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Exact numbers

Exact numbers refers to the relationship between an actual value and how that value is represented. A number is represented exactly when that representation has no difference between the representation and the value. It is exact when there is no error in the representation.

Integer numbers (-54, 0, 157, 1,943,661) are examples of exact numbers. Some, but not all, rational numbers (-2/10=-0.2, 968/1000 = 0.968, etc.) can be represented exactly as, indeed, integers are rational numbers. Some rational numbers cannot be represented exactly. Consider 1/3 = ~0.333 which is inexact and has an inexact error of ~0.000333.

Irrational numbers ( $π$ , e, etc.), however, can never be represented exactly.

Representation of exactness

Numbers represented without a decimal point are integers and can be presumed to be exact. Exact infinite representation of rational numbers can be ?lexical? represented with an overline over the repeated sequence of digits such as 1/3 = 0.3, 1/17 = 0.0588235294117647.

Numbers that might otherwise appear to be inexact, such as Planck's constant, which is defined to have the exact value h = 6.62607015×10⁻³⁴ J⋅s can be exactly represented as h = 6.626070150×10⁻³⁴ indicating that Plank's constant is followed by an infinite string of zeros.

This overline notation cannot be parsed in ASCII or Unicode expressions.

References to Exact number

Impedance of free space exactly
Approximation error exact value
Planck constant exact value
Vacuum permittivity exact number
Avogadro constant exactly
Gas constant exact value
Elementary charge value is exactly
Speed of light exact value
Electronvolt exact value
Coulomb constant exact numeric value
Moving sofa problem exact value

Bounded Floating Point

Bounded Floating Point is a method for representation of real numbers that includes information about the accuracy of the representation.^[1] The data structure for bounded floating point includes the international standard data structure and interpretation^[2] but adds an additional field that contains information about the error between the infinitely accurate real value represented and the value actually represented by the value of the floating point representation.^[3] This additional field contains a sub-field whose value is the number of bits of the of the floating-point representation that are insignificant and a sub-field whose value is the accumulated rounding error in fractions of units in the last place (ulps). Bounded floating point accommodates both cancellation and rounding error.^[4]

References

^ 10540143 B2 US patent US 10540143 B2, Alan A. Jorgensen, "Apparatus for Calculating and Retaining a Bound on Error during Floating Point Operations and Methods Thereof"
^ "ISO/IEC/IEEE 60559:2011 — Information technology — Microprocessor Systems — Floating-Point arithmetic". www.iso.org. Retrieved 4 April 2018.
^ US Appends a field to standard floating point providing loss of significant bits and accumulated rounding error., Jorgensen, Alan A., "Apparatus for Calculating and Retaining a Bound on Error During Floating Point Operations and Methods Thereof", issued 2020-01-17
^ Jorgensen, Alan A.; Masters, Connie R.; Guha, Ratan K.; Masters, Andrew C. (2021). "Bounded Floating Point: Identifying and Revealing Floating-Point Error" (PDF). Advances in Science, Technology and Engineering Systems Journal. 6 (1). ASTESJ: 519–531. doi:10.25046/aj060157. ISSN 2415-6698. Retrieved 2021-10-19.

[1] 10540143 B2 US patent US 10540143 B2, Alan A. Jorgensen, "Apparatus for Calculating and Retaining a Bound on Error during Floating Point Operations and Methods Thereof"

[2] "ISO/IEC/IEEE 60559:2011 — Information technology — Microprocessor Systems — Floating-Point arithmetic". www.iso.org. Retrieved 4 April 2018.

[3] US Appends a field to standard floating point providing loss of significant bits and accumulated rounding error., Jorgensen, Alan A., "Apparatus for Calculating and Retaining a Bound on Error During Floating Point Operations and Methods Thereof", issued 2020-01-17

[ASTESJ1-4] Jorgensen, Alan A.; Masters, Connie R.; Guha, Ratan K.; Masters, Andrew C. (2021). "Bounded Floating Point: Identifying and Revealing Floating-Point Error" (PDF). Advances in Science, Technology and Engineering Systems Journal. 6 (1). ASTESJ: 519–531. doi:10.25046/aj060157. ISSN 2415-6698. Retrieved 2021-10-19.

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