Final answer:
Any repeating decimal can be converted into a fraction because of the base-10 number system, where the repeated sequence can be captured as a geometric series and expressed as a fraction with a denominator of 9's matching the number of repeating digits. The original repeating decimal 0.333... is equivalent to the fraction 1/3.
Step-by-step explanation:
The reason that any decimal that repeats some pattern of digits forever can be converted into a fraction is because of the base-10 number system. This is not due to mathematical coincidence, prime factorization, or a characteristic of irrational numbers. Instead, it's a property of how we represent numbers in the decimal system. When a pattern of digits repeats indefinitely in a decimal, we can express that repeating part as the numerator over a denominator that contains the same number of 9's as there are digits repeating. This method works because it essentially captures the repeating sequence as a geometric series, which corresponds to a fraction.
Here is a step-by-step explanation using an example of a repeating decimal, 0.333...:
- Let x equal our repeating decimal, x = 0.333...
- Multiply both sides by a power of 10 that matches the number of repeating digits. In this case, multiply by 10 to get 10x = 3.333...
- Subtract the original number from this result: 10x - x = 3.333... - 0.333...
- This subtraction removes the repeating decimal part, leaving us with 9x = 3.
- Finally, solve for x by dividing both sides by 9, so x = 3/9, which simplifies to x = 1/3.
As a result, we can see that the original repeating decimal 0.333... is equivalent to the fraction 1/3.