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Physical quantities as data

A number becomes useful in physics only when its meaning is attached. A value of \(3.2\) might represent a length, a time, a temperature, or a count. To calculate safely, preserve the unit, the quantity being measured, and enough context to interpret the observation.

Quantity, unit, and measurement record

A physical quantity has a magnitude and a unit. For example, \(1.20\,\mathrm{m}\) is a length. The metre tells us how the numerical magnitude relates to an agreed standard. The dimension is length, written \(L\), which is independent of whether the unit is metres, centimetres, or kilometres.

A measurement record contains more information than the quantity alone. If a ruler reading is reported as

\[\ell=(1.20\pm0.01)\,\mathrm{m}\]

then \(1.20\,\mathrm{m}\) is the stated value and \(0.01\,\mathrm{m}\) is an uncertainty under some declared interpretation. The plus-minus notation alone does not say whether this is a standard uncertainty, an instrument resolution, or a coverage interval. A careful data record names that interpretation.

The record may also need the instrument, calibration version, timestamp, environmental conditions, sample identity, coordinate frame, and operator or data source. A room temperature without time and location may not describe the same state as a later reading. A vector position without an origin and basis is incomplete even when its components have units.

A data shape that preserves meaning

The exact schema depends on the project, but the separation between quantity and observation is broadly useful.

from dataclasses import dataclass
from typing import Generic, Literal, TypeVar

Unit = TypeVar("Unit", bound=str)
LengthUnit = Literal["m", "cm", "km"]


@dataclass(frozen=True)
class Quantity(Generic[Unit]):
    value: float
    unit: Unit


@dataclass(frozen=True)
class Measurement(Generic[Unit]):
    quantity: Quantity[Unit]
    standard_uncertainty: Quantity[Unit] | None
    method: str
    source: str
    observed_at: str
    conditions: str | None = None

The optional uncertainty is not an invitation to assume zero when it is missing. "Unknown" and "exact" are different states. Defined conversion factors and constants may be exact by definition, while an ordinary measured input is never made exact by omitting metadata.

Before constructing a record, validate finite values and supported units. Domain rules belong to the measured quantity: a signed coordinate may be negative, while a mass in a classical introductory model ordinarily must not be negative.

from math import isfinite

raw_value = 125.0
raw_unit = "cm"
uncertainty = 0.5
assert isfinite(raw_value)
assert uncertainty >= 0
assert raw_unit in {"m", "cm", "km"}
observation = Measurement(
    quantity=Quantity(raw_value, raw_unit),
    standard_uncertainty=Quantity(uncertainty, raw_unit),
    method="ruler reading",
    source="illustrative measurement, not experimental data",
    observed_at="2026-09-06",
)
print(observation)

Normalize without erasing the original

Programs often convert all values to a canonical unit before calculation. For length, metres are a convenient SI choice. Converting

\(35.0\,\mathrm{cm}\)

uses an exact metric-prefix relationship:

\[35.0\,\mathrm{cm}\left(\frac{1\,\mathrm{m}}{100\,\mathrm{cm}}\right)=0.350\,\mathrm{m}\]

The conversion changes the stored magnitude and unit together. It does not improve the measurement precision. Retaining the original observation and recording the conversion lets a reviewer distinguish source data from derived data.

def length_in_metres(
    length: Quantity[LengthUnit],
) -> Quantity[LengthUnit]:
    metres_per_unit = {
        "m": 1.0,
        "cm": 0.01,
        "km": 1000.0,
    }

    return Quantity(
        value=length.value * metres_per_unit[length.unit],
        unit="m",
    )

An exact conversion factor can still be implemented incorrectly, so useful tests include round trips and known identities. Converting one kilometre must give one thousand metres. Converting to metres and back should recover the original value within the rounding tolerance of the representation.

Compatibility is more than matching text

Two quantities may be added only if their dimensions are compatible. Values in metres and centimetres are compatible after conversion; a length and a duration are not. Multiplication and division form derived dimensions. For example,

\[(5.0\,\mathrm{m/s})(4.0\,\mathrm{s})=20\,\mathrm{m}\]

because the time dimension cancels.

Dimensional compatibility is necessary but not sufficient for semantic compatibility. Two positions may both use metres but refer to different coordinate origins. A distance and a displacement both have dimension length but are not interchangeable in every equation. Torque and energy share the SI dimensions of force times length, yet they represent different physical quantities and use different unit conventions. A unit system prevents some category errors; it cannot infer the intended meaning of the data.

