The fraction 2/9 is equivalent to a repeating decimal.
Understanding Repeating Decimals and Their Fractional Equivalents
Repeating decimals are numbers where one or more digits after the decimal point repeat infinitely. For instance, 0.3333... is a repeating decimal where the digit '3' repeats, and 0.121212... is a repeating decimal where the sequence '12' repeats.
Fractions, which represent a part of a whole, can always be converted into decimals by dividing the numerator by the denominator. While some fractions yield terminating decimals (like 1/2 = 0.5 or 3/4 = 0.75), others result in repeating decimals.
When Does a Fraction Become a Repeating Decimal?
A fraction, when simplified to its lowest terms, will produce a repeating decimal if its denominator contains any prime factors other than 2 or 5. If the denominator only has prime factors of 2 and/or 5, the decimal will terminate.
For example:
- 1/3: The denominator is 3. Since 3 is not 2 or 5, 1/3 results in 0.3333...
- 2/7: The denominator is 7. Since 7 is not 2 or 5, 2/7 results in 0.285714285714...
- 2/9: The denominator is 9 (which is 3 x 3). Since it has prime factors of 3, 2/9 results in 0.2222...
Common Examples of Repeating Decimals and Their Fractional Forms
Many common repeating decimals have straightforward fractional equivalents. Here are a few examples:
| Repeating Decimal | Equivalent Fraction |
|---|---|
| 0.2222... | 2/9 |
| 0.4444... | 4/9 |
| 0.5555... | 5/9 |
| 0.7777... | 7/9 |
These patterns often emerge when the denominator of the fraction is 9 or a multiple of 9, or other prime numbers not equal to 2 or 5.
Why Repetition Occurs
The phenomenon of repeating decimals stems from the long division process. When you divide the numerator by the denominator, you perform a series of subtractions and bring-downs, generating remainders. If, at any point, a remainder repeats, the sequence of digits in the quotient will also repeat indefinitely. This is bound to happen if the denominator has prime factors other than 2 or 5, because these prime factors cannot be "cancelled out" to make the remainder zero in the same way powers of 10 (which are solely composed of prime factors 2 and 5) can.
Converting Repeating Decimals to Fractions (A Glimpse)
While a full explanation is complex, a common method to convert a simple repeating decimal (like 0.DDD...) to a fraction involves setting the decimal equal to a variable (e.g., x = 0.DDD...). Then, you multiply by a power of 10 to shift the decimal, allowing for subtraction to isolate the repeating part. For example, for 0.2222..., if x = 0.2222..., then 10x = 2.2222.... Subtracting the first equation from the second (10x - x = 2.2222... - 0.2222...) gives 9x = 2, so x = 2/9.
