A decimal is another place-value form of a rational number: positions to the right of the point represent tenths, hundredths, and smaller powers of ten.
Use it to rewrite the same quantity in a form suited to estimation, comparison, or calculation.
The relationship
Write the rule before calculating
Long division tracks remainders to detect a cycle.
See the structure
What the calculation is doing
0 . 3 7 5ones · tenths · hundredths · thousandthsLong division tracks remainders to detect a cycle.
Worked interpretation
Read the result in context
1/6 = 0.1(6), while 3/8 = 0.375.
Interpret with care
Important boundary
A displayed finite decimal can be an approximation; repeating expansions need their repeating pattern preserved.
Use the Mathematics collection to move between connected concepts without duplicating the calculation.
Understand the subject
Why do some decimals terminate and others repeat?
Long division of integers either reaches remainder zero or eventually repeats a remainder. A repeated remainder creates a repeating decimal cycle.
Use the expansion to distinguish exact terminating decimals from recurring forms and to inspect the repeating block.
The relationship
The defining equation
remainderₖ₊₁ = 10·remainderₖ mod denominator
There are only finitely many possible nonzero remainders, so one must eventually repeat unless the process terminates.
A terminating reduced fraction has no prime denominator factors other than 2 and 5. Display limits may truncate a very long cycle without changing the exact fraction.
Quick guide
How to use this calculator
Enter values using the notation shown beside each field.
Choose a calculation mode when the tool offers more than one interpretation.
Read the primary answer first, then inspect the supporting details.
Calculation method
Apply the stated definition
Long division tracks remainders to detect a cycle.
The calculator validates the mathematical domain before returning a result and does not replace undefined states with zero.
Worked example
Worked example
1/6 = 0.1(6), while 3/8 = 0.375.
Long division tracks remainders to detect a cycle.
Supported inputs
Precision and limits
Finite, bounded input
Integers, finite decimals, fractions, and mixed numbers are evaluated with exact rational arithmetic wherever the operation permits.
Notation
Enter fractions as a/b and mixed numbers as w n/d. A denominator can never be zero.
