Significant Figures Rules: Measurement in Scientific Work - Physics (Grade 10)

A Calculator Does Not Know the Tool Precision

When a bottle cap diameter is read as $3.12 \text{ cm}$ , a calculator can produce many digits after the area is calculated. The measuring tool does not justify all of those digits. This is why we use significant figures.

Significant figures are the trusted digits in a measurement. The last digit is usually an estimated digit, meaning it is still reasonable based on the measuring tool.

\text{measurement result}=\text{certain digits}+\text{last estimated digit}

The rule is not about making numbers shorter. It keeps a report from looking more precise than the measurement tool.

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.
$1000 \text{ m}$	Without a decimal point or another marker, trailing zeros are usually not treated as significant.
$3.14 \times 10^{-5} \text{ m}$	The coefficient $3.14$ has $3$ significant figures

In scientific notation, a power such as $10^{-5}$ only gives the scale, not a new significant figure.

If a zero must be significant, write it in a way that removes ambiguity. Scientific notation is often the cleanest choice.

\begin{aligned} 1000 \text{ m} \text{ with } 1 \text{ significant figure} &= 1 \times 10^3 \text{ m} \\ 1000 \text{ m} \text{ with } 4 \text{ significant figures} &= 1.000 \times 10^3 \text{ m} \end{aligned}

Rounding Follows the Next Digit

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

First dropped digit	Decision
$\ge 5$	Increase the last kept digit.
$<5$	Keep the last kept digit unchanged.

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

Addition Follows Decimal Position

For addition and subtraction, precision is judged by decimal position. The roughest data decides the final decimal place.

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.

Multiplication Follows the Least Precise Data

For multiplication and division, the limiting value is the smallest number of significant figures among the measured data.

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 $3.12 \text{ cm}$ , constants such as $\pi$ and $\frac{1}{4}$ do not limit significant figures because they come from the formula, not from a measuring tool. In this example calculation, $\pi$ is approximated as $3.14$ .

\begin{aligned} A &= \frac{1}{4}\pi d^2 \\ &= \frac{1}{4}(3.14)(3.12 \text{ cm})^2 \\ &= 7.641504 \text{ cm}^2 \\ &\to 7.64 \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.

7.64 \text{ cm}^2 = 0.000764 \text{ m}^2 = 7.64 \times 10^{-4} \text{ m}^2

Rule references from OpenStax University Physics and NIST Technical Note $1297$ can be opened through openstax.org and nist.gov.