Preserve uncertainty through transformations

Formatting a measurement to fewer displayed digits should not mutate its stored value or uncertainty. Likewise, converting units must convert uncertainty by the same scale factor. If a length and its standard uncertainty are converted from centimetres to metres, both numerical values are multiplied by \(0.01\).

Derived quantities need an explicit uncertainty rule. If two independent length measurements are added, their variances may be added under common assumptions. If the measurements share a calibration error, treating them as independent understates the uncertainty. The data model must therefore be able to represent correlations or declare that the simplified propagation rule ignores them.

Raw samples should not be overwritten by rounded summaries. Store the observations, the calculation version, and identifiers for the inputs used in each derived value. This creates a provenance chain: a surprising result can be traced back to source records instead of being an orphaned decimal in a chart.

Display is not storage

A user interface might show the same stored length as \(1.2\,\mathrm{m}\), \(120\,\mathrm{cm}\), or a localized string. Those are presentations of one quantity. Changing display units should not change the canonical value used by later calculations.

Conversely, attaching "m" to a rendered number does not repair a value whose unit was discarded earlier. By the time a chart receives a primitive array, incompatible values may already have been combined. Unit checks are most useful at input and operation boundaries, not only at output.

Checks for a trustworthy record

A small quantity system should make its promises executable:

These checks verify data handling, not the experiment. A perfectly structured record can still contain a biased observation, a misidentified sample, or an incorrect calibration. Scientific review must examine measurement design as well as software.

Treating physical quantities as structured data makes assumptions visible at the point where code can enforce them. The structure does not replace judgment, but it prevents a bare number from silently acquiring whatever meaning a later function happens to need.

A worked Python notebook

Keep a ruler's original reading in centimetres and produce a separate value in metres. The question for this calculation is precise: how does length change when we change ruler reading, while keeping the other choices fixed? The earlier sections explain the physical relationship. Here we turn one part of that relationship into a calculation that you can run, inspect, and change in Google Colab.

Conversion preserves measurement uncertainty; it does not create more accurate data. Keep that sentence beside the results. A formula can return a number outside its useful range, and a plotting library will usually draw that number without asking whether the physical assumptions still hold. The selected interval is from 10 to 200 cm. It gives us a specific domain to discuss instead of leaving the graph's range to a software default.

Use the notebook link above the lesson, run the setup cell, and then run the cells in order. The code below is also included there. NumPy handles the arrays of numerical values, SymPy keeps the equation symbolic, and Matplotlib draws the results. If these libraries or notebook cells are unfamiliar, start with Your first physics notebook.

State the relationship before choosing a graph

In this small example, q stands for ruler reading, expressed in cm. The output is length, expressed in m. The equation below uses the numerical values in those units. Any fixed coefficients therefore carry the units needed to make the relation consistent. A value supplied in another unit must be converted before it enters this expression.

\[f(q) = \frac{q}{100}\]

This is a function of one changing input. It is not a statement that the physical system has only one variable. We deliberately hold the other inputs fixed so that one cause of variation is visible. Changing several inputs at once would make it harder to tell which change produced the output difference. Write down the held-fixed choices before trying a different experiment.

At 125 cm, the expression gives 1.25 m. This is our reference calculation. It is a prediction of the stated example, not a new measurement. Its displayed digits help us compare implementations. They should not be read as a claim that physical inputs are known to six significant figures.

import numpy as np
import sympy as sp
import matplotlib.pyplot as plt

q = sp.Symbol("q", real=True)
expression = sp.sympify("q/100", locals={"q": q})
reference_input = 125.0
reference_output = float(expression.subs(q, reference_input))
print("Input:", reference_input, "cm")
print("Predicted length:", reference_output, "m")

sp.Symbol declares a mathematical symbol. It does not assign a numerical measurement to that symbol. subs then replaces the symbol with our chosen input, and float converts the resulting symbolic number into a standard Python numerical value. Keeping these steps separate lets us inspect the equation before numerical substitution. It also prevents a later plot from becoming the only surviving record of the model.

Compare a few cases you can explain

Before drawing hundreds of points, calculate three. The lower endpoint gives 0.1 m. The midpoint, at 105 cm, gives 1.05 m. The upper endpoint gives 2 m. Read these together with the assumptions: they describe different choices of ruler reading within the same example, with the other inputs held fixed.

