๐ Repeating Decimal to Fraction Converter
Convert repeating decimals into exact fractions with step-by-step mathematical explanation and educational examples.
๐ Convert Repeating Decimal to Fraction
Enter a repeating decimal to convert it to an exact fraction:
๐ข Understanding Repeating Decimals
Repeating decimals occur when a fraction cannot be expressed exactly in decimal form. They represent rational numbers that have non-terminating decimal expansions with repeating patterns.
๐ Input Formats
๐ Three Dots Format
0.333...
0.142857...
0.090909...
Simple and intuitive
๐ถ Parentheses Format
0.3(3)
0.142857(142857)
0.09(09)
Mathematically precise
โก Mixed Format
0.16(6)
0.08(3)
Non-repeating + repeating
Most common in textbooks
๐ข Terminating Decimals
0.25
0.5
0.75
Also supported for completeness
๐งฎ Common Repeating Decimals
๐ก Mathematical Insight: Every repeating decimal represents a rational number (fraction). Terminating decimals are also rational numbers. Only irrational numbers like ฯ or โ2 have non-repeating, non-terminating decimal expansions.
๐ Conversion Method
๐ Pure Repeating
0.333... = 3/9 = 1/3
0.666... = 6/9 = 2/3
Repeating digit(s) over 9's
Simple pattern
๐ Mixed Decimal
0.16(6) = 166/990 = 83/495
Multiply by 10^(total digits)
Subtract non-repeating equation
More complex calculation
โ๏ธ Fraction Simplification
Always reduce to lowest terms
Find GCD of numerator/denominator
Divide both by the GCD
Ensure simplest form
๐ฏ Verification
Divide fraction to check decimal
Confirm repeating pattern
Validate mathematical accuracy
Ensure correctness
๐ Educational Examples
๐ง Why Repeating Decimals Exist
๐ข Prime Denominators
Primes other than 2 and 5
Cause repeating decimals
2 and 5 are only non-repeating primes
Fundamental mathematical property
โก Division Process
Long division with remainders
Same remainder repeats
Pattern emerges
Never terminates
๐จ Pattern Length
Depends on denominator
7 has 6-digit pattern
11 has 2-digit pattern
17 has 8-digit pattern
Varies by prime factors
๐ Mathematical Beauty
Patterns in number theory
Connection to fractions
Link between rationals and decimals
Elegant mathematical structure
๐ฌ Advanced Concepts
๐ Terminating vs Repeating
Terminating: 1/2 = 0.5
Repeating: 1/3 = 0.333...
Depends on denominator factors
Only 2 and 5 cause terminating
๐ Recurring Patterns
Single digit: 1/3 = 0.333...
Multiple digits: 1/7 = 0.142857...
Complex patterns exist
Can be predicted mathematically
โ๏ธ Rational Numbers
All fractions are rational
All repeating decimals are rational
All terminating decimals are rational
Irrational numbers don't repeat
๐ฏ Real Number Types
Rational: fractions, repeating/terminating
Irrational: ฯ, e, โ2, non-repeating
Transcendental: e, ฯ (special irrationals)
Algebraic: โ2 (solvable equations)