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#### Chapter 1: Number System

Exercise 1.3 Part 1

CLASS 9 NCERT SOLUTIONS NUMBER SYSTEMS

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

Sol. (i) 0.36, terminating.

(ii)

Hence, it is a non-terminating recurring number.

(iii) 4.125, terminating

non-terminating recurring

(v)

non-terminating recurring.

(vi) 0.8225, terminating.

1. You know that

Can you predict what the decimal expansions of

are, without actually doing the long division? If so, how?

Question 3: Express the following in the form p/q, where p and q are integers and q ≠ 0.

(i) 0.6
ii) 0.47

Let, x=0.66666…x=0.66666…

[Since only one digit is repeating, so multiply x with 10.]

So, 10x=10×6.66666….10x=10×6.66666….
Or, 10x=6+0.66666…10x=6+0.66666…
Or, 10x=6+x

10x=6+x
Since, x=0.66666…x=0.66666…
Or, 10x−x=610x-x=6
Or, 9x=69

x=6

Or, x=6/9=3×2/3×3=2/3

Thus, 0.6 =2/3

ii) 0.47

Ans: Let x = 0.47
Or, x=0.47777….x=0.47777….

[Since only one digit is repeating, so multiply x with 10.]

Since 10x=10×0.477777..10x=10×0.477777..
Or, 10x=4.777777…10x=4.777777…
Or, 10x=4.3+0.4777…10x=4.3+0.4777…
Or, 10x=4.3+x10x=4.3+x
Since x=0.47777…x=0.47777…
Or, 10x−x=4.310x-x=4.3
Or, 9x=4.39x=4.3

Or, x=4.3/9

x=4.39

Or, x=43/90

(iii) 0.001

Or, x=0.001001…x=0.001001…

Since, there are three repeating decimal digit, so multiply x with 1000

Or, 1000 x=1000×0.001001…1000 x=1000×0.001001…
Or, 1000 x=1.001…1000 x=1.001…
Or, 1000 x=1+0.001….1000 x=1+0.001….
Since x=0.001001…x=0.001001…
Thus, 1000 x=1+x 1000 x=1+x
Or, 1000 x−x=11000 x-x=1
Or, 999 x=1999 x=1
Or, x=1/999

Q 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.

Sol. x = 0.99999 … =

One digit is repeating, we multiply by 10.
10x = 9 9 . ⇒ 10x = 9 + x ⇒ 9x = 9
⇒ x = 1

. is infinitely close to 1, i.e., we can make the difference between 1 and
0.99 …… as small as we wish by taking enough 9’s.

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.

The maximum number of digits in the repeating block is 16 (< 17). Division gives

6. The repeating block has 16 digits. 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?

Sol. 2/5= 0 4
3/ 2 = 1 5
7/8 = 0 875
7/ 10 = 0.7

All the denominators are either 2 (or its power), 5 (or its power) or a combination of both.

7.Write three numbers whose decimal expansions are non-terminating non-recurring.
Sol. 7.314114111411114………. , 0.101002000300004………..,

π = 3.1416……..

Q 7: Write three numbers whose decimal expansions are non-terminating non-recurring.

Answer: non-terminating non-recurring numbers are known as irrational numbers. Irrational numbers cannot be expressed in the form of p/q where q≠0.
Following are the possible numbers:
0.72012001200012000001………
0.73013001300013000001…………
0.7501500150001500001………..

Q 8: Find three different irrational numbers between the rational numbers 5/7 and 9/11.

Ans: 5/7 = 0.714285714285…….. and 9/11 = 0.8181818………
Possible irrational numbers between them can be as follows:
0.72012001200012000001………
0.73013001300013000001…………
0.7501500150001500001………..
Note: non-terminating non-recurring numbers are known as irrational numbers. Irrational numbers cannot be expressed in the form of p/q where q≠0. Numbers given above cannot be expressed in the form of p/q and hence are irrational