These three values are a useful first check, but they do not establish the shape between them. Two endpoints can be equal even when the curve bends or oscillates in between. Likewise, a large change between endpoints does not tell us whether the relationship is linear. We need the equation and intermediate samples to answer those questions.

evaluate = sp.lambdify(q, expression, modules="numpy")
inputs = np.array([10.0, 105.0, 200.0])
outputs = np.asarray(evaluate(inputs), dtype=float)
outputs = np.broadcast_to(outputs, inputs.shape)
for input_value, output_value in zip(inputs, outputs):
    print(f"{input_value:.6g} cm -> {output_value:.6g} m")
assert np.all(np.isfinite(outputs))

lambdify turns the symbolic expression into a numerical function. We explicitly select NumPy as its numerical backend so it can accept an array. The same expression now has two useful forms: one for algebra and one for calculating many cases. The assertion checks that the chosen inputs produce finite outputs. It would catch an infinity or an undefined numerical result, but a finite value can still be physically inappropriate.

Make a graph that answers the question

The horizontal axis shows ruler reading; the vertical axis shows length. Every plotted point is a prediction of the same expression. We mark the reference case separately so that a reader can connect the earlier printed value to its place on the curve. The axis labels include units because a shape without a scale cannot tell us the size of a physical effect.

input_grid = np.linspace(10.0, 200.0, 301)
output_grid = np.asarray(evaluate(input_grid), dtype=float)
output_grid = np.broadcast_to(output_grid, input_grid.shape)
fig, ax = plt.subplots(figsize=(7, 4))
ax.plot(input_grid, output_grid, label="Stated model")
ax.scatter([reference_input], [reference_output], label="Reference case", zorder=3)
ax.set_xlabel("Ruler reading (cm)")
ax.set_ylabel("Length (m)")
ax.legend()
ax.grid(alpha=0.25)
fig.tight_layout()
plt.show()

Look for a sign change, a turning point, or a region where the curve becomes steep. Connect each feature to the expression rather than to the colour of the line. If the curve has a minimum or maximum, changing the input in opposite directions may have similar effects on the output near that point. If it is nearly straight over a short interval, a local linear approximation may be useful even when the complete expression is nonlinear.

Increasing the number of samples makes the line smoother and may reveal features that a coarse grid missed. It does not improve the physical inputs or enlarge the model's validity range. For quantities that can take only discrete values, use the line as a guide to the relationship and interpret only the allowed input values. A plotting function does not enforce such restrictions for you.

Measure the local sensitivity

A derivative asks a more specific question than the whole graph: how quickly does length change near the reference input? Its units are the output unit divided by the input unit. For this calculation, the derivative at 125 cm is 0.01 in those ratio units. Locally, the output increases as the input increases. This statement is local; the sign or size of the slope can be different elsewhere on the curve.

\[f'(q) = \frac{1}{100}\]

For a small input change, multiply the derivative by that change to estimate the output change. Then compare that estimate with a fresh evaluation of the full expression. The distinction between these two calculations matters: the derivative gives a tangent approximation, while substitution evaluates the chosen expression at the new point.

slope_expression = sp.diff(expression, q)
slope_at_reference = float(slope_expression.subs(q, reference_input))
changed_input = 125.95
linear_prediction = reference_output + slope_at_reference * (
    changed_input - reference_input
)
full_prediction = float(evaluate(changed_input))
print("Symbolic derivative:", slope_expression)
print("Tangent prediction:", linear_prediction, "m")
print("Full expression:", full_prediction, "m")
print("Approximation difference:", full_prediction - linear_prediction, "m")

Here the changed input is 125.95 cm. The tangent estimate is 1.2595 m, while the full expression gives 1.2595 m. If the two values are close, it means the chosen input change is small enough for this local approximation at this point. It does not mean the entire model is linear. If the slope is zero, a quadratic or higher-order term may supply the leading change instead.

This is also the beginning of uncertainty propagation. If the input has a small stated standard uncertainty, the magnitude of the slope times that uncertainty estimates one contribution to output uncertainty. Do not automatically use the same rule for a wide interval, a discontinuity, a turning point with zero first derivative, or several correlated uncertain inputs. Those situations need more of the model than one slope.

Check the derivative with numerical differences

We can estimate the same slope from values on either side of the reference point. Subtract the lower output from the upper output and divide by twice the input step. Parentheses are essential: the entire output difference must be divided by the entire input interval. Without them, Python can execute a different mathematical expression while still producing a plausible number.

