Long Division Calculator
Divide any two numbers and see the bus-stop work, step-by-step explanation, and decimal/fraction/mixed forms — including repeating-decimal detection.
Division inputs
Updates as you typeLong division work
Step-by-step
- Enter a dividend and divisor to see each step.
Tip — hover any step to highlight it in the bus-stop above.
Formula
- Dividend
- The number being split into equal parts
- Divisor
- The size of each part (or how many parts)
- Quotient
- How many times the divisor fits into the dividend
- Remainder
- What is left over when the division is not exact
- Dividend ÷ Divisor = —
- Integer quotient = —
- Remainder = Dividend − (Divisor × Quotient) = —
- Decimal expansion = —
- Check: Divisor × Quotient + Remainder = —
Repeating decimals happen when the same remainder appears twice during the division — from that point the digits cycle forever. Terminating decimals happen when (after reducing the fraction) the divisor's only prime factors are 2 and 5. Inputs with decimals are scaled to integers internally so the arithmetic stays exact.
Examples
How It Works
To go beyond an integer answer, append a decimal point and start bringing down implicit zeros. Each new digit is computed exactly the same way. This is also where repeating decimals come from: if you ever see the same remainder appear twice during the decimal phase, the digits between those two occurrences will repeat forever. For example, dividing 1 by 3 immediately produces remainder 1 again, so the digit 3 repeats — that is 0.3.
Any division can be written in four equivalent forms: as a decimal (possibly repeating), as an exact fraction in lowest terms, as a mixed number (whole part plus fractional part), or as a quotient with remainder ("23 R 5"). The R-form is the natural output of long division itself; the others are different ways of expressing the same exact value.
Tips & Best Practices
Frequently Asked Questions
What is long division?
Long division is a step-by-step algorithm for dividing one number by another by hand. It breaks the work into the same four-step loop — divide, multiply, subtract, bring down — until either the dividend runs out or the remainder cycles. The result is a quotient (and a remainder, if any), shown in the classic 'bus-stop' layout with the divisor outside the bracket and the quotient written on top.
What is a repeating decimal, and how do I write one?
A repeating decimal is a decimal expansion that has a block of digits which goes on forever. It happens when, during long division, the same remainder appears twice — from that point on the same quotient digits are produced in the same order. The standard notation is to draw a bar (vinculum) over the repeating block: 1/3 is 0.3 and 1/7 is 0.142857.
When does a decimal terminate?
After reducing the fraction to lowest terms, the decimal terminates exactly when the denominator's only prime factors are 2 and 5. So 1/8 = 0.125 terminates (8 = 2³), 7/20 = 0.35 terminates (20 = 2² × 5), but 1/6 = 0.16 does not, because 6 contains the prime factor 3.
How does this calculator handle decimal inputs?
The calculator scales both the dividend and divisor by the same power of 10 to convert them into integers, then runs long division on the integers and re-interprets the answer. This avoids floating-point rounding errors entirely. So 12.5 ÷ 0.4 is computed as 125 ÷ 4, which is exactly 31.25.
What's the difference between quotient + remainder and decimal?
They are two answers to two different questions. 'Quotient + remainder' answers 'how many full divisor-sized pieces fit, and what's left over?' — useful for problems involving discrete things (boxes, people, hours). The decimal (or fraction) answers 'what is the exact value of dividend ÷ divisor?' — useful for measurements and continuous quantities. 23 R 5 and 23.5 are both correct for 235 ÷ 10.
Can I do long division with negative numbers?
Yes — the magnitudes are divided as usual and the sign is determined by the rule for signed division: the result is negative if exactly one of the inputs is negative, and positive otherwise. The calculator handles this for you and shows a leading minus sign when appropriate.