Difficulty: Introductory Prerequisites: Companion notes on elements, units, and scientific notation (same lecture).
Significant figures are the rules that keep your reported answers honest about how precise your measurements really were. This section also covers how to round correctly after calculations, the difference between accuracy and precision (a distinction exams love to test), and density as your first real formula in the course. These concepts apply every time you report a numerical answer in chemistry, so they never go away.
Significant figures tell you how many digits in a measurement actually mean something. When you multiply or divide, your answer keeps the fewest sig figs of any input; when you add or subtract, your answer keeps the fewest decimal places. Density is mass divided by volume, reported in g/mL for solids and liquids or g/L for gases.
Significant figures (sig figs)
The digits in a measured value that carry meaning about the precision of the measurement. In simple terms, they are the digits you can trust.
Rounding
The process of dropping extra digits from a calculated result to match the precision your measurements actually support. If the first dropped digit is 5 or greater, round up; if 4 or less, round down (leave the last kept digit unchanged).
Accuracy
How close a measurement is to the true or accepted value. Think of it as hitting the bullseye.
Precision
How close repeated measurements are to each other, regardless of whether they are near the true value. Think of it as a tight grouping on the target, even if the grouping is off-centre.
Density
The ratio of an object's mass to its volume: density = mass / volume. In simple terms, it tells you how much stuff is packed into a given space.
All non-zero digits are significant. (456 g has 3 sig figs.)
Zeros between non-zero digits ("sandwiched zeros") are significant. (1003 has 4 sig figs.)
Leading zeros (zeros before the first non-zero digit) are never significant. They are just placeholders. (0.052 g has 2 sig figs.)
Trailing zeros after the decimal point are significant. (45.360 g has 5 sig figs.)
For numbers greater than 1 that contain a decimal point, all zeros to the right of the decimal are significant.
Trailing zeros in a whole number with no decimal point are ambiguous. Use scientific notation to clarify. (15,300 could be 3, 4, or 5 sig figs; writing 1.53 × 10⁴ makes clear it is 3.)
Value | Sig figs | Reasoning |
|---|---|---|
456 g | 3 | All non-zero digits |
15,300 g | 3 (ambiguous) | Trailing zeros, no decimal; assume 3 unless stated otherwise |
0.052 g | 2 | Leading zeros are not significant |
45.360 g | 5 | Trailing zero after decimal is significant |
0.5 mL | 1 | Leading zero not significant; only the 5 counts |
When you need to drop digits from a calculated answer:
Look at the first digit you are about to drop.
If it is 5 or greater, round the last kept digit up by one.
If it is 4 or less, leave the last kept digit as it is.
The point of rounding is to avoid implying false precision. If you add 0.05 mL to 200 mL, your answer is 200 mL, not 200.05 mL, because the 200 mL measurement was only precise to the ones place.
The rule: your answer has the same number of significant figures as the input with the fewest sig figs.
Example: D = 3.2167 g / 1.2 cm³
Calculator gives 2.680583... g/cm³
3.2167 has 5 sig figs; 1.2 has 2 sig figs.
Answer must have 2 sig figs: 2.7 g/cm³
The rule: your answer has the same number of decimal places as the input with the fewest decimal places.
Example: 82.236 cm + 4.1 cm
Calculator gives 86.336 cm.
82.236 is precise to the thousandths place (0.001). 4.1 is precise only to the tenths place (0.1).
Answer is rounded to the tenths place: 86.3 cm.
These two words are not interchangeable in science.
Accuracy means your measurement is close to the true value.
Precision means your repeated measurements are close to each other.
You can be precise without being accurate (a tight cluster that misses the target), accurate without being precise (shots scattered around the bullseye), both, or neither.
Density = mass / volume.
For solids and liquids, density is typically reported in g/mL (or equivalently g/cm³).
For gases, density is typically reported in g/L.
Density is an intensive property, meaning it does not depend on how much of the substance you have. A teaspoon of gold and a brick of gold have the same density.
Density:
D = m / V
where D = density, m = mass (g), V = volume (mL or L).
