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MATH- CALCULUS I

- DONE BY-
- Supatchaya, Piyate,and Sulakshmi

- PROBLEM 1
- Under what conditions
- will the decimal expansion p/q
- terminate? Repeat?
- PROBLEM 2
- Suppose that we are given
- the decimal expansion of a rational number. How

can we represent the decimal - in the rational form p/q?

PROBLEM 3

- Express each of the following repeating

decimals in the rational number form of p/q. - (a) 13.201 (b)0.27 (c)0.23 (d)4.163333..
- Show that the repeating decimal
- 0.9999. 0.9 represets the number 1.
- Also note that 1 1.0.
- Thus it follows that a rational number may have

more than one decimal representation. - Can you find any others?

SOLUTIONS

SOLUTION 1

- The decimal expansion of p/q can be represented

as p(1/q) - The cases when it will terminate is if the value

of p and q is equal to zero - The other cases when it will terminate is

explained as follows

EXPLANATION

- Lets assume that the value of q to be from 1 to

15. - In all the cases the value of p remains the same

.i.e. 1 - On calculating we get the following results

CALCULATION

- 1/1 1
- 1/2 0.5
- 1/3 0.333..
- 1/4 0.25
- 1/5 0.2
- 1/6 0.16666..
- 1/7 0.142857142857..
- 1/8 0.125
- 1/9 0.1111111.
- 1/10 0.1

- 1/11 0.090909
- 1/12 0.833333
- 1/13 0.0769230769230
- 1/14 0.0714285714..
- 1/15 0.06666..

CONCLUSION

- We notice in the above calculations that the

decimal expansion p/q will terminate when the

value of q is 2 or 5. The epansion will

terminate even if the value of q is a factor of 2

or 5. - The value of p will repeat when the value is

neither 2 or 5 or a multiple of 2 or 5.

SOLUTION 2

- We can represent all decimal function as a

rational number. - For Example - 7.9
- 5.55
- 998.21

79 10

555 100

99821 100

EXPLANATION

- In the case of repeating decimal numbers like
- 0.333333 or 0.0909090
- it is very impossible to present the decimal

function as a rational number as it is infinite. - So we use the geometric series to find the

result. - In this case we will use the decimal 0.33333
- and represent it as a rational number through

the geometric series.

CALCULATION

3 10

3 100

3 1000

3 10000

- 0.3 .
- GEOMETRIC SERIES , iff -1lt r lt1
- This is a geometric series with a and r

0.1 - So, 0.3

a 1-r

3 10

1 10

3/10 (1- 1/10)

3/10 9/10

- So the value of 0.3333.equalizes to
- 3/9 which on simplification is 1/3.
- CONCLUSION
- This is how we can represent the a repeating

decimal function in the rational form.

SOLUTION 3

- We will use the geometric series to solve

the following problem. - GEOMETRIC SERIES , iff -1lt r lt1
- a) 13.201201201
- Here a ,and r
- ___
- 13.201 13
- 13 / (1 -

) - 13
- 13

a 1-r

201 1,000

1 1,000

201 1,000

201 10,000

201 1,000

1 1,000

201 1,000

1,000 999

201 999

201 999

12987 999

4396 333

- b) 0.272727.
- Here a ,and r
- __
- 0.27
- / (1 - )

27 100

1 100

27 100

27 10,000

27 100

1 100

27 100

100 99

27 99

3 11

- c) 0.2323232323232..
- Here a ,and r
- __
- 0.23
- / (1 - )

23 100

1 100

23 100

23 10,000

1 100

23 100

23 100

100 99

23 99

- d) 4.1633333.
- Here a ,and r
- _
- 4.163 4

- 4

/ (1 - ) - 4

- 4
- 4

3 1,000

1 10

16 100

3 1,000

3 10,000

16 100

3 1,000

1 10

16 100

3 1,000

10 9

16 100

1 300

48 300

1200 48 300

1248 300

312 75

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