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Units, precision, and floating-point errors

A program can execute exactly as written and still produce a physically meaningless answer. The inputs may use incompatible units, the reported digits may exceed the measurement precision, or finite binary arithmetic may distort an otherwise correct formula. These are different problems and need different checks.

A physical quantity is more than a number

A physical quantity combines a numerical value with a unit. The statements \(5\,\mathrm{m}\) and \(5\,\mathrm{s}\) contain the same number but describe different kinds of quantity. Adding them has no physical meaning.

Dimensions describe the kind of quantity independently of a particular unit. Length has dimension \(L\) and time has dimension \(T\). Velocity has dimension \(LT^{-1}\) and may be reported in metres per second, kilometres per hour, or another compatible unit.

Unit conversion changes the numerical value and unit together while preserving the physical quantity:

\[72\,\mathrm{km/h}=72\,\frac{\mathrm{km}}{\mathrm{h}}\left(\frac{1000\,\mathrm{m}}{1\,\mathrm{km}}\right)\left(\frac{1\,\mathrm{h}}{3600\,\mathrm{s}}\right)=20\,\mathrm{m/s}\]

The conversion factors equal one, so they change the representation rather than the speed. A program that relabels \(72\) as metres per second without changing its value describes a different speed.

def kilometres_per_hour_to_metres_per_second(
    speed_kilometres_per_hour: float,
) -> float:
    metres_per_kilometre = 1000
    seconds_per_hour = 3600

    speed_metres_per_second = (
        speed_kilometres_per_hour * metres_per_kilometre / seconds_per_hour
    )

    return speed_metres_per_second

Descriptive names reduce mistakes, but runtime unit objects or static quantity types can provide stronger protection. Whatever representation is used, convert at an explicit boundary instead of scattering unexplained factors through formulas.

Accuracy, precision, and uncertainty

These words are related but not interchangeable. Accuracy describes agreement with a reference or accepted value. Precision describes the resolution of a measurement or the repeatability of results. Uncertainty quantifies a range associated with incomplete knowledge of a measured value.

A scale that always reads \(0.50\,\mathrm{kg}\) too high may give tightly grouped, precise measurements that are inaccurate because of systematic bias. Repeating the measurement can estimate random spread, but it does not automatically remove calibration error.

Significant figures communicate the precision justified by measured inputs. They do not create accuracy, and they are not a substitute for an explicit uncertainty analysis. OpenStax recommends retaining extra digits during intermediate calculations and rounding only the final result. The Physics Hypertextbook gives the same warning: rounding a partial result can make later arithmetic less accurate.

Suppose a measured length is reported as \(12.3\,\mathrm{cm}\). Writing a derived answer as

\(0.123000000\,\mathrm{m}\)

does not mean the original instrument resolved nine decimal places in metres. Unit conversion preserves the measurement's information; it does not add digits of knowledge.

Floating-point numbers are finite

Python 3's built-in float normally uses IEEE 754 binary64 arithmetic. Many simple decimal fractions have no finite binary expansion, just as one third has no finite decimal expansion. The stored value is the nearest available binary number under the active rounding rule.

total = 0.1 + 0.2

if total == 0.3:
    raise ArithmeticError("this branch is not reached in binary64")

In Python 3 the displayed result is approximately \(0.30000000000000004\). This is not evidence that arithmetic laws are false. It is the correctly rounded result of operations on binary approximations to the decimal inputs.

Not every expression involving an inexact fraction visibly differs from its expected decimal result. In Python 3, (1 / 3) * 3 rounds to exactly 1.0. It is therefore a poor demonstration of visible floating-point error. Tests must check the actual operation and range that matter instead of assuming every inexact input produces an inexact-looking output.

