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Output docs content for large language models.

---

export const metadata = {
  title: "Significant Figures Rules",
  description:
    "Learn how to identify significant figures, round measurement results, and report calculated values with the right precision.",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "04/25/2026",
  subject: "Measurement in Scientific Work",
};

## A Calculator Does Not Know the Tool Precision

When a bottle cap diameter is read as <InlineMath math="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.

<BlockMath math="\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 |
| :------------------ | :---------------------------------- |
| <InlineMath math="3.25 \text{ cm}" /> | Every nonzero digit counts, so there are <InlineMath math="3" /> significant figures. |
| <InlineMath math="1.004 \text{ cm}" /> | Zeros between nonzero digits count, so there are <InlineMath math="4" /> significant figures. |
| <InlineMath math="31.00 \text{ cm}" /> | Zeros after the decimal point count, so there are <InlineMath math="4" /> significant figures. |
| <InlineMath math="0.0026 \text{ kg}" /> | Zeros before the first nonzero digit only hold place value, so there are <InlineMath math="2" /> significant figures. |
| <InlineMath math="1000 \text{ m}" /> | Without a decimal point or another marker, trailing zeros are usually not treated as significant. |
| <InlineMath math="3.14 \times 10^{-5} \text{ m}" /> | The coefficient <InlineMath math="3.14" /> has <InlineMath math="3" /> significant figures |

In scientific notation, a power such as <InlineMath math="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.

<BlockMath math="\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 |
| :------------------ | :------- |
| <InlineMath math="\ge 5" /> | Increase the last kept digit. |
| <InlineMath math="<5" /> | Keep the last kept digit unchanged. |

For example, if a calculated area is <InlineMath math="52.976686625 \text{ cm}^2" /> and it must be written to <InlineMath math="4" /> significant figures, the next digit is <InlineMath math="6" />.

<BlockMath math="52.976686625 \text{ cm}^2 \to 52.98 \text{ cm}^2" />

If the calculated value is <InlineMath math="52.973376625 \text{ cm}^2" />, the next digit is <InlineMath math="3" />.

<BlockMath math="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 <InlineMath math="8.235 \text{ cm}" /> is joined to another rod of length <InlineMath math="4.5 \text{ cm}" />.

<BlockMath math="\begin{aligned}
8.235 \text{ cm}+4.5 \text{ cm} &= 12.735 \text{ cm} \\
&\to 12.7 \text{ cm}
\end{aligned}" />

The value <InlineMath math="12.735 \text{ cm}" /> is rounded to <InlineMath math="1" /> decimal place because <InlineMath math="4.5 \text{ cm}" /> has only <InlineMath math="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 <InlineMath math="12.455 \text{ cm}" /> and length <InlineMath math="35.2 \text{ cm}" />.

<BlockMath math="\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 <InlineMath math="12.455 \text{ cm}" /> has <InlineMath math="5" /> significant figures, while <InlineMath math="35.2 \text{ cm}" /> has <InlineMath math="3" /> significant figures. Therefore, the area is reported with <InlineMath math="3" /> significant figures.

For a bottle cap with diameter <InlineMath math="3.12 \text{ cm}" />, constants such as <InlineMath math="\pi" /> and <InlineMath math="\frac{1}{4}" /> do not limit significant figures because they come from the formula, not from a measuring tool. In this example calculation, <InlineMath math="\pi" /> is approximated as <InlineMath math="3.14" />.

<BlockMath math="\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 <InlineMath math="\text{cm}^2" /> to <InlineMath math="\text{m}^2" />. SI is the international standard for measurement units used in science.

<BlockMath math="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 <InlineMath math="1297" /> can be opened through [the OpenStax link](https://openstax.org/books/university-physics-volume-1/pages/1-6-significant-figures) and [the NIST link](https://www.nist.gov/pml/nist-technical-note-1297/nist-tn-1297-7-reporting-uncertainty).