check your answer.
Solution :
We need to find the number of digits in the recurring block ofNCERT solution for class 9 Maths Chapter-1 Number Systems/image077.png.

Let us perform the long division to get the recurring block ofNCERT solution for class 9 Maths Chapter-1 Number Systems/image077.png.

We need to divide 1 by 17, to get

We can observe that while dividing 1 by 17 we got the remainder as 1, which will continue to be 1 after carrying out 16 continuous divisions.

Therefore, we conclude thatNCERT solution for class 9 Maths Chapter-1 Number Systems/image078.png, which is a non-terminating decimal and recurring decimal.

 

Question 6. Look at several examples of rational numbers in the form NCERT solution for class 9 Maths Chapter-1 Number Systems/image041.png(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 :
Let us consider the examples of the formNCERT solution for class 9 Maths Chapter-1 Number Systems/image041.png that are terminating decimals.

NCERT solution for class 9 Maths Chapter-1 Number Systems/image079.png

NCERT solution for class 9 Maths Chapter-1 Number Systems/image080.png

NCERT solution for class 9 Maths Chapter-1 Number Systems/image081.png

NCERT solution for class 9 Maths Chapter-1 Number Systems/image082.png

NCERT solution for class 9 Maths Chapter-1 Number Systems/image083.png

We can observe that the denominators of the above rational numbers have powers of 2, 5 or both.

Therefore, we can conclude that the property, which q must satisfy inNCERT solution for class 9 Maths Chapter-1 Number Systems/image041.png, so that the rational numberNCERT solution for class 9 Maths Chapter-1 Number Systems/image041.pngis a terminating decimal is that q must have powers of 2, 5 or both.

 

Question 7. Write three numbers whose decimal expansions are non-terminating non-recurring.
Solution : The three numbers that have their expansions as non-terminating on recurring decimal are given below.

0.04004000400004  ….

0.07007000700007  ….

0.013001300013000013 ….

 

Question 8. Find three different irrational numbers between the rational numbers and   NCERT solution for class 9 Maths Chapter-1 Number Systems/image086.png.
Solution :
Let us convertNCERT solution for class 9 Maths Chapter-1 Number Systems/image086.pnginto decimal form, to get

NCERT solution for class 9 Maths Chapter-1 Number Systems/image087.pngand NCERT solution for class 9 Maths Chapter-1 Number Systems/image088.png

Three irrational numbers that lie betweenNCERT solution for class 9 Maths Chapter-1 Number Systems/image089.pngare:

0.73073007300073 ….

0.74074007400074 ….

0.76076007600076 ….

 

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

(i)√23

(ii)√225

(iii) 0.3796

 (iv) 7.478478...

(v) 1.101001000100001...

Solution :
(i) NCERT solution for class 9 Maths Chapter-1 Number Systems/image090.png

 

We know that on finding the square root of 23, we will not get an integer.

Therefore, we conclude thatNCERT solution for class 9 Maths Chapter-1 Number Systems/image090.pngis an irrational number.

(ii) NCERT solution for class 9 Maths Chapter-1 Number Systems/image091.png

We know that on finding the square root of 225, we get 15, which is an integer.

Therefore, we conclude thatNCERT solution for class 9 Maths Chapter-1 Number Systems/image091.pngis a rational number.

(iii) 0.3796

We know that 0.3796 can be converted intoNCERT solution for class 9 Maths Chapter-1 Number Systems/image041.png.

While, converting 0.3796 intoNCERT solution for class 9 Maths Chapter-1 Number Systems/image041.pngform, we get

NCERT solution for class 9 Maths Chapter-1 Number Systems/image092.png.

The rational numberNCERT solution for class 9 Maths Chapter-1 Number Systems/image093.pngcan be converted into lowest fractions, to getNCERT solution for class 9 Maths Chapter-1 Number Systems/image094.png.

We can observe that 0.3796 can be converted into a rational number.

Therefore, we conclude that 0.3796 is a rational number.

(iv) 7.478478….

We know that 7.478478…. is a non-terminating recurring decimal, which can be converted intoNCERT solution for class 9 Maths Chapter-1 Number Systems/image041.pngform.

While, converting 7.478478…. intoNCERT solution for class 9 Maths Chapter-1 Number Systems/image041.pngform, we get

NCERT solution for class 9 Maths Chapter-1 Number Systems/image095.png

NCERT solution for class 9 Maths Chapter-1 Number Systems/image096.png

While, subtracting (a) from (b), we get

NCERT solution for class 9 Maths Chapter-1 Number Systems/image097.png

We know thatNCERT solution for class 9 Maths Chapter-1 Number Systems/image098.pngcan also be written asNCERT solution for class 9 Maths Chapter-1 Number Systems/image099.png.

Therefore, we conclude that 7.478478…. is a rational number.

(v)1.101001000100001  ….

We can observe that the number 1.101001000100001…. is a non-terminating on recurring decimal.

We know that non-terminating and non-recurring decimals cannot be converted intoNCERT solution for class 9 Maths Chapter-1 Number Systems/image041.pngform.

Therefore, we conclude that 1.101001000100001…. is an irrational number.