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Modulo Remainders and Operation Ordering

Calculate the nonnegative residue and the retained alternative signed-remainder conventions for a nonzero divisor.

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Calculator inputs

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How to use this calculator

  1. Enter the known values in the units shown. Results update as you type.
  2. Where results are editable, change one to solve backwards. Lock a value to hold it fixed.
  3. Use the worked example to check the method. Reset restores the starting fields.

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Formula and method

For dividend −21 and divisor 4, the nonnegative residue is 3; a dividend-signed remainder is −1.

r = x − |y| × floor(x/|y|), with 0 ≤ r < |y|

Worked example

Enter these known values and leave the other values blank.

Dividend x
-21
Nonzero divisor y
4
Nonnegative Euclidean residue
3
Dividend-signed alternative remainder
-1

Assumptions and limitations

  • The divisor is nonzero. The Euclidean residue is nonnegative even when either input is negative.
  • The displayed alternatives use ceiling division for a negative dividend or floor division with the signed divisor. Programming-language remainder conventions can differ.
  • Modulo is not generally associative as a binary operation: (9 mod 4) mod 3 = 1, but 9 mod (4 mod 3) = 0. Parentheses determine the intended operation.
  • Real-valued inputs are supported by the captured floor-based expressions; modular integer arithmetic uses integer inputs.
  • Arithmetic uses 64 significant decimal digits, with separate rounding for displayed values.

Common questions

Why do programming languages disagree on a negative remainder?

They can choose different quotient rounding conventions. Floor, truncation and a nonnegative Euclidean residue need not give the same signed result.

References

The calculation equations, inverse formulas, units, and input rules were imported from this source. Bookify provides the interface and equation solver.

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