Project 1.2 Decimal Expansions of Rational Numbers

By

Steven Hedrick[W1][W2]

Mitchell Jordan

Nathan Walter

Summary:

This project determines under what conditions the fractions of rational numbers will repeat or terminate. It also shows that a repeating decimal can only be displayed as a rational number with a degree of error.

Steven Hedrick: summary, problem and solution for problem 1, and program

Mitchell Jordan: problem and solution for problem 2 and Power Point presentation

Nathan Walter: problem and solution for problem 3

Solutions:

  1. Under what conditions will the decimal expansion of p/q terminate? Repeat? Hint: Since p/q = p(1/q), it is sufficient to investigate the decimal expansions of 1/q. Calculate 1/q for enough positive integers q to form a conjecture as to whether the decimal expansion will terminate or repeat. What is your conjecture?

Values for which 1/q will terminate or repeat
1/1 / Term. / 1/11 / Repeat / 1/21 / Repeat / 1/31 / Repeat / 1/41 / Repeat
1/2 / Term. / 1/12 / Repeat / 1/22 / Repeat / 1/32 / Term. / 1/42 / Repeat
1/3 / Repeat / 1/13 / Repeat / 1/23 / Repeat / 1/33 / Repeat / 1/43 / Repeat
1/4 / Term. / 1/14 / Repeat / 1/24 / Repeat / 1/34 / Repeat / 1/44 / Repeat
1/5 / Term. / 1/15 / Repeat / 1/25 / Term. / 1/35 / Repeat / 1/45 / Repeat
1/6 / Repeat / 1/16 / Term. / 1/26 / Repeat / 1/36 / Repeat / 1/46 / Repeat
1/7 / Repeat / 1/17 / Repeat / 1/27 / Repeat / 1/37 / Repeat / 1/47 / Repeat
1/8 / Term. / 1/18 / Repeat / 1/28 / Repeat / 1/38 / Repeat / 1/48 / Repeat
1/9 / Repeat / 1/19 / Repeat / 1/29 / Repeat / 1/39 / Repeat / 1/49 / Repeat
1/10 / Term. / 1/20 / Term. / 1/30 / Repeat / 1/40 / Term. / 1/50 / Term.

1/q will terminate when q is in the form of 2x5y and x and y are positive integers.

  1. If a number is a repeating decimal when expanded, it cannot be easily displayed as a rational number, but it is possible, with a degree of error.

If a decimal repeats at a certain point, the number should be multiplied so that that repeating point becomes a ones-place number.

Ex: 3.135135135… should be multiplied by 1000

To become 3135.135135…

By assigning the original repeating expansion the variable r, we are able to calculate a way to display the number rationally

1)r = 3.135135…

2)1000r = 3135.135135…

3)1000r – r = 3132

4)Eg. 999r = 3132 and r = 3132/999 = 116/37

3. Problem: the purpose of problem 3 was to express various repeating decimals in the the rational number form of p/q. The problem would also go further to ask that the repeating decimal 0.99999… be shown to represent the number 1

  1. 13.201201…

First I set r=13.201201…

I then multiply r by 1000 to get 1000r=13201.201…

I then subtract r to get a whole number 1000r-r=13188

999r=13188  r/999=13188/999

So after simplifying  4396/333

  1. 0.2727

R=.2727…

100r=27.2727

100r-r=27

So finally we come to the rational number r= 27/100

  1. r=0.2323…

100r=23.2323…

100r-r=23

Therefore 99r would be equal to 23/100

  1. 4.16333333….

R=4.1633333….

1000r=4163.333333….

1000r-100r=3747

900r=3747

Finally we can conclude that r=3747/900 or  1249/300

The final part of problem number 3 asked that I prove 0.99999…. is another representation of the number 1. To solve this I will utilize the formula provided by the book for rationalizing a repeating decimal:

a)r=0.9999….

b)10r=9.9999

c)10r-r=9

d)9r(1/9)=9(1/9)

e)R=9/9 or 1

f)From this we can derive that r is both equal to 0.9999…. and 1, therefore 0.9999repeating is equal to 1.0. this rational number has more than one decimal representation but I could not find any others that are similar to this circumstance.

[W1]Good job

[W2]