Under What Conditions Will the Decimal Expansion of P/Q Terminate

Decimal Expansion of Rational Numbers

Arogya Singh Prashant Rajbhandari Seema KC SUMMARY

This project is based on the decimal expansion of rational numbers. Rational number is any number that can be expressed as the ratio of two integers p/q, where q  0. The resulting decimal expansion of the rational number will either terminate or repeat. The first problem in this project has pretty much dealing with the conditions of the decimal expansion of p/q to terminate or repeat. Our assumption in this problem is to satisfy the

n1 n2 nk equation of the product of prime factors that is (Q = P1 *P2 ………..…Pk ). The result is that a fraction terminates in its decimal form, if the prime factors of the denominator are only 2’s and 5’s or a product of prime factors, 2’s and 5’s. Otherwise it repeats. The

Problem 2 has represented the repeating decimal in the rational number form p/q. Our result for this problem is based on our assumption of five rational numbers that has different numbers of repeating numbers after the decimal. The pattern of the problem 3 is also similar to problem 2 but here our emphasis was on the given numbers.

Equal contribution has been made on this project by our team members. The group members that have been assigned for this project is Arogya Singh, Prashant Rajbhandari and Seema KC. Arogya Singh has illustrated his idea on how a repeating decimal can be represented in the rational number form P/Q. His main goal for this project was to elaborate as many examples as he could. Prashant Rajbhandari has illustrated his thought on “what conditions the decimal expansion of p/q will terminate or repeat?” Seema KC has come up with the idea that a fraction terminates in its decimal form, if the prime factors of the denominator are only 2’s and 5’s or a product of prime factors, 2’s and 5’s.

Otherwise it repeats. So this project is a combined effort of all the participants.

1 Problem 1

The decimal from of a rational number p/q can be obtained simply by dividing the denominator into the numerator. The result will be either a terminating decimal or a repeating decimal.

Under what conditions will the decimal expansion of p/q terminate? Repeat? a. The decimal expansion of a rational number can either terminates or repeats. A

rational number is any number that can be expressed as the ratio of two integers p/q,

where q  0. Therefore rational numbers include the integers. Some examples of

2 1 1  50 22 rational numbers are , , , , , 0, 25 , and 1.2. Rational numbers that 1 3 4 2 7

can be expressed in a decimal form either terminates or repeats. For Example;

3 = 0.6 (Terminate) 5

27 = 1.35 (Terminate) 20

2 = 0.6 (Repeat) 3

Note that the overbar indicates that 0.6 = 0.66666666….. b. A fraction (in simplest form/lowest terms) terminates in its decimal form, if the

prime factors of the denominator are only 2’s and 5’s or a product of prime factors,

2’s and 5’s. Otherwise it repeats. The product of prime factors can also be expressed

n1 n2 nk (Q = P1 *P2 ………..……………Pk ). In the examples listed below, factor the denominators

into the product of prime factors and see what we get! 2 Terminate Repeat 1 1 , where 2 in the denominator is the prime Here, 3 in the denominator is not the 2 3 factor of (2*1). prime factor of 2’s & 5’s. 4 1 , where 5 in the denominator is the prime Where, 15 in the denominator is not 5 15 factor of (5*1). the prime factor of 2’s & 5’s. 3 7 , where 10 in the denominator is the Where, 30 in the denominator is not 10 30 prime factor of (2*5). the prime factor of 2’s & 5’s. 4 1 ,where 25 in the denominator is the Where, 66 in the denominator is not 25 66 prime factor of (5*5). the prime factor of 2’s & 5’s. 3 7 , where 16 in the denominator is the Where, 21 in the denominator is not 16 21 prime factor of (2*2*2*2). the prime factor of 2’s & 5’s. 7 4 , where 20 in the denominator is the Where, 29 in the denominator is not 20 29 prime factor of (2*2*5). the prime factor of 2’s & 5s.

c. The decimal expansion of an irrational number neither terminates nor repeats. Since

every fraction has an equivalent decimal form, real numbers include rational

numbers. However, some real numbers cannot be expressed by fractions. These

numbers are called irrational numbers. Some examples of irrational numbers are:

2 = 1.414213562

 = 3.141592653

r = 0.10110111011110

3 What is your conjecture?

Since p/q = p(1/q), it is sufficient to investigate the decimal expansions of 1/q. Our

conjecture or assumption is that the decimal expansion of 1/q for enough positive

integers can either be terminates or repeats. As it has already been describe above that

a fraction (in simplest form/lowest terms) terminates in its decimal form if the prime

factors of the denominator are only 2’s and 5’s or a product of primes factors, 2’s and

5’s. Otherwise it repeats. The product of prime factors can also be expressed in the

equation of;

n1 n2 nk Q = P1 *P2 ………..……………Pk , Therefore

1/2 = 0.5 Terminate 1/3 = 0.3 Repeat 1/4 = 0.25 Terminate 1/5 = 0.20 Terminate 1/6 = 0.16 Repeat

1/7 = 0.142857 Repeat 1/8 = 0.125 Terminate 1/9 = 0.1 Repeat 1/10 = 0.1 Terminate

If we take 100 as a denominator it terminates because 100 is the product of prime

factors of 2*2*5*5. Similarly, if we take 66 it repeats because 66 is the product of

prime factors of 2*3*11. Therefore any number that is in the denominator and is the

4 combination of 2’s and 5’s is terminated. The denominators of the first few unit

fractions having repeating decimals are 3, 6, 7, 9, 11, 12, 13, 14, 15, 17, 18, 19, 21,

22, 23, 24, 26, 27, 28, and 29.

