# Unit conversion and significant figures

Canonical: https://duckyhelper.com/learn/physics/unit-conversion-and-significant-figures/
Updated: 2026-10-01

To convert units, multiply by conversion factors, fractions equal to 1 such as \(\tfrac{1\ \text{km}}{1000\ \text{m}}\), arranged so the old units cancel and only the new unit is left. Significant figures show how precise a measurement is. When you multiply or divide, round the answer to the fewest significant figures in the data. When you add or subtract, round to the fewest decimal places.

## Key ideas

### Conversion factors

A conversion factor is a fraction whose top and bottom are the same amount, so multiplying by it does not change the value. Put the unit you want to cancel on the bottom:

$$
5.0\ \text{km} \times \frac{1000\ \text{m}}{1\ \text{km}} = 5000\ \text{m}
$$

**Common conversions (the first four are exact by definition)**

| Conversion | Factor |
| --- | --- |
| inch to centimeter | 1 in = 2.54 cm |
| mile to meter | 1 mi = 1609.344 m |
| hour to second | 1 h = 3600 s |
| kilo, centi, milli | 1 km = 1000 m, 1 m = 100 cm, 1 m = 1000 mm |
| liter to cubic centimeter | 1 L = 1000 cm³ = 1000 mL |
| km/h to m/s | divide by 3.6 |

### Counting significant figures

- Nonzero digits always count: 4.56 has 3.
- Zeros between nonzero digits count: 1.005 has 4.
- Leading zeros never count: 0.0032 has 2.
- Trailing zeros count only if there is a decimal point: 2.500 has 4, but 2500 has 2. Write \(2.50 \times 10^3\) to show 3.
- Exact numbers, like counted objects or defined conversions (1 in = 2.54 cm), have unlimited significant figures.

### Rounding answers

- **Multiply or divide:** keep the fewest **significant figures** of any measurement. \(4.56 \times 1.4 = 6.384\), which rounds to 6.4 (2 significant figures).
- **Add or subtract:** keep the fewest **decimal places**. \(12.11 + 18.0 + 1.013 = 31.123\), which rounds to 31.1 (1 decimal place).
- Keep extra digits during a long calculation, and round only once at the end.

## Worked examples

**Example 1: miles per hour to meters per second**

Problem: A car drives at 65 mph. What is that in m/s?

1. Chain two factors: miles to meters, then hours to seconds.

   $$
   65\ \frac{\text{mi}}{\text{h}} \times \frac{1609.344\ \text{m}}{1\ \text{mi}} \times \frac{1\ \text{h}}{3600\ \text{s}} = 29.06\ \text{m/s}
   $$
2. The conversion factors are exact, so the 2 significant figures of 65 decide the answer.

Answer: 29 m/s

**Example 2: converting a unit that is squared or cubed**

Problem: Convert a density of 2.5 g/cm³ to kg/m³.

1. Grams to kilograms: divide by 1000. Cubic centimeters to cubic meters: 1 m = 100 cm, so 1 m³ = \(100^3 = 10^6\) cm³.

   $$
   2.5\ \frac{\text{g}}{\text{cm}^3} \times \frac{1\ \text{kg}}{1000\ \text{g}} \times \frac{10^6\ \text{cm}^3}{1\ \text{m}^3} = 2500\ \text{kg/m}^3
   $$
2. Show 2 significant figures with scientific notation.

Answer: \(2.5 \times 10^3\) kg/m³

**Example 3: counting significant figures**

Problem: How many significant figures are in 0.004050?

1. The leading zeros (0.00) only place the decimal point. They do not count.
2. 4 and 5 count, the zero between them counts, and the trailing zero counts because there is a decimal point.

Answer: 4 significant figures

**Example 4: a mixed calculation**

Problem: A runner covers 100.0 m, then 52.5 m, in a total time of 21.6 s. What is the average speed?

