Integer Division Zero Remainder Sign Tool

Description

Label exact integer remainder conventions, preserving dividend-signed zero metadata for truncation.

Exact integer arithmetic and finite numeric arithmetic have different range limits. A safe integer number is bounded by +/-9007199254740991, while supported decimal-integer text can represent larger values. Division and remainder also need a signed convention: truncating quotients and Euclidean nonnegative remainders can give different results.

Label exact integer remainder conventions, preserving dividend-signed zero metadata for truncation.

When to use this tool

Use Integer Division Zero Remainder Sign when you need this exact stated integer sign, bound, division, comparison, or factor component.

How to use the tool and how it works

Enter dividend, divisor, convention. Use the units and conventions below.
Run the tool. Correct any format, range, or cross-field input error before interpreting the output.
Read remainder, zero, zeroSign. Inspect zero before interpreting the associated values.

Parse signed decimal integers and select convention truncated or euclidean. Return the corresponding exact remainder as text and zero as a Boolean. For a zero remainder return zeroSign=negative only for truncated division when the original dividend text begins with -, otherwise positive; nonzero results use nonzero.^[1]

Input field	Meaning and units	Accepted input
dividend	Signed integer dividend in the representation declared below.	Required. Text; maximum UTF-16 length 8192. Integer syntax: optional + or - and 1-128 decimal digits; leading zeros accepted. Apply the sign restriction above.
divisor	Nonzero integer divisor in the representation declared below.	Required. Text; maximum UTF-16 length 8192. Integer syntax: optional + or - and 1-128 decimal digits; leading zeros accepted. Apply the sign restriction above.
convention	Convention	Required. Text; one of "truncated", "euclidean"; maximum UTF-16 length 8192.

Output field	Interpretation	Returned shape
remainder	Remainder under the explicitly selected signed-division convention.	Required. Text; maximum UTF-16 length 8192.
zero	Zero	Required. Boolean.
zeroSign	Textual signed-zero convention based on original dividend text and selected division.	Required. Text; maximum UTF-16 length 8192.

json

{
  "input": {
    "dividend": "-6",
    "divisor": "3",
    "convention": "truncated"
  },
  "output": {
    "remainder": "0",
    "zero": true,
    "zeroSign": "negative"
  }
}

Limitations and assumptions

ZeroSign is a textual provenance label rather than an actual signed BigInt zero. Nonzero Euclidean remainders are nonnegative, while truncated remainders follow the dividend sign.
Inputs must satisfy both the field types and the cross-field conditions above. Invalid input causes an error; a successful false flag or documented placeholder is a different result.

Alternative or Complementary methods

Use exact decimal-integer tools when the range exceeds safe numeric integers. State the division, remainder, midpoint, and zero convention before combining results from different operations.

References

Calculation and input contract: number/integer-division-zero-remainder-sign — tool.io

Similar or alternative tools

Signed Remainder Convention Residual
Return truncatedRemainder=dividend%divisor using truncation toward zero, euclideanRemainder=(r+abs(divisor)) modulo abs(divisor), and residual=truncated-euclidean, all as decimal strings.
Integer Euclidean Quotient Sign Case
Choose remainder in [0,abs(divisor)), then return quotient=(dividend-remainder)/divisor and remainder as exact decimal text.