Long Division Calculator

Dividend:

Enter the whole number under the division bar, e.g. 9876.

Divisor:

Enter the positive whole number to divide by, e.g. 37.

Decimal places:

Use 0 for whole-number division, or show up to 24 decimal digits.

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Long division turns a large division problem into a sequence of smaller place-value decisions. Each step asks how many copies of the divisor fit into the current working number, subtracts that product, and carries the leftover amount forward. The written layout is slower than a calculator answer, but it exposes the reason each quotient digit belongs in its position.

The method is useful when the remainder carries meaning. A warehouse case count, classroom group split, page batch, or serving plan may need complete groups plus leftovers, not only a rounded decimal. The same work also gives a built-in check: multiply the divisor by the quotient, add the remainder, and the result should return to the original dividend.

Dividend: The whole number being divided.
Divisor: The positive whole number used to make equal groups.
Quotient: The number of complete groups found.
Remainder: The leftover amount after the complete groups are removed.

Long division place-value cycle showing bring down, choose quotient digit, subtract, and carry remainder.

Division answers can be written in several correct forms. Quotient with remainder fits whole-number grouping. A reduced fraction keeps the exact ratio. A decimal helps with comparison, but it can end, repeat forever, or stop only because someone chose a display limit.

The easy mistake is reading a displayed decimal as a rounded final value. Long division appends decimal digits by continuing the remainder. When the same non-zero remainder appears again, the following digits repeat; when the work stops before a zero remainder or repeat appears, the shown decimal is a truncated continuation.

How to Use This Tool:

Enter the whole-number division problem, then use the answer check to confirm the exact quotient-and-remainder form before relying on the decimal display.

Type the Dividend, the whole number being divided. Commas, spaces, and underscores are ignored; after cleanup, the value must be a non-negative whole number with no more than 48 digits.
Type the Divisor, the number used for the grouping. It must be a positive whole number, so 0, signs, letters, and decimal points are rejected.
Set Decimal places from 0 to 24. Use 0 when the remainder form is enough, or choose a positive value to continue the remainder into decimal digits.
Read the summary answer first. It shows the remainder form, the decimal continuation, and badges for exact division, remainder division, selected decimal length, and detected repeat cycles.
Open Division Steps when you need the written work. Each row shows the working number, quotient digit, product, remainder, and note for that step.
Open Answer Check to confirm the Whole-number quotient, Remainder, Reduced fraction, Verification equation, and Decimal behavior.
Use Remainder Trail when you want to see whether the remainder falls to 0, cycles, or is still changing at the selected decimal limit. Fix any validation alert before copying a value or using an exported table.

Interpreting Results:

The most important check is the recomposition equation. If Verification equation multiplies the divisor by the Whole-number quotient, adds the Remainder, and returns the original dividend, the whole-number division is internally consistent.

Exact means the whole-number remainder is 0.
Remainder means the leftover amount is greater than or equal to 0 and less than the divisor.
Terminating decimal means decimal continuation reached a zero remainder within the selected places.
Repeating decimal means a non-zero remainder appeared again, so the decimal digits cycle.
Decimal continued means the selected decimal-place limit was reached before a zero remainder or repeat was shown.

Do not treat Decimal quotient alone as the exact answer when the decimal repeats or continues. Use Reduced fraction or the quotient-and-remainder form when the precise value matters, and use Remainder Trail to confirm whether the apparent pattern actually repeats within the selected limit.

Technical Details:

Whole-number division decomposes one non-negative integer into complete groups of a positive divisor plus a remainder. The quotient counts the complete groups. The remainder records what is left after those groups are removed, and it must be smaller than the divisor or another complete group would still fit.

Long division finds the quotient from left to right. Each place-value step brings down a digit, chooses the largest quotient digit whose product does not exceed the working number, subtracts that product, and carries the new remainder forward. Decimal continuation repeats the same rule after the whole-number places are finished by appending zeros to the remainder.

Formula Core:

The quotient and remainder satisfy this identity for a dividend N, divisor D, whole-number quotient Q, and remainder R:

\begin{array}{lll} N & = & D \times Q + R \\ 0 & \leq & R < D \end{array}

For 9876 ÷ 37, the quotient-and-remainder answer is 266 R 34. The check is 37 x 266 + 34 = 9,876, and 34 < 37, so the remainder is within the required range.

Decimal Continuation:

After the whole-number quotient is found, each decimal digit is produced by multiplying the current remainder by 10, choosing the next digit, and carrying a new remainder:

R_{i} \times 10 = D \times d_{i} + R_{i + 1}

A zero remainder ends the decimal. A repeated non-zero remainder begins a recurring cycle because the same remainder will produce the same next digit sequence again. If the selected decimal-place limit stops first, the displayed decimal is truncated to that many places rather than rounded, so the exact value is still better represented by the fraction or remainder form.

