Thursday, July 28, 2016

Chapter 6.10 - Representing Recurring decimals

In the previous section we saw how 13 is expressed as decimals. In this section, we will see another example.
Let us try to write 16 in decimal form. Just like in the case of 16, there is no natural number, which when multiplied with 6, will give any 'power of 10'. So we will use the other method:
1. We know that 16 = (1×10) (6×10)
2. Let us rearrange the right side: 16 = 110 × 106 
3. In the above result, we can write 106 as (1 + 4)
4. So (2) becomes 16 = 110 × (1 + 46 ). So we get:
5. 16 = 110 + 460
6. Look at the above result carefully. We have two fractions on the right side: 110 and 460 
    ♦ Out of these two, 110 is in an 'ideal form'. Because it has a 'power of 10' in the denominator. So it can be readily converted into a decimal form
    ♦ But the other fraction 460 is causing a problem. It cannot be readily converted into a decimal form
    ♦ If this 460 is very very small, then we can ignore it. In that case, (5) will become 16 = 110
    ♦ But unfortunately, 460 is not very small, and we cannot ignore it. 
• After reaching (5), if we write 16 = 0.1, we are ignoring 460
• That is not a good thing to do because, 460 is not a small quantity, that can be 'just ignored'
7. We arrived at (5) by writing 16 as (1×10) (6×10) in (1).  Now let us write it in a modified form: 

8. We know that 16 = (1×100) (6×100)
9. Let us rearrange the right side: 16 = 1100 × 1006 
10. In the above result, we can write 1006 as (16 + 46 )
11. So (9) becomes 16 = 1100 × (16 + 46 ). So we get:
12. 16 = 16100 + 4600
13. Look at the above result carefully. We have two fractions on the right side: 16100 and 4600
    ♦ Out of these two, 16100 is in an 'ideal form'. Because it has a 'power of 10' in the denominator. So it can be readily converted into a decimal form
    ♦ But the other fraction 4600 is causing a problem. It cannot be readily converted into a decimal form
    ♦ If this 4600 is very very small, then we can ignore it. In that case, (12) will become 16 = 16100
    ♦ But unfortunately, 4600 is not very small, and we cannot ignore it
• After reaching (12), if we write 16 = 0.16, we are ignoring 4600
• That is not a good thing to do because, 4600 is not a small quantity, that can be 'just ignored'
• It may be noted that 4600 is ten times smaller than 460 , which is causing the problem in (5) 
14. We arrived at (12) by writing 16 as (1×100) (6×100) in (8).  Now let us write it in a modified form:

15. We know that 16 = (1×1000) (6×1000)
16. Let us rearrange the right side: 16 = 11000 × 10006 
17. In the above result, we can write 10006 as (166 + 46 )
18. So (16) becomes 16 = 11000 × (166 + 46 ). So we get:
19. 16 = 1661000 + 46000
20. Look at the above result carefully. We have two fractions on the right side: 1661000 and 46000
    ♦ Out of these two, 1661000 is in an 'ideal form'. Because it has a 'power of 10' in the denominator. So it can be readily converted into a decimal form
    ♦ But the other fraction 46000 is causing a problem. It cannot be readily converted into a decimal form
    ♦ If this 46000 is very very small, then we can ignore it. In that case, (19) will become 16 = 1661000
    ♦ But unfortunately, 46000 is not very small, and we cannot ignore it
• After reaching (19), if we write 16 = 0.166, we are ignoring 46000
• That is not a good thing to do because, 46000 is not a small quantity, that can be 'just ignored'
• It may be noted that 46000 is ten times smaller than 4600 , which is causing the problem in (12)
• Also it is 100 times smaller than 460 , which is causing the problem in (5)
• So the fractional part is obviously decreasing with each step. It will keep on decreasing with each step, and reach very low values. How low can it reach? 
The lowest value possible is 'zero'. So, with each step, the fractional part gets closer and closer to zero 
21. We arrived at (19) by writing 16 as (1×1000) (6×1000) in (15)

