What "Exact" Means
A conversion factor is exact if it is fixed by an international agreement, a definitional standard, or a mathematical identity. For example, one inch is defined as exactly 2.54 centimeters. This is not a measurement of the inch — it is the definition. Because the definition fixes the ratio exactly, the conversion factor is exact to every decimal place, forever.
What "Approximate" Means
A factor is approximate if it depends on a measurement. Even the best measurements have finite precision. For example, the density of mercury at 0 °C is a measurable quantity with a certain uncertainty. A conversion factor that uses this density (like the inch of mercury pressure unit) is therefore approximate.
Exact Factors (Partial List)
- 1 inch = 2.54 cm (1959 international yard and pound agreement)
- 1 foot = 0.3048 m
- 1 yard = 0.9144 m
- 1 international mile = 1.609344 km
- 1 nautical mile = 1852 m
- 1 pound (avoirdupois) = 0.45359237 kg
- 1 ounce = 28.349523125 g
- 1 US liquid gallon = 3.785411784 L
- 1 imperial gallon = 4.54609 L
- 1 calorie (thermochemical) = 4.184 J
- 1 kilowatt-hour = 3,600,000 J
- 1 bar = 100,000 Pa
- 1 standard atmosphere = 101,325 Pa
- 1 electronvolt = 1.602176634 × 10⁻¹⁹ J (fixed by the 2019 SI redefinition)
- Speed of light in vacuum = 299,792,458 m/s (definitional)
- Planck constant = 6.62607015 × 10⁻³⁴ J·s (definitional)
Approximate Factors
- Inch of mercury at 0 °C = 3386.389 Pa (depends on conventional mercury density)
- Some gas-flow conversions that depend on pressure and temperature
- Ingredient densities for cooking (flour, sugar, etc.) — vary by brand, humidity, and packing
- The parsec (derived from au and the definition of the arcsecond, which involves π and tangent)
- Any conversion involving a measured gravitational constant, molar masses of specific compounds, or material properties
Mathematically Exact But Not Decimal-Exact
Some factors are exact in a mathematical sense but cannot be written as a finite decimal. Examples:
- 1 degree = π/180 radians. π is irrational, so no decimal representation is complete.
- 1 radian = 180/π degrees.
- 1 knot = 1852/3600 m/s exactly, which equals 0.5144̄ m/s (repeating 4).
In these cases, the underlying relationship is exact, but any decimal display is a rounded approximation. The precision depends on the display, not the underlying relationship.
Why This Matters
Engineering and science
When you multiply two exact quantities, the result is exact. When you multiply an exact quantity by an approximate one, the result inherits the approximation. If you are working with a measurement uncertainty budget, you need to know which inputs are exact and which are not.
Cooking
A cup of flour weighs about 125 grams, but the actual value can range from 110 to 150 grams depending on how the flour is packed. The "exact" answer for a specific cup of flour depends on the specific cup. This is why professional bakers weigh ingredients rather than using volume.
Everyday life
For most everyday conversions, the difference between exact and approximate does not matter. If you are converting miles to kilometers for a road trip, the difference between "1.609344" and "1.60934" is not meaningful. But knowing that the mile is exact and the inch of mercury is approximate is useful when the stakes are higher.
How M-Convert Labels This
Each converter page includes an accuracy note that states whether the underlying factor is exact or approximate, and if approximate, where the approximation comes from. The engine itself uses the exact factor when available.
Summary
- Exact: Fixed by definition or international agreement. Unlimited precision in principle.
- Approximate: Based on a measured value. Precision depends on the measurement.
- Mathematically exact but decimal-irrational: Exact in principle, rounded when displayed.
If you are unsure which category a factor falls into, check the source page on M-Convert or look at the SI Brochure or NIST SP 811 directly.