Long division,
every step shown.

Field guide

How long division works — the complete algorithm explained.

Long division is the standard algorithm for dividing two integers when the result cannot be computed mentally in a single step. It breaks a complex division into a sequence of simpler steps : estimate, multiply, subtract, bring down, repeated until all digits of the dividend are consumed.

While calculators make the final answer instant, understanding the long-division algorithm develops number sense, estimation skill, and algebraic thinking. It is the foundation for polynomial long division and synthetic division in algebra.

The four repeating operations

Every iteration of long division cycles through four operations:

1. Divide. Estimate how many times the divisor fits into the current working number. This is the quotient digit.
2. Multiply. Multiply the quotient digit by the divisor to find what gets subtracted.
3. Subtract. Subtract the product from the working number to find the remainder.
4. Bring down. Bring the next digit of the dividend down to join the remainder, forming the new working number.

Repeat until all dividend digits have been processed. The collected quotient digits form the final answer; the last remainder is the integer remainder.

Worked example: 952 ÷ 7

Step-by-step:

1. Working number = 9. 9 ÷ 7 = 1, 1 × 7 = 7, 9 − 7 = 2.
2. Bring down 5 → working number = 25. 25 ÷ 7 = 3, 3 × 7 = 21, 25 − 21 = 4.
3. Bring down 2 → working number = 42. 42 ÷ 7 = 6, 6 × 7 = 42, 42 − 42 = 0.

Quotient digits collected: 1, 3, 6 → quotient = 136, remainder = 0. Check: 136 × 7 = 952. ✓

Remainders vs. decimal extension

When the dividend is not exactly divisible by the divisor, you have two choices for expressing the result:

• Integer quotient with remainder. Write the result as Q remainder R, for example 446 ÷ 12 = 37 r 2. This form is exact and common in discrete mathematics, programming (the % operator), and practical problems like dividing objects into groups.
• Decimal quotient. Continue the algorithm by appending zeros to the remainder (decimal phase). Each appended zero generates one more decimal digit. For 446 ÷ 12: remainder 2 → 20 ÷ 12 = 1 r 8 → 80 ÷ 12 = 6 r 8 → 80 ÷ 12 = 6 r 8 … → 37.1666… = 37.1̄6̄ (the "6" repeats).

Repeating decimals

Whenever the remainder in the decimal phase matches a remainder already seen, the decimal expansion enters a cycle. The division will never terminate, the same sequence of digits repeats indefinitely.

This is not a quirk: every rational number (any integer divided by another) has either a terminating or repeating decimal expansion. A decimal terminates only when the denominator (in lowest terms) has no prime factors other than 2 and 5 — exactly the prime factors of 10 (our number base). All other fractions repeat.

The repeating part is written with an overline bar: 1/3 = 0.3̄, 1/7 = 0.142857̄, 1/6 = 0.16̄. The length of the repeating cycle for 1/n is at most n − 1 digits (achieved by primes like 7, which give the full 6-digit cycle 142857).

The bus-stop notation

The traditional long-division layout. Also, called the bus-stop or galley method — arranges the work vertically so that place values align:

• The divisor sits to the left of a vertical bar.
• The dividend is written to the right of the bar.
• The quotient is written above the dividend, each digit directly above the last digit it was derived from.
• Each multiple is written below the current working number, aligned to the right.
• A short horizontal line separates each multiple from the new remainder.

The calculator above generates this traditional display with exact character alignment for any dividend up to 9,999,999.

Connection to the division algorithm

Long division is a concrete implementation of the Division Algorithm: for any non-negative integer a and positive integer b, there exist unique integers q and r such that:

q is the quotient and r is the remainder. This identity is the foundation of modular arithmetic, the Euclidean algorithm for GCF, and all of number theory.

Long division in algebra: polynomial division

The same "divide, multiply, subtract, bring down" procedure extends directly to dividing polynomials. To divide x³ + 2x² − 5x + 6 by x − 2, you treat each term as a "digit" and perform identical steps. The result is a polynomial quotient (plus a remainder polynomial). Understanding integer long division is therefore a prerequisite for polynomial long division and synthetic division.

Common mistakes and how to avoid them

• Forgetting to write a zero. When a working number is smaller than the divisor, the quotient digit for that step is 0. This zero must be written in the quotient; omitting it shifts all subsequent digits. Example: 1029 ÷ 3 → 343, not 34 r 3.
• Estimation errors. If you overestimate the quotient digit (multiple > working number), your subtraction will go negative. Back down by 1 and recalculate. If you underestimate (remainder ≥ divisor), increase by 1.
• Place-value misalignment. Always right-align the multiple under the working number, not under the full dividend. Misalignment is the most common source of errors in the bus-stop layout.