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

[W1]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 numberthat can be expressed as the ratio of two integers p/q, where q0. 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 equation of the product of prime factors that is (Q = P1n1*P2n2………..…Pknk).The result isthat 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 byour team members. The groupmembers 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.

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?

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 q0. Therefore rational numbers include the integers. Some examples of rational numbers are, , , , , 0, , and 1.2. Rational numbers that can be expressed in a decimal form either terminates or repeats. For Example;

= 0.6 (Terminate)

= 1.35 (Terminate)

= 0. (Repeat)

Note that the overbar indicates that0. = 0.66666666…..

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 as

(Q = P1n1*P2n2………..……………Pknk).In the examples listed below, factor the denominators into the product of prime factors and see what we get!

Terminate / Repeat
, where 2 in the denominator is theprime
factor of (2*1). / Here, 3 in the denominator is not the
prime factor of 2’s & 5’s.
, where 5 in the denominator is the prime
factor of (5*1). / Where, 15 in the denominator is not
the prime factor of 2’s & 5’s.
, where 10 in the denominator is the
prime factor of (2*5). / Where, 30 in the denominator is not
the prime factor of 2’s & 5’s.
,where 25 in the denominator is the
prime factor of (5*5). / Where, 66 in the denominator is not
the prime factor of 2’s & 5’s.
, where 16 in the denominator is the
prime factor of (2*2*2*2). / Where, 21 in the denominator is not
the prime factor of 2’s & 5’s.
, where 20 in the denominator is the
prime factor of (2*2*5). / Where, 29 in the denominator is not
the prime factor of 2’s & 5s.

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:

= 1.414213562

= 3.141592653

r = 0.10110111011110

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;

Q = P1n1*P2n2………..……………Pknk, Therefore

1/2 = 0.5 / Terminate
1/3 = 0. / Repeat
1/4 = 0.25 / Terminate
1/5 = 0.20 / Terminate
1/6 = 0.1 / Repeat
1/7 = 0. / Repeat
1/8 = 0.125 / Terminate
1/9 = 0. / 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 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.. Note that the overbar indicates that the 3. is equal to 3.777777………

Set, r = 3.777777………….

Then10r = 37.77777……

And10r – r = 34

So9r = 34

Thereforer = Ans.

(b)Example 2

In this example we have taken two different repeating numbers after the decimal. Therefore we have chosen 5. this is equal to 5.1616161616…….

Set, r = 5.161616…………….

Then100r = 516.1616…………….

And100r – r = 511

So99r = 511

Thereforer = Ans.

(c)Example 3

In this example we have taken three different repeating numbers after the decimal. We have selected 4. this is equal to 4.312312312………………

Set, r = 4.312312312………………….

Then1000r = 4312.312312………………………

And1000r – r = 4308

So999r = 4308

Thereforer = = Ans.

(d)Example 4

In this example we have taken four different repeating numbers after the decimal. Therefore, we have selected 4.and this is equal to4.37613761………….

Set, r = 4.37613761………………..

Then10000r = 43761.37613761…………….

And10000r – r = 43757

So9999r = 43757

Thereforer = Ans.

(e)Example 5

In this example we have taken five different repeating numbers after the decimal. Therefore, we have selected 3. and this is equal to 3.152131521315213………

Set, r = 3.152131521315213…………….

Then100000r = 315213.1521315213…………….

And100000r – r = 315210

So99999r = 315210

Therefore r = Ans.

Problem 3

We have expressed each of the given repeating decimals in the rational number form p/q.

(a)13.

The overbar can also be expressed as 13.201201201……

Set r = 13.201201201…………….

Then1000r = 13201.201201………….

And1000r – r = 13188

So999r = 13188

Thereforer = Ans.

(b)0.

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

Set r = 0.272727………….

Then100r = 27.2727………..

And100r – r = 27

So99r = 27

Thereforer = = Ans.

The above bar can also be expressed as 0.2323232323……

Let r = 0.23232323………….

Then100r = 23.2323…………..

And100r – r = 23

So99r = 23

Thereforer = Ans.

(d)4.16

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

Let r = 4.163333……………..

Then 1000r = 4163.33………

And1000r – r = 4159.17

So999r = 4159.17

Again99900r = 415917

Thereforer = Ans.

Showing that the repeating decimal 0.999… = 0. 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.

Let r = 2.9999…

Then 10r = 29.9999….

And 10r – r = 27

So 9r = 27

Therefore, r = = 3 Ans.

b)9.9999…… = 9.

Let r = 9.9999….

Then 10r = 99.9999….

And 10r – r = 90

So 9r = 90

Therefore r = = 10 Ans.

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. = 10, 2. = 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

[W1]Great job!