Rational Numbers: Fractions to Terminating & Repeating Decimals - Seventh Grade Mathematics
Every rational number can be expressed as a ratio of integers, translating through long division into a decimal that either terminates cleanly or cycles in an infinite repeating pattern.
Defining the Set of Rational Numbers
The word rational originates from the mathematical concept of a ratio. A rational number is any number that can be written in the formal algebraic form p divided by q (or p/q), where p and q are both integers and the denominator q is not equal to zero.
THE SET OF RATIONAL NUMBERS (Q)
p / q where p, q are integers and q != 0
The set of rational numbers includes all whole numbers (since five can be written as 5/1), all integers (since negative seven can be written as -7/1), all proper and improper fractions, mixed numbers, and all decimals that either terminate or repeat.
+--------------------------------------------------------+
| Real Numbers |
| +--------------------------------------------------+ |
| | Rational Numbers (e.g., -3/4, 0.25, 0.333...) | |
| | +--------------------------------------------+ | |
| | | Integers (..., -3, -2, -1, 0, 1, 2, 3, ...) | | |
| | | +--------------------------------------+ | | |
| | | | Whole Numbers (0, 1, 2, 3, ...) | | | |
| | | +--------------------------------------+ | | |
| | +--------------------------------------------+ | |
| +--------------------------------------------------+ |
+--------------------------------------------------------+
Converting Fractions to Decimals Using Long Division
To convert any fraction p/q into decimal form, divide the numerator p by the denominator q. By placing a decimal point after the numerator and appending trailing zeros, long division reveals the exact decimal expansion.
Example: Convert 3/8 to a decimal
0.375
8 ) 3.000
- 2.4 (8 * 3 = 24)
---
60
- 56 (8 * 7 = 56)
--
40
- 40 (8 * 5 = 40)
--
0 <-- Remainder is 0: TERMINATING DECIMAL
When long division reaches a remainder of zero, the decimal terminates. The fraction 3/8 is exactly equal to zero and three hundred seventy-five thousandths (0.375).
Terminating Decimals and Prime Factorization
A simplified fraction p/q produces a terminating decimal if and only if the prime factorization of its denominator q contains no prime factors other than two, five, or both.
+------------------+-----------------------+-----------------------+
| Fraction (p/q) | Prime Factors of q | Decimal Representation|
+------------------+-----------------------+-----------------------+
| 7 / 20 | 20 = 2 * 2 * 5 | 0.35 (Terminates) |
| 9 / 25 | 25 = 5 * 5 | 0.36 (Terminates) |
| 11 / 16 | 16 = 2 * 2 * 2 * 2 | 0.6875 (Terminates) |
| 5 / 12 | 12 = 2 * 2 * 3 | 0.41666... (Repeats) |
| 2 / 7 | 7 is prime (not 2, 5) | 0.285714... (Repeats) |
+------------------+-----------------------+-----------------------+
Because our base-ten number system is built on powers of ten (whose prime factors are two and five), any denominator composed solely of twos and fives can be scaled to a power of ten, producing a terminating decimal. If any other prime factor (such as three, seven, or eleven) appears in the simplified denominator, the division will never terminate with a remainder of zero.
Repeating Decimals and Periodicity
When the simplified denominator contains prime factors other than two and five, the long division algorithm eventually produces a remainder that has appeared previously. Because there are only q possible remainders when dividing by q, the sequence of remainders must begin repeating within at most q minus one steps.
Example: Convert 5/11 to a decimal
0.4545...
11 ) 5.0000
- 4.4 (Remainder 6)
---
60
- 55 (Remainder 5 repeats!)
--
50
- 44 (Remainder 6 repeats!)
--
6
Once a remainder repeats, the quotient digits must repeat in the exact same sequence indefinitely. A repeating decimal uses a horizontal bar called a vinculum placed directly over the repeating block of digits. Thus, 5/11 is written as 0.45 with a bar over both four and five.
Converting Repeating Decimals to Fractions
Every repeating decimal can be converted back into a ratio of integers using an algebraic method that eliminates the infinite repeating tail.
Example: Convert 0.777... to a fraction
Step 1: Let x = 0.777...
Step 2: Multiply by 10 to shift one repeating period:
10x = 7.777...
Step 3: Subtract the original equation:
10x = 7.777...
- x = 0.777...
----------------
9x = 7.000...
Step 4: Solve for x:
x = 7/9
When a decimal has a multi-digit repeating cycle, multiply by ten raised to the power of the cycle length. If the repeating cycle is two digits long, multiply by one hundred; if three digits long, multiply by one thousand.
Exercises
Work through each problem carefully in your notebook before reading the provided solutions.
Exercise 1. Without performing long division, determine whether the fraction seven-sixteenths will produce a terminating decimal or a repeating decimal. Explain your reasoning using prime factorization.
Exercise 2. Use long division to convert the fraction five-sixths into decimal form, identifying the repeating digit and writing the result with proper bar notation.
Exercise 3. Use an algebraic equation to convert the repeating decimal zero and eighteen hundredths (where both one and eight repeat indefinitely) into a simplified fraction.
Exercise 4. Explain why the decimal expansion of any rational number p/q must either terminate or enter a repeating cycle of digits.
Solutions
Solution to Exercise 1. First ensure the fraction seven-sixteenths is in simplest form, which it is since seven and sixteen share no common factors other than one. Next examine the prime factorization of the denominator sixteen: sixteen equals two times two times two times two (two to the fourth power). Because the prime factorization contains exclusively factors of two and no other prime numbers, seven-sixteenths must produce a terminating decimal. Performing the division confirms this: seven divided by sixteen equals zero and four thousand three hundred seventy-five ten-thousandths (0.4375).
Solution to Exercise 2. Divide five by six using long division. Placing a decimal and zeros yields five point zero divided by six. Six goes into fifty eight times with a remainder of two. Six goes into twenty three times with a remainder of two. The remainder of two repeats continuously, producing an unending sequence of the digit three. Therefore, five-sixths equals zero point eight three with the vinculum bar placed exclusively over the digit three: 0.83 with bar over 3.
Solution to Exercise 3. Let x equal zero point eighteen eighteen eighteen repeating. Because the repeating block consists of two digits (one and eight), multiply both sides of the equation by one hundred: one hundred x equals eighteen point eighteen eighteen repeating. Now subtract the original equation: one hundred x minus x equals ninety-nine x, and eighteen point eighteen repeating minus zero point eighteen repeating leaves exactly eighteen. Dividing both sides by ninety-nine gives x equals eighteen ninety-ninths. Divide both numerator and denominator by their greatest common factor, nine, to obtain the simplified fraction two-elevenths.
Solution to Exercise 4. When dividing an integer p by a positive integer q, each step of the long division algorithm produces an integer remainder. The only possible remainders are zero and the positive integers from one up to q minus one. If a remainder of zero is reached, the decimal terminates immediately. If zero is never reached, there are at most q minus one distinct non-zero remainders available. Once all available remainders have appeared, a previously seen remainder must repeat by the pigeonhole principle. Once a remainder recurs, the subsequent dividend and quotient digits repeat in an identical cycle, guaranteeing that every rational number is either terminating or repeating.