Significant Figures and Reporting Measurements

Imagine measuring a board for a shelf. You record its length as $1.24\,\mathrm{m}$: one meter, two tenths of a meter, and four hundredths of a meter. The last digit is in the hundredths place, so your report gives the length to the nearest $0.01\,\mathrm{m}$, or one centimeter.

Now suppose someone types that reading into a calculator and copies it as $1.240000\,\mathrm{m}$. The board has not been measured again. Where did those extra decimal places come from? The calculator can display them, but the measurement does not justify them.

Significant figures help us keep that distinction clear. By the end of this lesson, you should be able to explain what the last reported digit means, decide which zeros count, and round a calculated result without pretending to have made a better measurement.

Start with the last reported place

Compare two reports of the same board:

These have the same numerical value. They give different information about how finely the length is being reported. In the second report, the final zero says that the centimeter place was included in the measurement report and rounded to zero. Erasing it loses that information.

For example, lengths of $1.16\,\mathrm{m}$ and $1.24\,\mathrm{m}$ both round to $1.2\,\mathrm{m}$ when reported to the nearest tenth. A report ending at the hundredths place distinguishes them. That is the practical difference an extra reported place can make.

Thus $1.24\,\mathrm{m}$ has three significant figures: 1, 2, and 4. Writing $1.2\,\mathrm{m}$ retains two. Writing $1.20\,\mathrm{m}$ retains three, including the final zero.

The instrument and measuring method must justify the reported places. Adding a zero to an existing reading does not improve the measurement. As you saw in the measurement lesson, a finely marked ruler can still be misaligned. The last reported place tells us how the value was written; it does not, by itself, establish the full uncertainty.

Decide what each zero is doing

Zeros have two jobs: some set the size of the number, while others record a measured place. Use these examples to distinguish them.

Leading zeros set the scale

Suppose a small washer has a measured thickness of $0.0045\,\mathrm{m}$. In millimeters, the same thickness is

$$0.0045\,\mathrm{m}=4.5\,\mathrm{mm}.$$

The zeros before 4 disappear when we change units. They were needed to say that the thickness is a small fraction of a meter; they did not describe additional measurement detail. Both forms have two significant figures, 4 and 5.

Rule: start counting at the first nonzero digit. Do not count earlier zeros.

A zero between nonzero digits records a place

Suppose a board is reported as $1.05\,\mathrm{m}$. It is one meter and five centimeters long. The zero records zero tenths of a meter, between the meter digit and the hundredths digit.

That zero belongs to the reported measurement: removing it would give $1.5\,\mathrm{m}$, a different length. The digits 1, 0, and 5 are three significant figures.

Rule: zeros between nonzero digits count.

A final decimal zero tells you where reporting stops

Suppose the washer is instead reported as $0.00450\,\mathrm{m}$, equivalent to $4.50\,\mathrm{mm}$. The zeros before 4 still set the scale. The zero after 5 records an additional place: hundredths of a millimeter.

So $4.5\,\mathrm{mm}$ has two significant figures, while $4.50\,\mathrm{mm}$ has three. The second report claims finer measurement detail. It is appropriate only if the measuring method supports that detail.

Rule: zeros at the end of a decimal measurement, after its first nonzero digit, count.

Make whole-number zeros unambiguous

A field notebook says a surveyed path is $1500\,\mathrm{m}$ long. Was that rounded to the nearest hundred meters, ten meters, or meter? The notation alone does not tell us which final zeros were measured places.

Scientific notation answers that question by separating the significant digits from the power of ten:

The factor $10^3$ sets the scale: multiply by 1000. Count the digits in $1.5$, $1.50$, or $1.500$, not the exponent. Choose the form that matches the measurement record; do not guess the precision of an unexplained whole number such as 1500.

Use measured inputs to decide how far to report a calculation

A calculator works with the numbers you enter. It does not know how they were measured. We therefore need a reporting rule after the arithmetic. For introductory problems without explicit uncertainty bounds, the rule depends on the operation.

