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Mathematics · pattern A, Numeric fields

Quotient and remainder, with the check.

Enter a dividend and divisor for the whole-number quotient, the remainder, the decimal result and the multiplication that verifies it.

Inputs

The formula used

dividend = divisor × quotient + remainder

The remainder is always smaller than the divisor. That identity is the check on any division and the definition the algorithm rests on.

Quotient

198

Remainder
13
Decimal quotient
198.56521739
Check: divisor × quotient + remainder
4,567
Remainder as a fraction
13/23
Divides exactly?
no
Digits in the quotient
3

The identity is the proof.

Any integer division can be written as dividend = divisor × quotient + remainder with the remainder smaller than the divisor. Checking that identity catches every slip in a long division, and it is the same statement underlying modular arithmetic and Euclid's algorithm.

Questions about long division

Why learn long division?
Because it generalises to polynomial division, which is needed throughout algebra, and it makes the structure of division visible.
What about negatives?
Conventions differ on the sign of the remainder. Mathematics takes it non-negative; many languages do not.
How does this relate to modulo?
The remainder is the modulo result for positive operands. Modular arithmetic keeps only the remainder.

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Arithmetic runs in double-precision floating point, so results beyond about fifteen significant figures are not exact. Where a question wants an exact fraction or surd, keep the exact form rather than a decimal.