First we used 10, then 100, and we used 1000 just above. We can proceed using 10000, 100000, etc.,
But we do not have to write the steps. A pattern has already emerged. Based on that pattern, we can write:
 1   =  110   +  460       =   0.1 + 460
 1   =  16100   +  4600      =   0.16 +  4600
 1   =  1661000   +  46000       =   0.166 + 46000
 1   =  166610000   +  460000       =   0.1666 + 460000
 16     =  16666100000   +  4600000        =   0.16666 + 4600000

All the above results are true. They are exact values of 16 . We can proceed further as long as we wish. But this much is sufficient for us to understand an important property:

When the number of digits on the 'right side of the decimal point' increases, the remaining fractional portion decreases. 
• For example, if we take 4 places on the right side of the decimal point, 1= 0.1666, the fractional part then is  460000
• If we take 5 places on the right side of the decimal point, 1= 0.16666, the fractional part then is 4600000, which is smaller than 4600000

As the fractional part becomes smaller and smaller, it can be ignored if :
We take sufficient number of places after the decimal point.

Another important point can also be noted from the above discussion:
• We have written 16 as the sum of a decimal value and a fractional value 
• The left side is always a constant, which is equal to 16
• So the right side must also be a constant. That is., the sum of the decimal value and the fractional value must also be a constant equal to 16
• But we saw that, when the number of places after the decimal point increases, the fractional value decreases
• When the fractional value decreases, the decimal portion must increase in value. Then only will the sum remain a constant. Thus we can say: 
■ As the number of decimal places increases, the decimal portion increases, and gets closer and closer to 16 . This is shown below:


So now we know that we cannot convert 16 into an exact decimal form. There will always be a small fraction remaining. As in the case of 13, here also we will use the special method to represent it.


In the final pattern that we derived above, we saw 0.1, 0.16, 0.166, 0.1666, and so on. The digit '6' will repeat forever. So 16, when converted into decimal form, will give a recurring decimal. We can represent the decimal by any one of the methods that we saw in the case of 13.

• In Method 1, three dots are placed after the decimal. It indicates that it is a recurring decimal
• In Method 2, a dot is placed above the digit which repeats forever. In our case, 6 repeats for ever. So, the dot is placed over it

• In Method 3, a line is placed above the digit which repeats forever. In our case, 6 repeats for ever. So, the line is placed over it

Now we will see some solved examples
Solved example 6.28
For each of the fractions given below, write fractions (with denominators powers of 10) getting closer and closer to the original value, and then write the decimal form
(i) 56 ,  (ii) 311 ,  (iii) 2311 ,  (iv) 113 
Solution:
(i) 56 : 1. We know that 56 = (5×10) (6×10)
2. Let us rearrange the right side: 56 = 510 × 106 
3. In the above result, we can write 106 as (1 + 4)
4. So (2) becomes 56 = 510 × (1 + 46 ). So we get:
5. 56 = 510 + 2060
Second cycle:
1. We know that 56 = (5×100) (6×100)
2. Let us rearrange the right side: 56 = 5100 × 1006 
3. In the above result, we can write 1006 as (16 + 46 )
4. So (2) becomes 56 = 5100 × (16 + 46 ). So we get:
5. 56 = 80100 + 20600
Third cycle:
1. We know that 56 = (5×1000) (6×1000)
2. Let us rearrange the right side: 56 = 51000 × 10006 
3. In the above result, we can write 10006 as (166 + 46 )
4. So (2) becomes 56 = 51000 × (166 + 46 ). So we get:
5. 56 = 8301000 + 206000
Fourth cycle:
1. We know that 56 = (5×10000) (6×10000)
2. Let us rearrange the right side: 56 = 510000 × 100006 
3. In the above result, we can write 100006 as (1666 + 46 )
4. So (2) becomes 56 = 510000 × (1666 + 46 ). So we get:
5. 56 = 83301000 + 2060000

Based on the above results we can write:
 5   =  510   +  2060       =   0.5 + 2060
 56    =  80100   +  20600      =   0.8 +  20600
 56    =  8301000   +  206000       =   0.83 + 206000
 56    =  833010000   +  2060000       =   0.833 + 2060000
The fractions 510801008301000833010000, etc., gets closer and closer to 56 . Based on this, we can write the decimal form of as:

The result obtained when we divide 5 by 6 using a calculator is shown below:


In the next section we will solve the second problem 311.

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