A scientific function is a promise about a map: which inputs are admitted, which output is produced, and how units, signs, and scales transform. The same algebraic symbols can describe different physical questions if those promises change.
A function is more than a formula¶
Newtonian gravity gives a familiar scalar magnitude,
Its domain excludes zero separation; its output has force units; doubling a mass doubles the force; increasing separation weakens it as an inverse square. The vector force additionally needs a direction convention. A correct scalar magnitude with the wrong direction is not a correct force model.
The Stefan–Boltzmann relation
maps stellar radius and effective temperature to bolometric luminosity under specific physical assumptions. The fourth power makes temperature scale more strongly than radius locally, but the equation alone does not make an atmosphere model or predict a band-limited flux.
Units expose structure¶
Units are not decoration added after a calculation. They reject impossible operations, reveal missing scale factors, and distinguish a coordinate from a dimensionless ratio. Nondimensionalization introduces reference scales so the numerical problem operates on order-of-magnitude values near unity while the physical interpretation remains explicit.
A quantity may be dimensionless without being meaningless. An angle in radians, a relative residual, and a probability are dimensionless for different reasons and should retain different semantic names.
Scales guide numerical choices¶
Before computing, estimate the order of magnitude of every term. Ask whether subtraction will cancel nearly equal values, whether a denominator can approach zero, and whether one variable dominates because of physics or merely because of its chosen units. Scaling can improve conditioning; it cannot repair an incorrect model.
Predict¶
For a gravity or luminosity calculation, write the expected units, sign, monotonic direction, limiting behavior, and dominant scale before evaluating the formula.
Compute¶
Choose an explicit unit system, convert once at a clear boundary, and compute with named variables. Preserve the distinction between physical inputs and any nondimensional coordinates supplied to a numerical kernel.
Audit¶
Check dimensional consistency, one analytic ratio, one limiting case, and one independent evaluation. For tabulated or iterative code, also inspect status and coverage rather than accepting a finite value alone.
State the warranted claim¶
“The implementation preserves this relation and its unit contract on the tested domain” is warranted by algebraic and numerical checks. “The relation adequately describes this star” requires observational and model evidence that this page does not provide.
Misconception check¶
Changing units changes the numerical value, not the physical quantity. Making variables order unity can improve a computation, but it does not add missing physics or make an ill-posed inverse problem identifiable.
Continue to What is a model? or the numerical Quantities, units, and dimensional boundaries module page.