Fraction to Decimal Converter | Repeating & Terminating

After simplifying the fraction to lowest terms using the GCD, check the prime factorisation of the denominator:

• • If the denominator only has 2s and 5s as prime factors, the decimal terminates, e.g., 8 = 2³, 20 = 2²×5, 40 = 2³×5 all terminate.
• • If the denominator has any other prime factor (3, 7, 11, 13…), the decimal repeats, e.g., 6 = 2×3, 15 = 3×5, 7 are all repeating.

This rule comes directly from the structure of base 10 (= 2×5): only fractions whose denominators divide a power of 10 will terminate in base 10.

The period length is the number of digits in the repeating block. For example:

• • 1/3 = 0.(3), period 1 (just the digit 3 repeats forever)
• • 1/7 = 0.(142857), period 6 (the block “142857” repeats)
• • 1/11 = 0.(09), period 2 (the block “09” repeats)

For any fraction 1/p where p is prime and p ≠ 2, 5, the period divides p−1 (by Fermat's little theorem). The period of 1/7 is 6 = 7−1, which is the maximum possible.

For terminating decimals, the process is exact and straightforward:

• • Count the number of decimal places k
• • Write the decimal as an integer over 10ᵏ, e.g., 0.625 → 625/1000
• • Simplify by dividing both by GCD, GCD(625, 1000) = 125 → 5/8

For repeating decimals the algebraic method is used, let x equal the decimal, multiply by 10^(period length), subtract to eliminate the repeating tail, then simplify the resulting integer fraction.

Yes, every rational number (fraction of two integers) can be expressed as either a terminating or eventually repeating decimal. This is a theorem in number theory.

• • Terminating: finite decimal representation, e.g., 3/8 = 0.375
• • Repeating: eventually periodic, e.g., 1/3 = 0.333… The period always ends because there are only finitely many possible remainders (0 through d−1), so by the Pigeonhole Principle a remainder must repeat within d steps.

Irrational numbers (like √2 or π) cannot be expressed as fractions at all, and their decimal expansions are infinite and non-repeating.

They are equal. 0.(9) = 1 exactly, not approximately, but identically equal. Here's the proof:

• • Let x = 0.999… → 10x = 9.999… → 10x − x = 9 → 9x = 9 → x = 1
• • Alternatively: 1/3 = 0.(3), so 3 × 1/3 = 3 × 0.(3) = 0.(9) = 1

This is a rigorous mathematical fact, not a paradox. The decimal 0.999… represents the limit of the infinite geometric series 9/10 + 9/100 + 9/1000 + … = 9/10 × 1/(1 − 1/10) = 1.

Use the Mixed Number → Decimal tab, or do it manually:

• • Convert the mixed number to an improper fraction: 3¼ = (3×4+1)/4 = 13/4
• • Divide: 13 ÷ 4 = 3.25
• • The whole part (3) becomes the integer part of the decimal, and ¼ = 0.25 is added

For negative mixed numbers like −2½: the whole part carries the sign, so −2½ = −(2×2+1)/2 = −5/2 = −2.5. Enter a negative value in the whole-number field.

Computers use IEEE 754 binary floating-point. The decimal 0.1 (= 1/10) cannot be represented exactly in binary because the denominator 10 = 2×5 contains the factor 5, and 5 is not a power of 2.

• • In binary, 0.1 = 0.0(0011), a repeating binary fraction, stored as an approximation
• • The rounding errors from 0.1 and 0.2 do not cancel out when added, so the result differs from the stored approximation of 0.3
• • In JavaScript: 0.1 + 0.2 === 0.30000000000000004

This is why financial software uses fixed-point integer arithmetic (storing cents as integers, not dollars as floats) to avoid rounding surprises.

For a fraction n/d in lowest terms, the decimal eventually repeats with a period of at most d − 1 digits.

• • This follows from the Pigeonhole Principle: long division tracks remainders in {0, 1, …, d−1}, so after at most d steps a remainder must reappear
• • For prime denominators p, the maximum period is exactly p−1 (these are called full reptend primes, e.g., 7, 17, 19, 23)
• • 1/97 has period 96, the longest possible for that denominator

Before the decimal repeats there can also be a non-repeating “pre-period” if the denominator has factors of 2 or 5 mixed with other primes, e.g., 1/6 = 0.1(6) where the 1 is the non-repeating part.