Long Division Calculator

Perform long division with complete step-by-step working, quotient, remainder, decimal conversion, and interactive visual explanations.

Long division — divisor ) dividend

Integers up to 60 digits, positive or negative. Divisor cannot be zero.

What is the Long Division Calculator?

Long division is the school-taught procedure for dividing one whole number by another, digit by digit, while showing every subtraction and bring-down on the page. The Long Division Calculator runs the procedure for you and renders the working in classic paper-style layout — divisor on the left, dividend under the bar, quotient on top, and a stack of subtractions descending below — so you can see exactly how the answer was assembled.

Every step is captured: which digits the divisor fits into, the multiplied product written underneath, the subtraction line, and the digit brought down for the next step. The result panel reports the quotient, the remainder, the mixed-number form, the simplified improper fraction, the decimal expansion (with the repeating block isolated if applicable), and the percentage form, all in one tap.

How the Long Division Calculator works

1. Parse and validate

Both the dividend and divisor are read as exact integers (up to 60 digits) using BigInt arithmetic, so there is no rounding error even for very large numbers.

2. Run the digit loop

Iterate the dividend left-to-right. At each digit, append it to the running value, compute the next quotient digit as floor(current / divisor), and subtract.

3. Build the paper layout

Each step's product and subtraction are placed in a monospaced grid lined up under the digit being divided, so the layout matches what a student would draw on paper.

4. Show every alternate form

Convert the (quotient, remainder) pair into a mixed number, an improper fraction, a decimal expansion (with repeating block detection), and a percentage.

Five ways to use the Long Division Calculator

Homework checker

Compare your hand-worked paper to the calculator's layout to find the exact step where you made an arithmetic slip.

Decimal converter

Run any fraction through long division to get its full decimal expansion, including the repeating block.

Remainder reasoning

Decide whether a quantity divides evenly (remainder 0) or how many full groups fit before the leftover.

Mental-math reference

Walk through the four sub-steps — divide, multiply, subtract, bring down — to internalise the pattern for sketching divisions in your head.

Tutoring aid

Project the visual on a screen and step through each subtraction as a worked example for a student.

Best practices when dividing by hand

Always estimate first

Round both numbers to one significant digit and divide mentally to sanity-check your final answer's magnitude.

Underline the running value

Drawing a line under each subtraction's result prevents accidentally re-using a stale digit.

Write the decimal point as soon as you place it

Once the decimal point is in the quotient, write zeros explicitly when you bring down a zero — this keeps the columns aligned.

Verify with the identity

After finishing, recompute divisor × quotient + remainder. If it equals the original dividend, your work is correct.

Why long division still matters

Long division is the foundational algorithm for converting fractions to decimals, simplifying algebraic expressions, performing polynomial division, computing modular inverses in cryptography, and reasoning about divisibility. The procedure also generalises directly — the polynomial long-division algorithm has the same shape, as does synthetic division and even certain integer-factorisation methods.

Beyond the algorithmic value, long division builds quantitative intuition. Students who practice the algorithm learn to estimate magnitudes, spot arithmetic errors, and reason about scale — skills that transfer to budgets, scientific units, and engineering tolerances long after the explicit procedure is replaced by a calculator.

Tricky cases the calculator handles for you

Bringing down zeros

When the running value is smaller than the divisor and you must bring down a zero, the quotient gets a 0 placeholder. The calculator inserts these automatically so your final quotient has the right number of digits.

Long repeating decimals

Some fractions produce repeating blocks dozens of digits long (e.g. 1/97 has period 96). The expansion algorithm detects the repeat by tracking remainders and stops as soon as one recurs.

Negative dividends and divisors

The quotient takes the combined sign; the remainder follows the dividend's sign (school convention), while modulo always returns a non-negative residue.

Very large integers

Up to 60-digit values are supported via JavaScript BigInt, so even astronomical or cryptographic-sized numbers divide exactly.

Core division formulas

Euclidean division identity

a = b × q + r, 0 ≤ r < |b|

Foundational identity — every integer pair has a unique quotient q and remainder r satisfying this.

Long division per step

currentₙ = currentₙ₋₁ × 10 + digit

Each step appends the next dividend digit to the running value before the next division.

Verify identity

dividend = divisor × quotient + remainder

Plug your answer back into this equation to verify the division is correct.

Modulo (non-negative)

a mod b ∈ [0, b) for b > 0

The math convention; differs from the C/JavaScript % when a is negative.

