Convert a decimal with a stated repeating block into an exact reduced fraction. The calculation trail makes the reported exact fraction easier to reproduce.
Exact fraction form is useful when a recurring display such as 0.1666… must enter later algebra without premature rounding.
Enter digits only in the two decimal fields. A repeating block is required; leading zeros inside that block are meaningful. If the Repeating Decimal to Fraction assumptions do not fit, consider reduce the result.
A repeating decimal is rational because shifting it by a power of ten aligns a second copy of its repeating tail; subtraction then removes the infinite part.
A correct exact fraction is reliable for Repeating Decimal to Fraction only when its setup matches the relationship.
For 0.1(6), let x = 0.1666…. Then 100x − 10x = 16.666… − 1.666… = 15, so 90x = 15 and x = 1/6. This Repeating Decimal to Fraction example can be compared with decimal expansion.
Rework Repeating Decimal to Fraction without copying Exact fraction. Begin with Whole-number part and preserve Nonrepeating decimal digits. Predict the scale of the Repeating Decimal to Fraction answer, then compare it with Exact fraction. Store a changed Repeating block as another Repeating Decimal to Fraction case.
Build a numerator by subtracting the nonrepeating prefix number from the number formed by prefix plus one repeat. Use one 9 per repeating digit and one 0 per nonrepeating digit in the denominator, then reduce.
For Repeating Decimal to Fraction, a quick boundary test can be made by choosing a simple value for Whole-number part while holding Nonrepeating decimal digits fixed. Work out the expected direction of Exact fraction first; the calculator should follow that direction unless the formula reaches a defined limit or changes branch.
Yes, including leading zeros.
It converts exactly to 1.
It can be written with repeating zeros, but the ordinary decimal-to-fraction converter is clearer.
They mark exactly which digits repeat indefinitely.
Yes; enter them in the separate prefix field.