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Significant figures and rounding errors

Significant figures communicate the precision justified by measured data. They are a reporting convention, not a property that every intermediate arithmetic value should be forced to share. Rounding too early discards information; printing extra digits at the end invents an appearance of knowledge that the measurements did not provide.

Which digits are significant?

The significant digits of a written measurement begin with its first nonzero digit and include digits justified by the measurement's precision. Zeros need context:

Scientific notation removes that ambiguity. The forms \(1.5\times10^3\,\mathrm{m}\), \(1.50\times10^3\,\mathrm{m}\), and \(1.500\times10^3\,\mathrm{m}\) state two, three, and four significant figures respectively.

Exact counts and defined conversion factors do not limit significant figures. There are exactly four sides on a rectangle, and the factor

\(100\,\mathrm{cm}=1\,\mathrm{m}\)

is exact. A measured length written as one hundred centimetres is a different situation; its precision depends on how it was measured and reported.

Multiplication and division

An introductory rule reports a product or quotient with no more significant figures than the least precise measured factor. Suppose a rectangle has measured sides

\(\ell=12.3\,\mathrm{cm}\)

and \(w=4.56\,\mathrm{cm}\). The unrounded area is

\[A=\ell w=(12.3\,\mathrm{cm})(4.56\,\mathrm{cm})=56.088\,\mathrm{cm^2}\]

The first length has three significant figures and the second also has three, so the introductory rule reports \(A=56.1\,\mathrm{cm^2}\). The intermediate value should remain available for later calculations; the rounded form is the reported result.

This digit-count rule is an approximation to uncertainty propagation. If actual uncertainties and correlations are known, propagate them instead. Two measurements with the same digit count can have different uncertainty distributions.

Addition and subtraction

For sums and differences, the limiting issue is place value rather than the total number of significant figures. Adding

\[12.3\,\mathrm{m}+0.456\,\mathrm{m}=12.756\,\mathrm{m}\]

uses one input stated to tenths of a metre and another stated to thousandths. The usual rule reports the sum to tenths: \(12.8\,\mathrm{m}\). Reporting \(12.756\,\mathrm{m}\) would imply finer knowledge of the first measurement than was supplied.

Subtraction can be especially sensitive. If two large measured values are close, their difference may have high relative uncertainty even when each input has several significant figures. Counting digits alone does not reveal this loss of information; an uncertainty calculation does.

Round once, at the end

Rounding an intermediate result changes the input to every later operation. Consider a circular area computed from a measured diameter. Dividing the diameter by two, rounding the radius, squaring it, and rounding again can create a different answer from retaining guard digits and rounding only the final area.

length_metres = 1.234
width_metres = 0.567
area_square_metres = length_metres * width_metres
print(f"Unrounded area: {area_square_metres!r} m^2")
print(f"Reported area: {area_square_metres:.3g} m^2")
# Keep the unrounded value if another calculation needs it.
volume_cubic_metres = area_square_metres * 0.20
print(f"Volume: {volume_cubic_metres:.2g} m^3")

OpenStax, Light and Matter, and the Physics Hypertextbook all warn against rounding partial calculations. "Keep full precision" means retain the available machine value; it does not mean claim that all stored digits are experimentally meaningful.

Rounding itself has a rule

A statement such as "round to two places" is incomplete at a halfway case unless the tie-breaking rule is known. Common policies include ties away from zero and ties to the nearest even final digit. The latter reduces systematic upward bias over many balanced rounding operations.

Python formatting also acts on the stored binary approximation, which may lie just above or below the intended decimal halfway value. Financial or legal decimal rules may require decimal arithmetic rather than binary floating point. A scientific report should state any nonstandard rounding convention that materially affects results.

from math import isfinite


def format_measurement(
    value: float,
    decimal_places: int,
    unit: str,
) -> str:
    if not isfinite(value):
        raise ValueError("measurement value must be finite")

    if not isinstance(decimal_places, int) or decimal_places < 0:
        raise ValueError("decimal places must be a nonnegative integer")

    return f"{value:.{decimal_places}f} {unit}"

This function formats; it should not feed its string back into the calculation. Its use of a fixed-point format also needs tests for the project's chosen decimal cases.

