How do you convert a terminating decimal to a fraction?

Write the decimal over the appropriate power of ten, then reduce. For 0.375, write 375/1000, find GCD(375, 1000) = 125, and divide both by 125 to get 3/8. The calculator uses the continued-fraction algorithm to handle all cases without rounding errors.

How do you convert a repeating decimal to a fraction?

Use the algebraic shortcut. For 0.142857142857... with a 6-digit repeat block, the numerator equals the full digit string minus the non-repeating prefix, and the denominator is 10^(non-repeating digits) times (10^(repeating digits) - 1). Here that gives 142857 / 999999, which simplifies to 1/7.

What is the difference between a terminating and a repeating decimal?

A terminating decimal has a finite number of digits after the decimal point (e.g. 0.5, 0.25, 0.0625). A repeating decimal has a block of digits that cycles forever (e.g. 0.333..., 0.142857142857...). Every rational number is either terminating or eventually repeating.

What is the continued-fraction algorithm?

It repeatedly takes the integer part (floor) of a value, subtracts it, and inverts the remainder to get the next term. The convergents produced at each step are the best rational approximations to the target value. This gives the exact simplest fraction for any terminating decimal and an excellent approximation for other reals.

What does "lowest terms" mean?

A fraction is in lowest terms (also called simplest form or fully reduced) when the numerator and denominator share no common factor other than 1. The tool always divides both by their GCD before displaying the result.

Can the calculator handle negative decimals?

Yes. The sign is detected, the conversion runs on the absolute value, and the negative sign is carried through to the final fraction. So -0.75 becomes -3/4.

Is the repeating-decimal entry format flexible?

Enter only the digits, without any dot or special characters. Put the non-repeating digits after the decimal point in the first box, and the cycling block in the second box. For 0.3(142857), enter 3 and 142857. For a pure repeat like 0.(3), leave the first box empty and enter 3.

Decimal to Fraction Calculator

A decimal to fraction calculator that handles both terminating decimals (like 0.375) and repeating decimals (like 0.142857142857…), converts them to the exact simplest fraction in lowest terms, and shows every step of the working. The result is displayed as an improper fraction, a mixed number where applicable, and a decimal round-trip verification. For denominators up to 24 a shaded bar diagram gives a visual sense of the proportion.

How it works

Terminating decimals — the continued-fraction method

For a terminating decimal the calculator uses the continued-fraction (CF) algorithm:

Separate the whole-number part from the fractional part.
Apply the CF expansion to the fractional part: at each step take a = floor(b), update the convergent numerator h and denominator k using the recurrence h(i) = a * h(i-1) + h(i-2), stop when k exceeds 1,000,000 or the remainder is exactly zero.
Reduce the result with GCD(numerator, denominator).

This is more numerically stable than the naive “multiply by a power of ten” method, because floating-point arithmetic can introduce a spurious 9 or 0 at the end of a decimal representation.

Repeating decimals — the algebraic shortcut

For a repeating decimal 0.nonRepeat(repeat) the formula is:

numerator   = (nonRepeat + repeat) as integer  −  (nonRepeat) as integer
denominator = 10^len(nonRepeat) × (10^len(repeat) − 1)

For example, 0.3(142857) gives numerator = 3142857 − 3 = 3142854 and denominator = 10 × 999999 = 9999990. Dividing both by GCD(3142854, 9999990) = 285714 yields 3142854 / 9999990 = 11/35.

A simpler example: 0.(3) gives numerator = 3 − 0 = 3, denominator = 1 × 9 = 9, which simplifies to 1/3. This is the clean algebraic proof that 0.333… = 1/3.

Worked example — terminating decimal

Convert 1.625:

Whole part = 1, fractional part = 0.625.
CF expansion of 0.625: a=0, next b = 1/(0.625−0) = 1.6, a=1, etc. — convergents approach 5/8.
Combine: 1 + 5/8 = 8/8 + 5/8 = 13/8.
GCD(13, 8) = 1, so lowest terms = 13/8 (or mixed: 1 5/8).

Worked example — repeating decimal

Convert 0.142857142857… (the repeating expansion of 1/7):

Non-repeating digits after decimal: none (leave blank)
Repeating block: 142857

Formula: numerator = 142857 − 0 = 142857, denominator = 1 × (10^6 − 1) = 999999. GCD(142857, 999999) = 142857, so 142857/999999 = 1/7.

Decimal	Fraction	Form
0.5	1/2	terminating
0.375	3/8	terminating
1.625	13/8	terminating
0.333…	1/3	repeating, block “3”
0.666…	2/3	repeating, block “6”
0.142857…	1/7	repeating, block “142857”

Formula note

The GCD used throughout is computed by Euclid’s algorithm: GCD(a, b) = GCD(b, a mod b) until b = 0, then a is the answer. The LCM (used in some fraction-arithmetic contexts) is LCM(a, b) = (a / GCD(a,b)) * b. All arithmetic stays within JavaScript’s safe integer range (up to 2^53 − 1) for any realistic decimal input.

Everything runs in your browser — no numbers are uploaded or stored anywhere.