# Number System

## Exercise 1.3 Part 2

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

(i) 0.6

Let, x=0.66666…

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

So, 10x=10xx6.66666….
Or, 10x=6+0.66666…
Or, 10x=6+x
Since, x=0.66666…
Or, 10x-x=6
Or, 9x=6
Or, x=6/9=(3xx2)/(3xx3)=2/3

Thus, 0.6 =2/3

#### Alternate method:

Given, 0.6

Step: 1 - Omit the decimal and recurring symbol (repeating symbol i.e. bar)
Step: 2 – Put repeating decimal as numerator and one 9 as denominator for one repeating decimal digit.

Or, 6/9=(3xx2)/(3xx2)=2/3

Thus, 0.6 =2/3

(ii) 0.47

Or, x=0.47777….

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

Since 10x=10xx0.477777..
Or, 10x=4.777777…
Or, 10x=4.3+0.4777…
Or, 10x=4.3+x
Since x=0.47777…
Or, 10x-x=4.3
Or, 9x=4.3
Or, x=(4.3)/9

Or, x=43/90

#### Alternate method:

Given, 0.47

Step: 1 – Take 47 as numerator and subtract 4 (non repeating decimal digit) from it.
Step: 2 - Put one 9 for one repeating decimal digit
Step: 3 - Place one zero (0) after 9 for one non repeating decimal digit as denominator.
Thus, we get

(47-4)/90=43/90

(iii) 0.001

Or, x=0.001001…

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

Or, 1000x=1000xx0.001001…
Or, 1000x=1.001…
Or, 1000x=1+0.001….
Since x=0.001001…
Thus, 1000x=1+x
Or, 1000x-x=1
Or, 999x=1
Or, x=1/999

#### Alternate method:

Given, 0.001

Step: 1- Take 001 as numerator
Step: Since there are three repeating decimal digits, so take three 9, i.e. 999 as denominator.

Thus, 0.001 = 001/999=1/999

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. Since, there is only one repeating digit after decimal, thus multiply x with 10. #### Alternate method: Step: 1 – Take 9 as numerator.
Step: 2 – Since there is only one repeating decimal digit, so take one 9 as denominator.
After doing above steps we get, 