Compare with an appropriate tolerance

For computed real-valued quantities, exact equality is often the wrong test. A useful comparison combines an absolute tolerance near zero with a relative tolerance for larger magnitudes:

\[|a-b|\leq \varepsilon_{\mathrm{abs}}+\varepsilon_{\mathrm{rel}}\max(|a|,|b|)\] 
def nearly_equal(
    left: float,
    right: float,
    absolute_tolerance: float,
    relative_tolerance: float,
) -> bool:
    difference = abs(left - right)
    scale = max(abs(left), abs(right))

    return difference <= (absolute_tolerance + relative_tolerance * scale)

The tolerances are not magic constants. They should reflect measurement uncertainty, algorithmic error, accumulated rounding, and the consequence of accepting a mismatch. A tolerance copied from an unrelated problem can hide a serious defect or reject a perfectly adequate result.

The order of computation matters

Real-number algebra permits many equivalent forms. In floating-point arithmetic, those forms can have different errors. Subtracting nearly equal large numbers can destroy useful leading digits, a problem called cancellation.

For large positive \(x\), consider

\[\sqrt{x+1}-\sqrt{x}\]

At \(x=10^{16}\), binary64 cannot represent the increment from \(x\) to \(x+1\) at that magnitude, so a direct evaluation can become zero. Rationalizing the expression gives

\[\sqrt{x+1}-\sqrt{x}=\frac{1}{\sqrt{x+1}+\sqrt{x}}\]

The right side avoids subtracting nearby values and returns a small nonzero result. Symbolically equivalent does not mean numerically equivalent on finite hardware.

Long sums also accumulate error. Adding many values from smallest magnitude to largest often loses less information than the reverse order. Compensated summation can track part of the low-order information that ordinary addition discards. The appropriate technique depends on the required accuracy and data scale.

Separate the error sources

When a prediction disagrees with a reference, ask which layer is responsible:

Printing more digits addresses none of the first four. Arbitrary-precision arithmetic can reduce representation error, but it cannot repair a wrong unit or an inadequate model. Conversely, rounding every intermediate value to match the final measurement precision needlessly increases computational error.

A reliable numerical boundary

At input, parse the value and unit together, convert to one consistent unit system, and retain the original uncertainty. During calculation, keep unrounded values and monitor known numerical hazards. At output, choose units meaningful to the reader, round in a way consistent with uncertainty, and keep enough context to reproduce the calculation.

That workflow treats a number as evidence about a physical quantity rather than as a free-floating decimal. It is the difference between code that merely runs and a result that can be interpreted.

A worked Python notebook

Convert road speeds to SI units before using them in a motion calculation. The question for this calculation is precise: how does converted speed change when we change speed, 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.

The conversion factor is exact; the measured speed may still have uncertainty. 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 120 km/h. 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 speed, expressed in km/h. The output is converted speed, expressed in m/s. 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.

\[f(q) = \frac{5 q}{18}\]

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 72 km/h, the expression gives 20 m/s. 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*1000/3600", locals={"q": q})
reference_input = 72.0
reference_output = float(expression.subs(q, reference_input))
print("Input:", reference_input, "km/h")
print("Predicted converted speed:", reference_output, "m/s")

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/s. The midpoint, at 60 km/h, gives 16.6667 m/s. The upper endpoint gives 33.3333 m/s. Read these together with the assumptions: they describe different choices of speed 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, 60.0, 120.0])
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} km/h -> {output_value:.6g} m/s")
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 speed; the vertical axis shows converted speed. 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, 120.0, 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("Speed (km/h)")
ax.set_ylabel("Converted speed (m/s)")
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 converted speed change near the reference input? Its units are the output unit divided by the input unit. For this calculation, the derivative at 72 km/h is 0.277778 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.

\[f'(q) = \frac{5}{18}\]

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 = 72.6
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/s")
print("Full expression:", full_prediction, "m/s")
print("Approximation difference:", full_prediction - linear_prediction, "m/s")

Here the changed input is 72.6 km/h. The tangent estimate is 20.1667 m/s, while the full expression gives 20.1667 m/s. 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 = 120.0
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 speed inside the stated interval, predict whether converted speed 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: The conversion factor is exact; the measured speed may still have uncertainty. 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/s: 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(120.0) - evaluate(0.0))
for points in [101, 201, 401]:
    grid = np.linspace(0.0, 120.0, 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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