SI units as data types

Units play a role similar to types: they constrain which operations have physical meaning. The values \(3\,\mathrm{m}\) and \(3\,\mathrm{s}\) contain the same numerical magnitude, but one is a length and the other is a duration. A program that stores both as unlabelled numbers cannot distinguish them.

The SI foundation

The International System of Units, abbreviated SI, organizes measurements around seven base quantities and base units. Modern SI definitions fix exact numerical values for seven defining constants, providing reproducible ways to realize the units.

Base quantity	SI base unit	Symbol	Dimension symbol
time	second	s	\(T\)
length	metre	m	\(L\)
mass	kilogram	kg	\(M\)
electric current	ampere	A	\(I\)
thermodynamic temperature	kelvin	K	\(\Theta\)
amount of substance	mole	mol	\(N\)
luminous intensity	candela	cd	\(J\)

The dimension symbols are bookkeeping notation, not unit symbols. The metre and kilometre are different units for the same length dimension. The second is both an SI base unit and a unit representing the time dimension.

Most quantities are derived from the base quantities. Velocity has SI unit metres per second and dimension \(LT^{-1}\). Force has the derived SI unit newton:

\[1\,\mathrm{N}=1\,\mathrm{kg\,m/s^2}\]

Energy has the derived SI unit joule:

\[1\,\mathrm{J}=1\,\mathrm{N\,m}=1\,\mathrm{kg\,m^2/s^2}\]

Named derived units improve communication, while their base-unit forms make dimensions available to a checker.

Represent dimensions as exponent vectors

For mechanics, a dimension can be represented by exponents of mass, length, and time. Velocity is \(M^0L^1T^{-1}\) and force is \(M^1L^1T^{-2}\).

from dataclasses import dataclass


@dataclass(frozen=True)
class MechanicalDimension:
    mass: int
    length: int
    time: int


length = MechanicalDimension(mass=0, length=1, time=0)
velocity = MechanicalDimension(mass=0, length=1, time=-1)

Multiplication adds exponents, division subtracts them, and raising a quantity to a power multiplies every exponent by that power.

def multiply_dimensions(left, right):
    return MechanicalDimension(
        mass=left.mass + right.mass,
        length=left.length + right.length,
        time=left.time + right.time,
    )


def divide_dimensions(numerator, denominator):
    return MechanicalDimension(
        mass=numerator.mass - denominator.mass,
        length=numerator.length - denominator.length,
        time=numerator.time - denominator.time,
    )


duration = MechanicalDimension(0, 0, 1)
assert multiply_dimensions(velocity, duration) == length
assert divide_dimensions(length, duration) == velocity

Multiplying velocity by time produces length because

\[(LT^{-1})(T)=L\]

This representation scales to all seven SI base dimensions. Use fixed fields or a fixed-order tuple so absent exponents cannot be confused with missing data.

Addition needs compatible dimensions and units

Addition is stricter than multiplication. Two lengths may be added, but their numerical values must first be expressed in compatible units:

\[3.00\,\mathrm{m}+25.0\,\mathrm{cm}=3.00\,\mathrm{m}+0.250\,\mathrm{m}=3.25\,\mathrm{m}\]

Adding the raw numbers to obtain \(28\) has no defined unit and does not represent the physical sum. A unit-aware addition function should either convert one operand by a known conversion rule or require callers to normalize units first.

from dataclasses import dataclass
from typing import Literal


@dataclass(frozen=True)
class Length:
    value: float
    unit: Literal["m", "cm"]


def metres(length: Length) -> float:
    if length.unit == "m":
        return length.value

    return length.value / 100


def add_lengths(left: Length, right: Length) -> Length:
    return Length(
        value=metres(left) + metres(right),
        unit="m",
    )

This small union deliberately supports only two length units. Rejecting an unsupported unit is safer than guessing a conversion.

Static types and runtime data solve different problems

Python type checkers can distinguish wrappers such as Length and Duration during development. At runtime, however, JSON, form fields, and files arrive as untrusted data. A type annotation is erased and does not validate the value. The input boundary must check the unit string, numerical finiteness, and any domain constraints before constructing the trusted type.

from math import isfinite

record = {"value": 125.0, "unit": "cm"}
if not isinstance(record["value"], (int, float)):
    raise ValueError("The value must be a number.")
if not isfinite(record["value"]):
    raise ValueError("The value must be finite.")
if record["unit"] not in {"m", "cm"}:
    raise ValueError("Use metres or centimetres.")
trusted_length = Length(record["value"], record["unit"])
print(f"Length: {metres(trusted_length):.2f} m")

Branded types can prevent accidental mixing inside a program, while runtime unit records support dynamic calculations. A robust system often needs both.

