Terminating & Repeating Decimals - Sixth Grade Mathematics
Every rational number can be expressed as either a terminating decimal or a repeating decimal, revealing deep connections between fraction denominators and base-ten arithmetic.
Classifying Decimal Behaviors
When you convert a fraction into a decimal using long division, only two outcomes are mathematically possible: either the division terminates with a remainder of zero, or the remainders enter an infinite repeating loop.
+-------------------------------------------------------------------------+
| TERMINATING VERSUS REPEATING DECIMALS |
+-------------------+-----------------------+-----------------------------+
| Type | Definition | Examples |
+-------------------+-----------------------+-----------------------------+
| Terminating | A decimal that ends; | 1/2 = 0.5 |
| Decimal | division reaches a | 3/4 = 0.25 |
| | remainder of strictly | 7/8 = 0.875 |
| | zero. | 11/16 = 0.6875 |
+-------------------+-----------------------+-----------------------------+
| Repeating | A decimal that never | 1/3 = 0.3333... = 0.[3] |
| Decimal | ends; a digit or block| 2/3 = 0.6666... = 0.[6] |
| | of digits repeats | 5/11 = 0.4545... = 0.[45] |
| | infinitely. | 1/6 = 0.1666... = 0.1[6] |
+-------------------+-----------------------+-----------------------------+
The Denominator Prime Factorization Rule
You can predict whether a simplified fraction will terminate or repeat WITHOUT performing long division, simply by analyzing the prime factors of its denominator.
The Fundamental Rule:
A fully simplified fraction a/b produces a TERMINATING decimal
if and only if the prime factorization of its denominator b
contains NO prime factors other than 2 and/or 5.
If the simplified denominator contains ANY prime factor other than 2 or 5
(such as 3, 7, 11, 13, etc.), it produces a REPEATING decimal!
Why Only 2 and 5?
Our number system is base ten. The prime factors of 10 are 2 and 5 (10 = 2 * 5). Every power of 10 (100 = 2^2 * 5^2, 1,000 = 2^3 * 5^3) consists exclusively of 2s and 5s. If a denominator contains only 2s and 5s, you can always multiply it by appropriate powers of 2 or 5 to turn it into a power of 10, producing a terminating decimal. If a denominator contains a factor of 3, 7, or 11, it can never divide evenly into any power of 10; thus, the division can never terminate.
Testing Fractions using Prime Factorization:
- 7/40: Denominator 40 = 2^3 * 5. Only 2 and 5 appear --> TERMINATES! (0.175)
- 9/25: Denominator 25 = 5^2. Only 5 appears --> TERMINATES! (0.36)
- 5/24: Denominator 24 = 2^3 * 3. Contains factor 3! --> REPEATS! (0.208333...)
- 21/70: First simplify! 21/70 = 3/10. Denom 10 = 2*5 --> TERMINATES! (0.3)
Representing Repeating Decimals with Bar Notation
In formal mathematics, repeating decimals are represented using bar notation (a horizontal bar called a vinculum placed directly over the repeating block of digits).
Bar Notation Conventions:
- 0.3333... --> 0._3 (bar over 3)
- 0.454545... --> 0._4_5 (bar over 45)
- 0.16666... --> 0.1_6 (bar over 6 ONLY, because 1 does not repeat!)
- 0.142857142857... --> 0._1_4_2_8_5_7 (bar over the entire 6-digit block)
The Length of the Repeating Period
The sequence of repeating digits is called the period. When dividing a whole number by a denominator d, there are only d - 1 possible non-zero remainders. Therefore, the repeating period can never exceed d - 1 digits. When dividing by 7, the remainders must cycle through at most 6 digits before repeating, which is why 1/7 produces the repeating 6-digit period 0.142857...
Exercises
Exercise 1. Without performing long division, determine whether each of the following fractions terminates or repeats: (a) 7/16; (b) 5/18; (c) 14/35; (d) 11/80.
Exercise 2. Use long division to find the decimal representation of 7/12. Express the final answer using proper bar notation.
Exercise 3. Convert the repeating decimal 0.777... (where 7 repeats infinitely) into a simplified fraction using algebraic reasoning.
Solutions
Solution to Exercise 1. For (a) 7/16: 7/16 is fully simplified. The denominator is 16 = 2^4. Because its only prime factor is 2, it Terminates. For (b) 5/18: 5/18 is simplified. The denominator is 18 = 2 * 3^2. Because it contains the prime factor 3, it Repeats. For (c) 14/35: simplify first by dividing top and bottom by 7, giving 2/5. The simplified denominator is 5. Because its only prime factor is 5, it Terminates. For (d) 11/80: 11/80 is simplified. Denominator 80 = 2^4 * 5. Because its only prime factors are 2 and 5, it Terminates.
Solution to Exercise 2. Set up long division for 7 divided by 12: 12 into 7.000 goes 0.5 times (5 * 12 = 60; remainder 10). Bring down 0 to make 100. 12 into 100 goes 8 times (8 * 12 = 96; remainder 4). Bring down 0 to make 40. 12 into 40 goes 3 times (3 * 12 = 36; remainder 4). The remainder 4 repeats indefinitely, generating repeating 3s. The decimal is 0.58333... Written in bar notation, the bar is placed strictly over the repeating digit 3: 0.583 with a bar over 3.
Solution to Exercise 3. Let x = 0.7777... Multiply both sides by 10 to shift the repeating block one place to the left: 10x = 7.7777... Subtract the original equation (x = 0.7777...) from the new equation: 10x - x = 7.7777... - 0.7777... This simplifies to 9x = 7, because the infinite repeating decimals subtract to zero. Divide both sides by 9: x = 7/9. Therefore, 0.777... equals 7/9.