1. Add first. Both distances have 1 decimal place, so the sum keeps 1.

   $$
   100.0 + 52.5 = 152.5\ \text{m}
   $$
2. Then divide. 152.5 has 4 significant figures and 21.6 has 3, so the answer gets 3.

   $$
   \frac{152.5\ \text{m}}{21.6\ \text{s}} = 7.060\ \text{m/s}
   $$

Answer: 7.06 m/s

## Common mistakes and how to fix them

- **Flipping the conversion factor.** Fix: write the units on every number. If the unit does not cancel, turn the fraction over.
- **Converting squared units with the plain factor.** 1 m² is 10,000 cm², not 100. Fix: square or cube the factor too.
- **Using the sig fig rule for multiplication when adding.** Fix: addition and subtraction go by decimal places.
- **Rounding in the middle.** Fix: keep one or two extra digits until the final answer.
- **Counting the digits of an exact number.** "3 trials" or "1 h = 60 min" never limit your answer.

**Practice problems**

1. How many significant figures are in 0.00320?
   A. 2
   B. 3
   C. 5
   D. 6

   Answer: 3. The leading zeros do not count. 3 and 2 count, and the final zero counts because there is a decimal point.

2. How many significant figures are in 1500 (no decimal point)?
   A. 2
   B. 3
   C. 4

   Answer: 2. Trailing zeros without a decimal point are not significant. To show 4, write 1500. or \(1.500 \times 10^3\).

3. Convert 72 km/h to m/s.

   Answer: 20. m/s. \(72 \times \frac{1000\ \text{m}}{1\ \text{km}} \times \frac{1\ \text{h}}{3600\ \text{s}} = 20.\) m/s. The decimal point shows that both digits are significant.

4. Calculate \(3.0 \times 2.26\) with the correct significant figures.
   A. 6.78
   B. 6.8
   C. 7
   D. 6.780

   Answer: 6.8. \(3.0 \times 2.26 = 6.78\). 3.0 has only 2 significant figures, so round to 6.8.

5. A pencil is 5.00 inches long. How long is it in centimeters?

   Answer: 12.7 cm. \(5.00\ \text{in} \times 2.54\ \text{cm/in} = 12.7\) cm. 2.54 is exact, so 5.00 sets 3 significant figures.

## Frequently asked questions

### Why do significant figures matter?

They show how precise a measurement is. If a ruler measures to the nearest millimeter, writing a length to a millionth of a meter claims precision you do not have. Answers keep the precision of the least precise measurement, so they never look more exact than the data.

### Do I round 2.5 up or down?

Most high school classes round 5 up, so 2.5 becomes 3. Some science courses and computers round a 5 to the nearest even digit instead, so 2.5 becomes 2. Follow your teacher's rule. It rarely matters, because you should keep extra digits until the end anyway.

### What is dimensional analysis?

It is another name for converting with conversion factors and checking that units cancel. It also catches mistakes: if a speed problem ends in seconds per meter instead of meters per second, a step is upside down. Chemistry uses the same method for [moles and molar mass](https://duckyhelper.com/learn/chemistry/mole-and-molar-mass/).

## Sources

- [OpenStax College Physics 2e, 1.2 Physical Quantities and Units](https://openstax.org/books/college-physics-2e/pages/1-2-physical-quantities-and-units), accessed 2026-10-01
- [OpenStax College Physics 2e, 1.3 Accuracy, Precision, and Significant Figures](https://openstax.org/books/college-physics-2e/pages/1-3-accuracy-precision-and-significant-figures), accessed 2026-10-01
- [NIST, Guide for the Use of the SI (Appendix B conversion factors)](https://www.nist.gov/pml/special-publication-811), accessed 2026-10-01

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## Try asking Ducky

- "Count the significant figures in my lab data table and tell me how many my answer should have."
- "I converted cm² to m² and got a huge number. Check my conversion factor."
- "Quiz me on sig fig rules for adding versus multiplying."

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