Exact fraction and mixed-number form
The fraction has a signed integer numerator and a positive denominator. The common divisor shows how the initial fraction was reduced. A negative mixed number means the negative of the complete whole-plus-fraction amount, not a negative whole part plus a positive fraction.
To add, subtract, multiply, or divide the resulting fractions, use the Fraction Calculator .
The formula
For a terminating decimal, N is the number with its decimal point removed and m is the number of decimal digits. For repeating input, P contains the whole and non-repeating digits, m counts the non-repeating decimal places, R is the repeating block, and r counts its digits, including leading zeros. Apply the sign after constructing the magnitude.
A non-repeating digit before the recurring part
For 0.1(6), P = 1, m = 1, R = 6, and r = 1. The initial fraction is (1 × 9 + 6)/(10 × 9) = 15/90. Dividing by 15 gives 1/6. In comparison, the finite value 0.16 is 16/100 = 4/25.
How to use this calculator
- Type one decimal using a period as the decimal separator.
- If a final digit block repeats forever, enclose that block in parentheses; do not use an ellipsis.
- Calculate and open the integer construction steps to check the denominator and reduction.
A rounded decimal carries different information
A displayed 0.33 is exactly 33/100 under terminating notation. It does not tell the calculator that the intended value was one third. Writing 0.(3) supplies that missing recurring pattern. Converting a measured or rounded value produces the fraction for the characters entered; it cannot recover the original quantity or measurement uncertainty.
Leading zeros in the recurring block matter
The number of positions in a block determines the power of ten, even when the block begins with zero. For example, 0.(09) gives 9/99 = 1/11. Removing the initial zero and entering 0.(9) would instead give 1. The subtraction construction cancels the infinite tail; no fixed number of repeated digits is approximated.
Whole values and negative mixed numbers
Integer input is accepted and returned over denominator 1. Decimal trailing zeros reduce normally, so 2.500 and 2.5 have the same simplified fraction. For -2.375 the improper fraction is -19/8 and the mixed form is -2 3/8, meaning -(2 + 3/8). Use the signed numerator when entering the value into further arithmetic.
Assumptions & limitations
What this calculation assumes
- Parenthesized digits form one final block that repeats forever.
- Typed decimal digits are exact, rather than inferred from a floating-point approximation.
What to keep in mind
- Accepts up to 64 digits in one value. Scientific notation, commas, slashes, percent symbols, and ellipses are rejected.
- No nearest-denominator approximation, irrational-number conversion, or uncertainty estimate.
Common questions
How do I enter 0.1666…?
Enter 0.1(6). The 1 is non-repeating and the 6 repeats forever.
Why does 0.(9) become 1/1?
The repeating decimal equals 9/9 exactly. This is different from any finite string such as 0.999.
Can very large integers lose digits?
The fraction construction and reduction use BigInt integer arithmetic. Values within the digit limit retain their exact digits rather than passing through floating-point numbers.