Description
Return exact ceiling quotient and remainder=numerator-quotient*denominator as decimal strings.
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 exact ceiling quotient and remainder=numerator-quotient*denominator as decimal strings.
When to use this tool
Use Integer Ceiling Division Sign Case when you need this exact stated integer sign, bound, division, comparison, or factor component.
How to use the tool and how it works
- Enter numerator, denominator. Use the units and conventions below.
- Run the tool. Correct any format, range, or cross-field input error before interpreting the output.
- Read quotient, remainder. Use the output convention below.
Parse signed decimal integers and a nonzero denominator. Truncate division first and increment the quotient only for a nonintegral positive ratio. Return exact ceiling quotient and remainder=numerator-quotient*denominator as decimal strings.1
| Input field | Meaning and units | Accepted input |
|---|---|---|
| numerator | Numerator in the numeric or exact integer-text 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. |
| denominator | Denominator or exact integer denominator text under the stated sign restriction. | Required. Text; maximum UTF-16 length 8192. Integer syntax: optional + or - and 1-128 decimal digits; leading zeros accepted. Apply the sign restriction above. |
| Output field | Interpretation | Returned shape |
|---|---|---|
| quotient | Integer quotient under the stated division rule. | Required. Text; maximum UTF-16 length 8192. |
| remainder | Numerator minus ceiling quotient times denominator; not Euclidean remainder. | Required. Text; maximum UTF-16 length 8192. |
{
"input": {
"numerator": "7",
"denominator": "3"
},
"output": {
"quotient": "3",
"remainder": "-2"
}
}Limitations and assumptions
- For nonzero remainder its sign is opposite the denominator, unlike a Euclidean remainder. The tool returns ceiling-division bookkeeping rather than a nonnegative modulo result or a rounded fractional value.
- 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
- Rational Floor Ceiling Gap
Use exact integer division, lower the quotient by one for a nonintegral negative ratio, and return floor and ceiling as decimal text plus gap zero for integral ratios or one otherwise.
- Integer Euclidean Quotient Sign Case
Choose remainder in [0,abs(divisor)), then return quotient=(dividend-remainder)/divisor and remainder as exact decimal text.