Significant Figures Rules

Measurement Resolution Determines the Reported Digits

When a bottle cap diameter is read as $3.12 \text{ cm}$ , a calculator can produce many digits after the area is calculated. The resolution of the measuring tool limits how many of those digits can be reported. Significant-figure rules determine which digits belong in the final result.

Significant figures are the digits retained to communicate the resolution of a measured value. Write the value only to the decimal place supported by the instrument. The last reported digit shows the limit of the reading and can change when the value is rounded.

The significant-figure rule keeps a report from claiming more precision than the measurement tool supports.

Situation	Question to answer
Reading one measurement	Which digits are actually supported by the tool?
Addition and subtraction	Which data has the smallest number of decimal places?
Multiplication and division	Which data has the smallest number of significant figures?

Zero Can Be a Digit or Hold Place Value

Zeros are tricky because their meaning depends on where they are written.

Measurement writing	How to read the significant figures
$3.25 \text{ cm}$	Every nonzero digit counts, so there are $3$ significant figures.
$1.004 \text{ cm}$	Zeros between nonzero digits count, so there are $4$ significant figures.
$31.00 \text{ cm}$	Zeros after the decimal point count, so there are $4$ significant figures.
$0.0026 \text{ kg}$	Zeros before the first nonzero digit only hold place value, so there are $2$ significant figures.
$1000 \text{ m}$	The trailing zeros are ambiguous unless the resolution or significant figures are stated.
$3.14 \times 10^{-5} \text{ m}$	The coefficient $3.14$ has $3$ significant figures

In scientific notation, a power such as $10^{-5}$ gives the scale. It adds no significant figures.

Scientific notation states exactly which trailing zeros are significant because every digit in its coefficient is counted.

With $1$ significant figure:

1000\text{ m}=1\times10^3\text{ m}

With $4$ significant figures:

1000\text{ m}=1.000\times10^3\text{ m}

Round Once Using the First Dropped Digit

After the allowed number of digits is known, inspect the first digit that will be dropped.

First dropped digit	Decision
$<5$	Keep the last retained digit unchanged.
$>5$ , or $5$ followed by a nonzero digit	Increase the last retained digit by $1$ .
Exactly $5$ followed only by zeros	Use round-to-even: keep an even last digit, or increase an odd last digit by $1$ .

For example, if a calculated area is $52.976686625 \text{ cm}^2$ and it must be written to $4$ significant figures, the next digit is $6$ .

52.976686625 \text{ cm}^2 \to 52.98 \text{ cm}^2

If the calculated value is $52.973376625 \text{ cm}^2$ , the next digit is $3$ .

52.973376625 \text{ cm}^2 \to 52.97 \text{ cm}^2

For an exact halfway case, round-to-even avoids a long series of values being biased upward.

\begin{aligned} 7.450 &\to 7.4 \\ 7.550 &\to 7.6 \end{aligned}

Round directly from the unrounded value to the final precision. Do not round through several intermediate values.

Rounding Sums by Decimal Position

For addition and subtraction, the result keeps the decimal place of the measurement with the fewest decimal places.

Suppose an iron rod of length $8.235 \text{ cm}$ is joined to another rod of length $4.5 \text{ cm}$ .

\begin{aligned} 8.235 \text{ cm}+4.5 \text{ cm} &= 12.735 \text{ cm} \\ &\to 12.7 \text{ cm} \end{aligned}

The value $12.735 \text{ cm}$ is rounded to $1$ decimal place because $4.5 \text{ cm}$ has only $1$ digit after the decimal point.

Rounding Products by the Fewest Significant Figures

When a calculation tracks only significant figures, multiplication and division results are rounded to the fewest significant figures among the measured values. This shortcut does not replace a calculation of measurement uncertainty.

Suppose a rectangle has width $12.455 \text{ cm}$ and length $35.2 \text{ cm}$ .

\begin{aligned} A &= l \times w \\ &= 35.2 \text{ cm} \times 12.455 \text{ cm} \\ &= 438.416 \text{ cm}^2 \\ &\to 438 \text{ cm}^2 \end{aligned}

The value $12.455 \text{ cm}$ has $5$ significant figures, while $35.2 \text{ cm}$ has $3$ significant figures. Therefore, the area is reported with $3$ significant figures.

For a bottle cap with diameter $3.12 \text{ cm}$ , exact formula factors such as $\pi$ and $\frac{1}{4}$ do not limit significant figures because they are not measured data. Use the calculator value of $\pi$ and keep guard digits until the final line.

\begin{aligned} A &= \frac{1}{4}\pi d^2 \\ &= \frac{1}{4}\pi(3.12 \text{ cm})^2 \\ &= 7.645379881\ldots \text{ cm}^2 \\ &\to 7.65 \text{ cm}^2 \end{aligned}

If the International System of Units (SI) is required, convert $\text{cm}^2$ to $\text{m}^2$ . SI is the international standard for measurement units used in science.

\begin{aligned} 7.65 \text{ cm}^2 &= 0.000765 \text{ m}^2 \\ &= 7.65 \times 10^{-4} \text{ m}^2 \end{aligned}

For addition and subtraction, round by decimal position. For multiplication and division, round by the fewest significant figures among the measured values. Round-to-even and one-step rounding prevent repeated rounding from biasing the result.