Multiplication and division: count significant figures

For a product or quotient, use the fewest significant figures among the measured inputs. If the time had been $0.370\,\mathrm{s}$, the distance $0.84\,\mathrm{m}$ would still limit this report to two significant figures.

Why count digits here? Multiplication and division combine measurements through ratios and scale factors. Their uncertainties are naturally compared as fractions of the measured values. Significant-figure counts provide a rough shortcut for that comparison; they are not a full uncertainty calculation.

Addition and subtraction: compare place values

For addition or subtraction, first express the inputs in the same unit. Then report to the coarsest last reported place. Here the answer has two significant figures even though $0.3$ has only one. Counting significant figures is the multiplication/division rule; adding lengths requires comparing their place values.

Round once, and recognize exact numbers

To round a positive result, identify the last place you will retain and look immediately to its right. A next digit of 0–4 leaves the retained digit unchanged; 5–9 increases it by one. This lesson uses the convention that an exact halfway case rounds upward. For example, $1.245$ rounded to three significant figures is $1.25$. Other contexts may specify a different tie rule.

Keep extra calculator digits during intermediate steps and round the final report. Those extra digits help prevent rounding errors from accumulating; they do not become extra significant figures in the answer.

An exact count behaves similarly. If you count five boards, that count does not have the uncertainty of a length measurement written as $5\,\mathrm{m}$. Dividing a sum of five readings by the exact count of five does not restrict the mean to one significant figure.

Check your understanding

1. A notebook records a board length as $1.20\,\mathrm{m}$. What information would be lost by copying it as $1.2\,\mathrm{m}$?
2. A washer thickness is reported as $0.0060\,\mathrm{m}$. Write it in millimeters and explain which zeros count.
3. A path length was reported to the nearest ten meters as 1500 meters. Write it in scientific notation so that the reporting place is clear.
4. A cart travels straight from a start mark to a finish mark $1.26\,\mathrm{m}$ away in $0.50\,\mathrm{s}$. Calculate and report its average speed. Explain your rounding.
5. Two measured board lengths are $2.36\,\mathrm{m}$ and $0.4\,\mathrm{m}$. Report their combined length. Which rule applies?
6. Does converting $1.20\,\mathrm{m}$ to centimeters permit you to report $120.00\,\mathrm{cm}$? Explain.

Reveal suggested answers

Suggested answers

1. The final zero reports the hundredths-of-a-meter place, or centimeters. Writing $1.2\,\mathrm{m}$ reports only tenths of a meter and discards that finer detail.
2. $0.0060\,\mathrm{m}=6.0\,\mathrm{mm}$, with two significant figures. The leading zeros locate the decimal point; the final zero records the tenths-of-a-millimeter place.
3. $1.50\times10^3\,\mathrm{m}$. The coefficient has three significant figures; its hundredths place becomes tens of meters after multiplication by 1000.
4. $1.26/0.50=2.52$, in meters per second. The time has two significant figures, so report $2.5\,\mathrm{m/s}$. The third digit is 2, so the second digit stays unchanged.
5. The arithmetic sum is $2.76\,\mathrm{m}$; report $2.8\,\mathrm{m}$. Addition uses the coarsest reported place, here tenths of a meter. The hundredths digit is 6, so round up.
6. No. The conversion is exact, but it does not add measured places. Use $1.20\times10^2\,\mathrm{cm}$ to show the same three significant figures clearly. The report still ends at whole centimeters, not hundredths of a centimeter.

Connections and references

Significant figures give a compact way to report measurement detail. A statement such as a measured length plus or minus a justified uncertainty gives more explicit information. Continue with Uncertainty Bounds and Comparing Results to calculate with those bounds.

Reference basis: Raymond A. Serway and Chris Vuille, College Physics, volume 1, ninth edition, section 1.4, “Uncertainty in Measurement and Significant Figures,” pages 7–10. Measurement scenarios and practice values in this lesson are invented for teaching. Worked examples are original.