Units with offsets need special care

Many conversions use a scale factor, but temperature scales show why a generic value * factor function is incomplete. Converting a temperature from degrees Celsius to kelvins uses an offset:

\[T_{\mathrm{K}}=T_{^\circ\mathrm{C}}+273.15\]

A temperature difference is different: a change of

\(1\,^{\circ}\mathrm{C}\)

has the same magnitude as a change of \(1\,\mathrm{K}\) and should not receive the absolute-temperature offset. Libraries must distinguish absolute temperatures from temperature intervals.

Logarithmic units, calendar durations, and context-dependent units introduce other rules. Unit conversion is an operation with semantics, not a string replacement.

Dimensionless does not mean meaningless

Some quantities have all base-dimension exponents equal to zero. Ratios of two lengths, strain, refractive index, and many coefficients are dimensionless. Angles measured in radians are also dimensionless in SI because a radian is a ratio of arc length to radius.

Still, programmers may want distinct semantic types for angle, strain, and a plain count. Passing a dimensionless strain into a trigonometric function would satisfy a dimensional checker but usually violate intent. Retaining a quantity kind above the dimension layer catches more mistakes.

The same warning applies to energy and torque. Both reduce to \(ML^2T^{-2}\), but they are not interchangeable concepts. SI convention writes energy in joules and torque in newton metres to preserve that distinction.

What unit types can and cannot prove

A correct unit system can reject adding time to length, detect a missing power of time, and automate exact prefix conversions. It can help verify that both sides of an equation have the same dimensions. Those are powerful, inexpensive checks.

It cannot tell whether two positions use the same origin, whether a measured mass belongs to the intended object, whether a force law applies in the chosen regime, or whether the uncertainty is adequate. It also cannot prove an equation correct: many false equations are dimensionally consistent.

Use units as one layer of the data contract. Preserve physical meaning, provenance, and model assumptions in the layers above it. The goal is not to make illegal states impossible in every conceivable sense; it is to prevent unit information from being silently discarded before the calculation needs it.

Preserve precision through conversion

An exact unit conversion does not make a measured value more precise. Converting \(2.4\,\mathrm{m}\) to

\(240\,\mathrm{cm}\)

changes the unit and displayed magnitude but preserves the information in the original measurement. Extra trailing digits should not appear merely because the program stores binary64 values.

Uncertainty must be converted with the quantity. If a length is \((2.40\pm0.05)\,\mathrm{m}\), the corresponding centimetre record is \((240\pm5)\,\mathrm{cm}\). Converting only the central value leaves internally inconsistent data.

Tests should therefore check more than the nominal magnitude. Verify the output unit, the converted uncertainty, round-trip behavior, and the treatment of exact conversion constants. This turns the unit type from a decorative label into an enforced data boundary.

A worked Python notebook

A speed stored as 3 metres per second is multiplied by a duration stored in seconds. The question for this calculation is precise: how does distance change when we change duration, 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.

Only compatible time units make this product a distance in metres. 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 0 to 10 s. 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 duration, expressed in s. The output is distance, 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.

\[f(q) = 3 q\]

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 2 s, the expression gives 6 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("3*q", locals={"q": q})
reference_input = 2.0
reference_output = float(expression.subs(q, reference_input))
print("Input:", reference_input, "s")
print("Predicted distance:", 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 m. The midpoint, at 5 s, gives 15 m. The upper endpoint gives 30 m. Read these together with the assumptions: they describe different choices of duration 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([0.0, 5.0, 10.0])
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} s -> {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 duration; the vertical axis shows distance. 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(0.0, 10.0, 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("Duration (s)")
ax.set_ylabel("Distance (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 distance change near the reference input? Its units are the output unit divided by the input unit. For this calculation, the derivative at 2 s is 3 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.

\[f'(q) = 3\]

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 = 2.05
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 2.05 s. The tangent estimate is 6.15 m, while the full expression gives 6.15 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 = 10.0
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 duration inside the stated interval, predict whether distance 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: Only compatible time units make this product a distance in metres. 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(10.0) - evaluate(0.0))
for points in [101, 201, 401]:
    grid = np.linspace(0.0, 10.0, 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

• OpenStax, University Physics Volume 1, Section 1.2, "Units and Standards".
• OpenStax, University Physics Volume 1, Section 1.3, "Unit Conversion".
• MIT OpenCourseWare, 8.01SC Classical Mechanics, Chapter 2, "Units, Dimensional Analysis, Problem Solving, and Estimation".
• The Physics Hypertextbook, "International System of Units".