Problem 2

In a decimal expansion of a rational number, it is easy to represent terminating

decimal in the form p/q. For example, 3.74 = 374/100 = 187/50 and 1.2516 =

12,516/10,000 = 3129/2500. Similarly, to represent a repeating decimal in the rational

number form p/q, we have taken few examples.

(a) Example 1

To represent the repeating decimal in the form of p/q, first we have taken 3. 7 . Note

that the overbar indicates that the 3.7 is equal to 3.777777………

Set, r = 3.777777………….

Then 10r = 37.77777……

And 10r – r = 34

So 9r = 34

34 Therefore r = Ans. 9

(b) Example 2

In this example we have taken two different repeating numbers after the decimal.

Therefore we have chosen 5.16 this is equal to 5.1616161616…….

Set, r = 5.161616…………….

Then 100r = 516.1616…………….

5 And 100r – r = 511

So 99r = 511

(c) Example 3

In this example we have taken three different repeating numbers after the decimal.

We have selected 4.312 this is equal to 4.312312312………………

Set, r = 4.312312312………………….

Then 1000r = 4312.312312………………………

And 1000r – r = 4308

So 999r = 4308

4308 1436 Therefore r = = Ans. 999 333

(d) Example 4

In this example we have taken four different repeating numbers after the decimal.

Therefore, we have selected 4.3761 and this is equal to 4.37613761………….

Set, r = 4.37613761………………..

Then 10000r = 43761.37613761…………….

And 10000r – r = 43757

So 9999r = 43757

6 (e) Example 5

In this example we have taken five different repeating numbers after the decimal.

Therefore, we have selected 3.15213 and this is equal to 3.152131521315213………

Set, r = 3.152131521315213…………….

Then 100000r = 315213.1521315213…………….

And 100000r – r = 315210

So 99999r = 315210

Problem 3

We have expressed each of the given repeating decimals in the rational number form

(a) 13. 201

The overbar can also be expressed as 13.201201201……

Set r = 13.201201201…………….

Then 1000r = 13201.201201………….

And 1000r – r = 13188

So 999r = 13188

7 (b) 0. 27

The above bar can also be expressed as 0.2727272727……..

Set r = 0.272727………….

Then 100r = 27.2727………..

And 100r – r = 27

So 99r = 27

27 3 Therefore r = = Ans. 99 11

(c) 0. 23

The above bar can also be expressed as 0.2323232323……

Let r = 0.23232323………….

Then 100r = 23.2323…………..

And 100r – r = 23

So 99r = 23

(d) 4.163

The above bar can also be expressed as 4.163333333…….

Let r = 4.163333……………..

Then 1000r = 4163.33………

And 1000r – r = 4159.17 8 So 999r = 4159.17

Again 99900r = 415917

Showing that the repeating decimal 0.999… = 0. 9 represents the number 1 and also

that 1=1.0, it follows that a rational number may have more than one decimal

representation. To follow this criterion, we have assumed few other rational numbers

that have more than one decimal representation. For example we have taken

2.99999… and 9.9999….. a) 2.9999…..= 2.9

Let r = 2.9999…

Then 10r = 29.9999….

And 10r – r = 27

So 9r = 27

27 Therefore, r = = 3 Ans. 9

b) 9.9999…… = 9.9

Let r = 9.9999….

Then 10r = 99.9999….

And 10r – r = 90

So 9r = 90

90 Therefore r = = 10 Ans. 9

Conclusion

The first new thing we gained from this project was from the solution of

problem1. “A fraction terminates in its decimal form, if the prime factors of the

denominators are only 2s and 5s or a product of the prime factors, 2s and 5s.” This

statement alone gave fractions a new look. From now on we can easily state if a

fraction terminates in its decimal form just by looking at it.

Apart from the above gain, we also learned that “a rational number may have

more than one decimal representation.” We got this from the last part of problem 3.

0.9 = 10, 2.9 = 3. All three of us, Seema KC, Prashant Rajbhandari, and Arogya

Singh, are business majors, so even 0.1 makes a lot of difference in our fields.

Therefore this very problem and solution amazed us in the field of Mathematics.

Signature,

Seema KC

Prashant Rajbhandari

Arogya Singh