Common long-division mistakes

✗ Forgetting to bring down a zero

✓ Fix — If the running value is smaller than the divisor, you still write a 0 in the quotient before bringing down the next digit.

✗ Misaligned columns under the bar

✓ Fix — Each subtraction's product must sit directly beneath the digits it was computed from — use ruled paper or graph paper to keep columns straight.

✗ Choosing the wrong quotient digit

✓ Fix — If divisor × digit overshoots the running value, decrease the digit by 1 and try again. Estimating first prevents this.

✗ Stopping too early

✓ Fix — Continue until every digit of the dividend has been brought down; otherwise you'll under-report the quotient.

How we built and tested this calculator

Every division — integer, decimal, and fractional — is computed with BigInt arithmetic, never floating point, so there is no rounding error for inputs up to 60 digits. The long-division algorithm and the repeating-decimal detector are unit-tested against canonical values (1/3, 1/7, 22/7, 100/97, 355/113) and against textbook hand-worked examples (Knuth's TAOCP §4.3.1, Hardy & Wright §9.6).

The paper-style layout is rendered as a monospaced grid with one CSS column per dividend digit; this guarantees pixel-perfect alignment of every subtraction line regardless of font, browser, or screen size. The result panel and step list are wired to scroll into view after every Calculate press, and the full report can be copied, shared via URL, or printed without losing the layout.

Frequently Asked Questions

Long division is the standard step-by-step procedure for dividing one number (the dividend) by another (the divisor). At every step you ask 'how many times does the divisor fit?', write that quotient digit on top, subtract divisor × quotient-digit from the working number, then bring down the next digit and repeat. The final answer is a quotient and a remainder, which can also be written as a mixed number, a fraction, a decimal, or a percentage.

The quotient is the whole-number answer of a division — how many full copies of the divisor fit into the dividend. In 100 ÷ 7 = 14 R 2, the quotient is 14 because seven fits into one hundred fourteen full times before the leftover (the remainder) is reached. The quotient appears on top of the long-division bar.

The remainder is the amount left over after the quotient has taken as many full divisors as possible. For 100 ÷ 7, the quotient is 14 and 14 × 7 = 98, so the remainder is 100 − 98 = 2. The Euclidean division identity says dividend = divisor × quotient + remainder, with 0 ≤ remainder < |divisor|.

Write the dividend under the bar and the divisor outside. (1) Look at the leftmost digits of the dividend until they're large enough to be divided by the divisor. (2) Divide — write the result on top. (3) Multiply that result by the divisor and write the product below. (4) Subtract to get a new working value. (5) Bring down the next digit. Repeat steps 2–5 until every digit has been brought down. The number on top is the quotient and the final difference is the remainder.

When the remainder is zero the division is exact — the divisor divides the dividend cleanly with nothing left over, and the result is a whole-number quotient. For example, 56 ÷ 7 = 8 R 0, so 56 / 7 is exactly 8. In fraction form the result is just the integer; in decimal form there are no digits after the decimal point.

Move the decimal point of the divisor right until it becomes a whole number, then shift the dividend's decimal point the same number of places. Now divide as normal long division — the decimal point of the quotient sits directly above the new position in the dividend. The Decimal Division tab automates the alignment and shows the shift visually.

To divide a/b by c/d, flip the second fraction (the reciprocal becomes d/c) and multiply: (a/b) × (d/c) = (a × d) / (b × c). Then simplify by dividing numerator and denominator by their greatest common divisor. The Fraction Division tab does both steps and also converts the answer to a mixed number and decimal.

Modulo is the mathematical name for the remainder after division, usually defined to be non-negative. For 100 mod 7 = 2, because 100 = 14 × 7 + 2. Modulo is fundamental in computer science, cryptography, hashing, calendar arithmetic, and number theory. The Remainder tab reports modulo alongside the school remainder so you see both conventions.

Use the Euclidean identity: dividend = divisor × quotient + remainder. If the equation holds, the answer is correct. For 100 ÷ 7 with quotient 14 and remainder 2, check 7 × 14 + 2 = 98 + 2 = 100 ✓. The calculator runs this verification automatically on every long-division result.

Long division is the foundation for converting fractions to decimals, simplifying algebraic expressions, polynomial division in algebra, computing modular inverses in cryptography, integer factorisation, gear-ratio analysis, accounting splits, network subnetting, scaled production runs, and any time you need to know how many full groups of one quantity fit into another.

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