Rounding error and measurement uncertainty differ

Rounding error is the numerical difference introduced by replacing a value with a coarser representation. Measurement uncertainty expresses incomplete knowledge of a physical quantity. A ruler with millimetre markings does not become more capable when its reading is stored in binary64 or printed with twelve decimal places.

Likewise, reducing display precision does not reduce uncertainty. If a value is recorded as \((8.37\pm0.12)\,\mathrm{m}\), formatting it as \(8.4\,\mathrm{m}\) does not change the underlying observation. The display and the measurement model must not be stored as one mutable string.

Systematic errors also resist digit rules. A miscalibrated instrument can return highly repeatable readings with many digits while remaining inaccurate. Repetition estimates some random effects; calibration and comparison with standards address systematic ones.

Report uncertainty when it matters

If an uncertainty estimate is available, round the uncertainty to a sensible number of significant digits and round the central value to the same decimal place. For example, a computed record might be reported as

\((12.35\pm0.08)\,\mathrm{s}\)

rather than attaching a long central value to a short uncertainty.

The interpretation must be stated. A standard uncertainty, standard deviation, standard error, and ninety-five-percent coverage interval are not synonyms. The number of digits cannot communicate which quantity was calculated.

Tests for the reporting boundary

A reliable formatter should be pure: calling it must not alter the stored measurement. Test values below, above, and at tie boundaries; negative values; zero; very large and small magnitudes; and unit conversions. Reject NaN and infinities unless the output format explicitly defines them.

Also compare early and late rounding on a deliberately sensitive multi-step example. The test should confirm that later physics calculations consume the unrounded value, while tables and prose use the justified display. This is a data-flow invariant, not merely a typography preference.

Significant figures are useful when detailed uncertainty information is unavailable, but they should remain an honest shorthand. Preserve the measurement evidence, perform the computation without avoidable rounding, and let the final display communicate no more precision than the evidence supports.

A worked Python notebook

Multiply a measured length by a width of 0.567 metres and postpone rounding until display. The question for this calculation is precise: how does rectangle area change when we change measured length, 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.

Treating the width as fixed isolates the effect of length; both measurements would matter in a full uncertainty budget. 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 1 to 1.5 m. 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 measured length, expressed in m. The output is rectangle area, expressed in m^2. 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.

\[f(q) = 0.567 q\]

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 1.234 m, the expression gives 0.699678 m^2. 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*0.567", locals={"q": q})
reference_input = 1.234
reference_output = float(expression.subs(q, reference_input))
print("Input:", reference_input, "m")
print("Predicted rectangle area:", reference_output, "m^2")

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 0.567 m^2. The midpoint, at 1.25 m, gives 0.70875 m^2. The upper endpoint gives 0.8505 m^2. Read these together with the assumptions: they describe different choices of measured length 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([1.0, 1.25, 1.5])
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} m -> {output_value:.6g} m^2")
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 measured length; the vertical axis shows rectangle area. 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(1.0, 1.5, 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("Measured length (m)")
ax.set_ylabel("Rectangle area (m^2)")
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 rectangle area change near the reference input? Its units are the output unit divided by the input unit. For this calculation, the derivative at 1.234 m is 0.567 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.

\[f'(q) = 0.567\]

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 = 1.2365
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^2")
print("Full expression:", full_prediction, "m^2")
print("Approximation difference:", full_prediction - linear_prediction, "m^2")

Here the changed input is 1.2365 m. The tangent estimate is 0.701095 m^2, while the full expression gives 0.701095 m^2. 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 = 0.5
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 measured length inside the stated interval, predict whether rectangle area 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: Treating the width as fixed isolates the effect of length; both measurements would matter in a full uncertainty budget. 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^2: 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(1.5) - evaluate(1.0))
for points in [101, 201, 401]:
    grid = np.linspace(1.0, 1.5, 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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