# Nakafa Learning Content > For AI agents: use [llms.txt](https://nakafa.com/llms.txt) for the site index. Markdown versions are available by appending `.md` to content URLs or sending `Accept: text/markdown`. URL: https://nakafa.com/en/subjects/physics/measurement/significant-figures Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/material/lesson/physics/measurement/significant-figures/en.mdx Learn how to identify significant figures, round measurement results, and report calculated values with the right precision. --- ## 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**. Visible text: When a bottle cap diameter is read as , 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. ```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 | | :------------------ | :---------------------------------- | | $$3.25 \text{ cm}$$ | 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 | Visible text: | Measurement writing | How to read the significant figures | | :------------------ | :---------------------------------- | | | Every nonzero digit counts, so there are significant figures. | | | Zeros between nonzero digits count, so there are significant figures. | | | Zeros after the decimal point count, so there are significant figures. | | | Zeros before the first nonzero digit only hold place value, so there are significant figures. | | | Without a decimal point or another marker, trailing zeros are usually not treated as significant. | | | The coefficient has significant figures | In scientific notation, a power such as $$10^{-5}$$ only gives the scale, not a new significant figure. Visible text: In scientific notation, a power such as 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. ```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 | | :------------------ | :------- | | $$\ge 5$$ | Increase the last kept digit. | | $$<5$$ | Keep the last kept digit unchanged. | Visible text: | First dropped digit | Decision | | :------------------ | :------- | | | Increase the last kept digit. | | | 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 $$6$$. Visible text: For example, if a calculated area is and it must be written to significant figures, the next digit is . ```math 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 $$3$$. Visible text: If the calculated value is , the next digit is . ```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 $$8.235 \text{ cm}$$ is joined to another rod of length $$4.5 \text{ cm}$$. Visible text: Suppose an iron rod of length is joined to another rod of length . ```math \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. Visible text: The value is rounded to decimal place because has only 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 $$35.2 \text{ cm}$$. Visible text: Suppose a rectangle has width and length . ```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 $$12.455 \text{ cm}$$ has $$5$$ significant figures, while $$35.2 \text{ cm}$$ has $$3$$ significant figures. Therefore, the area is reported with $$3$$ significant figures. Visible text: The value has significant figures, while has significant figures. Therefore, the area is reported with 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 $$3.14$$. Visible text: For a bottle cap with diameter , constants such as and do not limit significant figures because they come from the formula, not from a measuring tool. In this example calculation, is approximated as . ```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 $$\text{cm}^2$$ to $$\text{m}^2$$. SI is the international standard for measurement units used in science. Visible text: If the International System of Units (SI) is required, convert to . SI is the international standard for measurement units used in science. ```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 $$1297$$ 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). Visible text: Rule references from OpenStax University Physics and NIST Technical Note 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).