The calculation below uses three decreasing step sizes. Each estimate is compared with the symbolic derivative, so the reference is independent of the numerical difference formula. Near an endpoint, it is acceptable here to evaluate a small extension only because the displayed mathematical expression is smooth there; such a check does not assert that extrapolated values describe additional physical cases.

input_span = 190.0
for fraction in [1e-2, 1e-3, 1e-4]:
    step = input_span * fraction
    upper = float(evaluate(reference_input + step))
    lower = float(evaluate(reference_input - step))
    numerical_slope = (upper - lower) / (2 * step)
    difference = numerical_slope - slope_at_reference
    print(f"step={step:.6g}: slope={numerical_slope:.6g}, difference={difference:.6g}")

For a smooth function, reducing the step often reduces the approximation error at first. Very small steps can eventually make subtraction of nearly equal floating-point values less reliable. Therefore “smaller” is not an unlimited guarantee of “better.” Compare the observed behaviour over a useful range of steps instead of choosing the smallest number the computer can represent.

An absolute slope difference is useful when the expected slope is near zero. A relative difference would divide by a very small reference and could look enormous even when the absolute discrepancy is tiny. Away from zero, a relative difference can help compare quantities with very different scales. The tolerance should reflect the question and the numerical method, rather than a universal decimal threshold copied into every test.

Change the example and preserve the reasoning

Make one copy of the reference calculation before editing it. Choose a second value of ruler reading inside the stated interval, predict whether length will rise or fall, and then evaluate the expression. Place the prediction and the result in a text cell. If they disagree, identify whether the mistake was in the sign, the unit conversion, the algebra, or the assumption about how the curve behaves.

Next choose a shorter plotting interval around the reference point. Compare the local curve with its tangent approximation. Then widen the interval again. The difference shows why a useful local estimate need not describe the whole range. Keep the axis labels and numerical scale visible when comparing the plots; automatic axis limits can make two different effects appear equally large.

Finally, return to the physical limit stated at the start: Conversion preserves measurement uncertainty; it does not create more accurate data. Describe one change to the setup that would violate that limit. State what new input, equation, or observation would be needed before trusting an answer in the changed situation. This last step connects the Python experiment to the physics instead of treating successful execution as the only criterion for a useful result.

Before saving, restart the runtime and run all cells in order. A result that depends on an earlier, now-deleted input cell is not reproducible from the saved notebook. Keep unrounded values for later calculations, round only the reported output, and label any new data as measured, assumed, or generated. The graph and printed values should remain traceable to the same expression and the same choices of inputs.

Recover the output change by integrating the slope

There is another way to check that the expression and its derivative describe the same relationship. Add up the small output changes predicted by the slope across the chosen input interval. In the continuous limit, this integral equals the output at the upper endpoint minus the output at the lower endpoint. The result has units of m: the input unit in the integration cancels the input unit in the denominator of the slope.

This statement is about the mathematics of this particular relationship. It does not assign a new physical meaning to an arbitrary area under a graph. For example, an area under force versus distance represents work, while an area under a different pair of quantities may represent something else or have no standard name. Always carry the axis units through the calculation before naming an integral.

slope_function = sp.lambdify(q, slope_expression, modules="numpy")
endpoint_change = float(evaluate(200.0) - evaluate(10.0))
for points in [101, 201, 401]:
    grid = np.linspace(10.0, 200.0, points)
    slopes = np.asarray(slope_function(grid), dtype=float)
    slopes = np.broadcast_to(slopes, grid.shape)
    integrated_change = np.trapezoid(slopes, grid)
    print(
        points,
        "integrated change:",
        integrated_change,
        "endpoint change:",
        endpoint_change,
    )

The trapezoidal rule joins neighbouring slope samples with straight segments and adds their signed areas. A negative contribution remains negative; replacing every slope by its magnitude would calculate a different quantity. Increasing the sample count provides a practical refinement check. If the values do not settle toward the endpoint difference, first inspect the interval, singularities, and array ordering.

Do not conclude that two agreeing calculations prove the model is physically complete. Both use the same selected relationship, so an omitted effect would be omitted from both. Their agreement is evidence that the numerical tools implement that relationship consistently. Comparing with an independent experiment or a more appropriate physical model is a separate test.

Keep this comparison in the notebook when you change the expression. It is particularly useful after adding a coefficient, changing a sign convention, or converting an input unit. A code cell that produces a smooth graph can conceal these mistakes; a second calculation of the same output change gives you another place to notice them.

Sources

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