Binary Fraction Grid Step
Return an exact binary fraction grid step as integer strings, including the unit step at zero fractional bits.
Description
Return an exact binary fraction grid step as integer strings, including the unit step at zero fractional bits.
A binary fixed-point format with b fractional bits has spacing 1/2^b. Exact fractions preserve that step at large widths.
When to use this tool
Return an exact binary fraction grid step as integer strings, including the unit step at zero fractional bits.
- Prepare the supplied fractionalBits fields for this operation.
- Compare the complete example with the representation used by your own input before carrying the result into another step.
How to use the tool and how it works
- Enter fractionalBits. Use the exact text syntax, units and index conventions explained below.
- Run the tool. If input validation fails, correct the supplied fields before interpreting any result.
- Read result.numerator, result.denominator. Interpret false, null and empty lists using the distinctions below; they can be valid outputs.
Require fractionalBits from 0 to 128. Return numerator string one and denominator string 2^fractionalBits.1
| Input field | Meaning and representation | Accepted input |
|---|---|---|
| fractionalBits | fractional bits. | Integer from 0 to 128 inclusive |
| Output field | Interpretation |
|---|---|
| result | Object containing the named outputs below. |
| result.numerator | Reduced signed numerator as decimal integer text. |
| result.denominator | Positive denominator as decimal integer text, under the normalization or common-denominator convention above. |
{
"input": {
"fractionalBits": 3
},
"output": {
"result": {
"numerator": "1",
"denominator": "8"
}
}
}Limitations and assumptions
- The step alone does not define signedness, total bits, exponent range or rounding behavior.
- Exact decimal-spelling operations require digits before the optional decimal point and allow a signed exponent. Decimal text is bounded to 256 code units and effective scale to minus 256 through 256; integer-string fields allow a sign and up to 256 digits.
- Every declared top-level field is required and unexpected fields are rejected. Numeric inputs must be finite; ordinary text fields are bounded to 8,192 UTF-16 code units, with tighter operation-specific limits where stated.
- Lists are limited to 128 entries. JSON text must meet bounded size and depth checks, and numeric outputs beyond the safe finite magnitude are rejected. Exact large integers use decimal strings.
- Indexes are zero-based and spans have an exclusive end unless explicitly stated otherwise. A valid false, null or empty-list result is distinct from malformed input.
Alternative or Complementary methods
Preserve decimal and integer strings when exact comparisons matter. Select signedness, scale, rounding and overflow policies separately, and use the matching encoder/decoder rather than a floating-point approximation.
References
Similar or alternative tools
- continued fraction
Find the finite simple continued-fraction coefficients of a rational number by Euclidean division, including correct floor behavior for negative numerators.
- Decimal Rounding Error Fraction
Exact decimal quantum rounding error with ties toward positive infinity; signed arithmetic uses floor rather than truncation.