DECIMAL EXPANSIONS OF RATIONAL NUMBERS Created By: Matthew Marple Zac Campbell Heather Schlichter

INTRODUCTION: A rational number, expressed in terms of p/q, can easily be set into decimal form by dividing q into p. However, to turn the decimal form into the form p/q requires several steps. A rational number, expressed in terms of p/q, can easily be set into decimal form by dividing q into p. However, to turn the decimal form into the form p/q requires several steps.

TERMINOLOGY: Terminating decimal-any number in which a product of 10 will result in a whole number, a number with out a decimal. Terminating decimal-any number in which a product of 10 will result in a whole number, a number with out a decimal. Repeating decimal-any number in which a pattern of the numbers repeats continuously without terminating. Repeating decimal-any number in which a pattern of the numbers repeats continuously without terminating. Both terminating and repeating decimals can be represented in the form p/q Both terminating and repeating decimals can be represented in the form p/q

THE FIRST PROBLEM: How do we know if the term p/q will terminate or repeat? How do we know if the term p/q will terminate or repeat? Answer: Replace the term p with the number 1. If 1/q terminates, than any number p over q shall terminate. If 1/q is a repeating decimal, than any number p over q will repeat. Answer: Replace the term p with the number 1. If 1/q terminates, than any number p over q shall terminate. If 1/q is a repeating decimal, than any number p over q will repeat.

EXAMPLE 1: Terminating: 3/4 = p/q where 3=p and 4=q ~replace p with the number 1 p/q=1/4=.25 ~the number can be multiplied by 100 to become the whole number 25, thus it is a terminating decimal. Since that terminates, any term p/4, or ¾, will terminate. ¾=.75 Terminating: 3/4 = p/q where 3=p and 4=q ~replace p with the number 1 p/q=1/4=.25 ~the number can be multiplied by 100 to become the whole number 25, thus it is a terminating decimal. Since that terminates, any term p/4, or ¾, will terminate. ¾=.75

EXAMPLE 2: Repeating: 2/3=p/q where 2=p and 3=q ~replace the number p with 1 p/q=1/3=.33 ~the number 3 repeats continuously and therefore is a repeating decimal and is unable to be multiplied by a product of 10 to become a whole number. Since the term 1/3 repeats, any p over 3 will repeat. p/3=2/3=.66 Repeating: 2/3=p/q where 2=p and 3=q ~replace the number p with 1 p/q=1/3=.33 ~the number 3 repeats continuously and therefore is a repeating decimal and is unable to be multiplied by a product of 10 to become a whole number. Since the term 1/3 repeats, any p over 3 will repeat. p/3=2/3=.66

THE SECOND PROBLEM: Part A How can we represent a terminating decimal in the rational form p/q? How can we represent a terminating decimal in the rational form p/q? Answer: Set the decimal equal to the letter r. Multiply both sides by a product of 10 until r becomes a whole number. Solve for r with a fraction on the other side. Answer: Set the decimal equal to the letter r. Multiply both sides by a product of 10 until r becomes a whole number. Solve for r with a fraction on the other side.

THE SECOND PROBLEM: Part B How can we represent the repeating decimal in the rational form p/q? How can we represent the repeating decimal in the rational form p/q? Answer: Set the rational number equal to r. For each digit in the repeating pattern multiply both sides by 10 so that the repeating pattern will align itself to the original number r. Subtract r from both sides to terminate the decimal*. Simplify r into fraction form by dividing the coefficient of r into the new whole number. Answer: Set the rational number equal to r. For each digit in the repeating pattern multiply both sides by 10 so that the repeating pattern will align itself to the original number r. Subtract r from both sides to terminate the decimal*. Simplify r into fraction form by dividing the coefficient of r into the new whole number. *If the number still has a decimal, multiply both sides by a product of 10 until it becomes a whole number AFTER the subtraction of r.

THE THIRD PROBLEM: Testing the Theory Example A: Example A: ___ =r ___ 1000r= r-r= r=13188 r=13188/999=4396/333 ___ =r ___ 1000r= r-r= r=13188 r=13188/999=4396/333

THE THIRD PROBLEM: Testing the Theory Example B: __ Example B: __ 0.27=r __ 100r= r-r=27 99r=27 r=27/99=3/11

THE THIRD PROBLEM: Testing the Theory Example C: __ 0.23=r __ 100r= r-r=23 99r=23 r=23/99 Example C: __ 0.23=r __ 100r= r-r=23 99r=23 r=23/99

THE THIRD PROBLEM: Testing the Theory Example D: _ 4.163=r _ 100r= r= r=41217 r=41217/9900=13739/3300 Example D: _ 4.163=r _ 100r= r= r=41217 r=41217/9900=13739/3300

FLAWS IN THE SYSTEM: There is a slight “flaw” in the calculation process which can be easily recognizable. This is done by concluding that a number such as.99 is so close to the number 1 that it is the number one. Another example is 1 x for that number is so small it actually represents the number zero. There is a slight “flaw” in the calculation process which can be easily recognizable. This is done by concluding that a number such as.99 is so close to the number 1 that it is the number one. Another example is 1 x for that number is so small it actually represents the number zero.

THE SUMMARY: It is easy to turn a fraction in the form of p/q into a decimal. It is also possible, with a couple more steps, to turn a terminating or repeating decimal into the fraction form p/q. It is easy to turn a fraction in the form of p/q into a decimal. It is also possible, with a couple more steps, to turn a terminating or repeating decimal into the fraction form p/q.