Why do some fractions recur and others do not?

Because we write in base ten, which factors into two and five. A denominator built only from those primes divides cleanly; any other prime factor leaves a remainder cycle that never closes.

Why can I not divide by zero?

Because no number multiplied by zero gives anything other than zero, so the identity that defines division has no solution. It is not a missing feature, it is that the question has no answer.

Math & Statisticsarithmetic

Long Division Calculator

Whole-number division with the remainder and the identity that checks it. The division algorithm guarantees exactly one quotient and remainder for any pair, and this shows both.

Also called: division with remainder, quotient and remainder.

Quotient

1,234 divided by 56 is 22 remainder 2. As a decimal that is 22.04, and the check is that 22 times 56 plus 2 returns 1,234.

Remainder

As a decimal

22.04

Check: quotient times divisor plus remainder

1,234

Does the decimal recur

The decimal recurs forever.

How this is calculated

Method and background

How this is calculated

Division with remainder is defined by one identity: the dividend equals the quotient times the divisor plus a remainder smaller than the divisor. That identity is shown as a check, because it is the only verification needed. Whether the decimal terminates depends entirely on the divisor: once reduced, a denominator whose only prime factors are two and five terminates, and anything else recurs forever.

dividend = quotient * divisor + remainder, with the remainder smaller than the divisor

a: Dividend
b: Divisor
q: Quotient
r: Remainder

Worked examples

Each of these is asserted on every build. If a change to the engine ever moved one of these answers, the build would fail before the page could print it.

a division with a remainder

Dividend: 1,234
Divisor: 56

Quotient22

The identity that defines division, checked directly

Open this example

a divisor of only twos and fives terminates

Dividend: 1
Divisor: 8

Quotient0

boundary: eight is two cubed

Open this example

Method and limits

What it assumes

Integer division for the quotient and remainder; the decimal is shown separately.

What it deliberately does not model

The recurring block itself is not printed, only whether one exists.

Formula version 1.0.0 · definition 1.0.0 · United States · Report a problem with this calculator

Frequently asked questions

Why do some fractions recur and others do not?: Because we write in base ten, which factors into two and five. A denominator built only from those primes divides cleanly; any other prime factor leaves a remainder cycle that never closes.
Why can I not divide by zero?: Because no number multiplied by zero gives anything other than zero, so the identity that defines division has no solution. It is not a missing feature, it is that the question has no answer.