Rearranged: m = D × V, and V = m / D.
Sig fig rules at a glance:
Operation | Rule for the answer |
|---|---|
Multiplication / division | Same number of sig figs as the input with the fewest sig figs |
Addition / subtraction | Same number of decimal places as the input with the fewest decimal places |
Density is how you identify an unknown substance in a lab: measure the mass and volume, calculate the density, and compare it to known values. It is also why oil floats on water (oil is less dense) and why helium balloons rise (helium is less dense than air).
Significant figures matter in any field that depends on measurement. An engineer reporting a bridge span to the wrong precision, or a pharmacist calculating a dosage with too many assumed decimal places, introduces real risk. The rules exist to prevent false confidence in a number.
Students often mix up the sig fig rules for multiplication and addition. Multiplication/division uses the fewest significant figures; addition/subtraction uses the fewest decimal places. These are different rules.
Leading zeros trip people up. In 0.052, the zeros before the 5 are not significant. That number has 2 sig figs, not 3.
Students sometimes think "precise" and "accurate" mean the same thing. They do not. You can have one without the other.
A common error with density is forgetting to match units. If mass is in grams and volume is in litres, you will get g/L, not g/mL. Always check that your units are consistent with what the problem expects.
⚠️ "How many significant figures does this number have?" is a near-certainty on exams. Practise with numbers that include trailing zeros, leading zeros, and sandwiched zeros until classification is instant.
⚠️ Expect a calculation problem where you must apply the correct sig fig rule (multiplication vs. addition) and round properly. Getting the maths right but rounding wrong still loses marks.
⚠️ The accuracy-vs.-precision distinction is a classic multiple-choice or short-answer question. Know the dartboard analogy cold.
⚠️ Density problems often appear as: "given mass and volume, find density" or "given density and one of mass/volume, find the other." Rearranging D = m/V is expected.
How many significant figures in 0.00340?
True or false: Precision describes how close a measurement is to the true value.
Round 4.8651 to 3 significant figures.
Fill in the blank: Density = ______ / ______.
True or false: When adding 12.1 and 3.456, the answer should have 1 decimal place.
Answers: 1. Three (the 3, 4, and trailing 0 are significant). 2. False (that is accuracy). 3. 4.87. 4. mass / volume. 5. True (12.1 has 1 decimal place, the fewer of the two).
Q: How many significant figures are in 0.020500?
A: Five. The leading zeros (before the 2) are not significant. The 2, 0 (sandwiched), 5, and two trailing zeros after the decimal are all significant.
Q: A student measures the density of a sample three times and gets 2.71, 2.73, and 2.72 g/mL. The accepted value is 2.90 g/mL. Is the student's work accurate, precise, both, or neither?
A: Precise but not accurate. The three values are very close to each other (good precision), but they are all well below the accepted value of 2.90 g/mL (poor accuracy).
Q: Calculate (6.221 cm) × (4.0 cm) and report with the correct number of significant figures.
A: 6.221 × 4.0 = 24.884 on a calculator. 6.221 has 4 sig figs; 4.0 has 2 sig figs. The answer must have 2 sig figs: 25 cm².
Q: Add 101.2 g + 0.056 g and report with the correct number of decimal places.
A: 101.2 + 0.056 = 101.256 on a calculator. 101.2 is precise to 1 decimal place; 0.056 is precise to 3 decimal places. The answer is rounded to 1 decimal place: 101.3 g.
Q: A block of metal has a mass of 45.2 g and a volume of 16.5 mL. What is its density?
A: D = 45.2 g / 16.5 mL = 2.7394... g/mL. Both inputs have 3 sig figs, so the answer is 2.74 g/mL.
Significant figures will follow you through every quantitative chapter in this course. In stoichiometry (Chapter 3+), every mole-to-gram or gram-to-mole calculation requires proper sig fig handling. Density appears again in gas laws, where you will calculate the density of gases at different temperatures and pressures. The accuracy-vs.-precision distinction carries into lab work, where you will evaluate your own experimental results against accepted literature values.
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