Description
Return the exact symmetric residue in [-m/2,m/2), choosing the negative representative at even-modulus ties.
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.
Return the exact symmetric residue in [-m/2,m/2), choosing the negative representative at even-modulus ties.
When to use this tool
Use Number: Modulus Symmetric Representative when you need this exact stated integer sign, bound, division, comparison, or factor component.
How to use the tool and how it works
- Enter integer, modulus. Use the units and conventions below.
- Run the tool. Correct any format, range, or cross-field input error before interpreting the output.
- Read result. Expand result to inspect the returned fields.
Parse signed integer text and a positive modulus, reduce to a nonnegative residue, then return residue-modulus when twice the residue>=modulus, otherwise residue. Return representative as exact decimal text inside result.1
| Input field | Meaning and units | Accepted input |
|---|---|---|
| integer | Safe integer number or exact signed decimal-integer text as declared below. | Required. Text. Integer syntax: optional + or - and 1-128 decimal digits; leading zeros accepted. Apply the sign restriction above. Implementation limit: 8192 Unicode code points and 16384 UTF-16 code units. |
| modulus | Modulus | Required. Text. Integer syntax: optional + or - and 1-128 decimal digits; leading zeros accepted. Apply the sign restriction above. Implementation limit: 8192 Unicode code points and 16384 UTF-16 code units. |
| Output field | Interpretation | Returned shape |
|---|---|---|
| result | Object containing the calculated result members described below. | Required. Object. |
| result.representative | Exact symmetric residue text with negative halfway tie. | Calculated result member: string. |
{
"input": {
"integer": "8",
"modulus": "10"
},
"output": {
"result": {
"representative": "-2"
}
}
}Limitations and assumptions
- For an even modulus the exact halfway residue maps to the negative representative. This fixed tie policy differs from choosing a positive midpoint and does not infer the convention required by another application.
- Outputs are nested inside result. 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
Similar or alternative tools
- Modular Signed Residue Representative
Return representative=residue-modulus when residue>modulus/2, or when exactly halfway and positiveTie=false; otherwise preserve the residue.
- 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.