Long division result fields and checks
Result field	Meaning	Best check
Whole-number quotient	The quotient before decimal continuation.	Use it with the divisor and remainder in the recomposition equation.
Remainder	The leftover value after whole-number division.	Confirm `0 <= R < D`.
Decimal quotient	The quotient continued to the selected number of decimal places.	Check Decimal behavior before treating it as final.
Reduced fraction	The exact fraction after dividing numerator and denominator by their greatest common divisor.	Use it when a repeating or unfinished decimal should not be rounded.
Remainder Trail	A plotted sequence of remainders from the whole-number and decimal steps.	Look for a drop to zero, a repeated remainder, or a non-zero value at the display limit.

Long division input boundaries
Input	Accepted value	Rejected cases
Dividend	A non-negative whole number up to `48` digits after commas, spaces, and underscores are removed.	Missing values, signs, decimal points, letters, and longer values.
Divisor	A positive whole number up to `48` digits after cleanup.	`0`, missing values, signs, decimal points, letters, and longer values.
Decimal places	A whole number from `0` through `24`.	Negative values, fractions, non-numeric values, and values above `24`.

Advanced Tips:

Use Decimal places 0 for whole-number grouping tasks, then increase it only when a decimal comparison is actually needed.
When Decimal behavior reports a repeat, prefer Reduced fraction or the repeating notation in Answer Check instead of copying only the truncated decimal.
Check Remainder Trail after increasing the decimal limit. A longer trail can reveal a repeat that was not visible at a shorter limit.
For very large numbers, keep each cleaned input within the 48-digit limit so the written step table remains usable.
Use Verification equation as the final arithmetic check when transferring the result into homework, notes, or another calculation.

Worked Examples:

Repeating Decimal:

With Dividend 9876, Divisor 37, and Decimal places 6, the summary gives 266 R 34 and Decimal quotient 266.918918. Verification equation is 37 x 266 + 34 = 9,876. Decimal behavior marks a repeating cycle, so the decimal digits should not be read as a rounded six-place answer.

Terminating Decimal:

For Dividend 1250, Divisor 16, and Decimal places 3, the whole-number answer is 78 R 2. Decimal continuation reaches Decimal quotient 78.125, and Decimal behavior reports that the decimal terminates within the selected digits.

Divisor Larger Than Dividend:

Set Dividend to 1, Divisor to 7, and Decimal places to 6. The Whole-number quotient is 0, the Remainder is 1, and Decimal quotient is 0.142857. The repeated remainder explains why the decimal cycles instead of ending.

Fixing an Entry Error:

A Divisor of 0 shows a validation alert because the divisor must be greater than zero. A Decimal places value such as 30 is also rejected because the allowed range is 0 to 24. Correct the highlighted input, then check Verification equation again.

FAQ:

Can I enter commas or spaces?

Yes. Commas, spaces, and underscores are removed before the number is read. The cleaned Dividend and Divisor still must contain only digits and stay within the 48-digit limit.

Does the decimal quotient round the answer?

No. Decimal quotient is continued to the selected number of places. Repeating or unfinished decimals are truncated to that display length, while terminating decimals may show zeros after the remainder reaches 0.

Why can the answer have a remainder and a decimal?

The remainder form records whole-number division. Decimal continuation starts from that remainder, multiplies it by 10, and keeps choosing more quotient digits until it reaches the selected limit, a zero remainder, or a repeat.

Why did I get an input error?

Common causes are a blank value, a negative sign, a decimal point, a letter, a Divisor of 0, more than 48 cleaned digits, or a Decimal places value outside 0 through 24.

Are my numbers sent away for calculation?

The division, step table, remainder chart data, and exports are calculated in the browser session from the numbers you enter. There is no separate server calculation path for the arithmetic.

Glossary:

Dividend: The whole number being divided.
Divisor: The positive whole number used to make equal groups.
Quotient: The number of complete divisor groups found.
Remainder: The leftover value after subtracting all complete divisor groups.
Terminating decimal: A decimal continuation that reaches a zero remainder.
Repeating decimal: A decimal continuation where a non-zero remainder returns and the digit sequence cycles.

References:

Theory of divisors, Encyclopaedia Britannica.
Euclidean algorithm, Encyclopaedia Britannica, Mar. 10, 2026.
Repeating Decimal, Wolfram MathWorld.
Multiplication and division, Khan Academy.