Decimal Expansion of Rational Numbers



Decimal Expansion of Rational Numbers

A rational number, expressed in terms of p/q, can easily be set into decimal form by dividing q into p. Both terminating and repeating decimals can be represented in the form p/q. However, to turn the decimal form into the form p/q requires several steps. In order to solve this dilemma, the three of us worked together, each added our own points of view and thoughts collectively to achieve the answers to our problems.

The first problem presented is the conditions in which the decimal with terminate or repeat. To determine if the term p/q will terminate, 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. Thus we were able to determine the probable outcome and the proper procedures in which to receive that outcome.

Now that it can be determined if the decimal is terminating or repeating we can define a formula in which to get the solution. For a terminating decimal, 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.

With a repeating decimal, however, different actions must be taken. 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. If the number still has a decimal, multiply both sides by a product of 10 until it becomes a whole number. Simplify r into fraction form by dividing the coefficient of r into the new whole number. This is the fractional answer of r in the form of p/q.

There were several problems to assist in the explanation of the formulas shown above. All of these examples are displayed in our power point presentation with a detailed display of every step. These problems conclude that our theory is correct and that the formulas discovered are functional. There are some slight flaws in the formula, however. One of the major flaws is the misconception of some whole numbers. For instance the repeating decimal .99… is an actual representation of the number one. Another example would be 1.0 x 10-24, which equals zero. In conclusion, we have developed a better understanding on how rational fractions can be represented in decimal form just as terminating and repeating decimals can be represented in a rational p/q form. This will not only help us presently, but in the future of calculus and all math problems that are to come by being able to picture the outcome of the rational numbers.

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