Slide presentation
Significant Figures
General Chemistry • Matter, Its Properties, and Measurement
Matter, properties, and measurement
Significant figures show how precise a measurement really is
Chemistry calculations are only as reliable as the measurements used to make them. Significant figures communicate which digits are known with confidence and which final digit is estimated.
Learning target
- Identify significant digits in measured values.
- Distinguish measured values from exact numbers.
- Apply rounding rules in calculations.
- Connect precision to reliable chemical data.
Why it matters
Significant figures protect calculations from false precision
A calculator may display many digits, but laboratory instruments do not measure with unlimited precision. Significant figures keep the final answer honest.
Report what was measured
A balance reading of 12.40 g is more precise than 12.4 g because the trailing zero is measured.
Round at the end
Rounding rules prevent a final answer from claiming more certainty than the data support.
Communicate precision
Significant figures help readers understand the resolution of the instrument and the reliability of the result.
Measurement precision
The instrument controls how many digits are meaningful.
Final answer precision
The calculated result must be rounded to match the data.
Core concept
A significant figure is a meaningful measured digit
Significant figures include all certain digits plus one estimated digit. They do not include placeholder zeros that only show the position of the decimal point.
Measured values include uncertainty
The last significant digit is not random. It is the best estimate allowed by the instrument scale.
Exact numbers, such as counted objects or defined conversion factors, do not limit significant figures.
Vocabulary and rules
Zeros are the main source of significant-figure decisions
Nonzero digits are significant. Zeros may or may not be significant depending on where they appear.
| Case | Rule | Example | Significant figures |
|---|---|---|---|
| Nonzero digits | Always significant. | \(347\) | 3 |
| Zeros between nonzero digits | Always significant. | \(1007\) | 4 |
| Leading zeros | Not significant; they locate the decimal point. | \(0.00452\) | 3 |
| Trailing zeros after a decimal | Significant because they show measured precision. | \(2.300\) | 4 |
| Trailing zeros without a decimal | Ambiguous unless scientific notation or a decimal point clarifies them. | \(1500\) | ambiguous |
Scientific notation removes ambiguity
\(1.50 \times 10^{3}\) clearly has 3 significant figures, while \(1.500 \times 10^{3}\) clearly has 4 significant figures.
Main calculation rules
Different operations use different precision rules
Addition and subtraction are controlled by decimal places. Multiplication and division are controlled by the number of significant figures.
Two major rules
Round the final answer to the least number of decimal places in the measured values.
Example: \(12.11 + 18.0 = 30.1\)
Round the final answer to the least number of significant figures in the measured values.
Example: \(6.20 \times 3.1 = 19\)
Keep guard digits during work
Do not round too early. Carry extra digits through the calculation and round the final result.
Do not limit precision
Counting 3 trials or using \(1000\ \text{mL} = 1\ \text{L}\) exactly does not restrict significant figures.
Interactive digit analyzer
Test a measured value and round it to a chosen precision
Enter a number or choose an example. Significant digits are highlighted, and the rounded value updates when you change the target number of significant figures.
Leading zeros only locate the decimal point. The digits 4, 5, 3, and final 0 are significant.
Rule comparison
Choose the operation before choosing the rounding rule
The same measurements can be limited in different ways depending on the operation. Use decimal places for addition and subtraction; use significant figures for multiplication and division.
Compare calculation types
For addition and subtraction, round to the least number of decimal places. Example: 12.11 + 18.0 = 30.1.
Worked example
Use significant figures in a density calculation
A student measures a mass of \(18.64\ \text{g}\) and a volume of \(23.5\ \text{mL}\). Calculate density and report the answer with correct significant figures.
-
Identify the operation.
Density is calculated by division: \(d = m/V\).
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Count significant figures in each measured value.
\(18.64\ \text{g}\) has 4 significant figures. \(23.5\ \text{mL}\) has 3 significant figures.
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Calculate without rounding too early.
\(d = 18.64\ \text{g} / 23.5\ \text{mL} = 0.793191...\ \text{g/mL}\).
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Apply the multiplication/division rule.
The final answer must have 3 significant figures because the volume has the fewest significant figures.
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Round the final result.
\(0.793191...\ \text{g/mL}\) becomes \(0.793\ \text{g/mL}\).
Final answer
The density is \(0.793\ \text{g/mL}\), reported with 3 significant figures.
Common misconception
More calculator digits do not mean more precision
A calculator displays mathematical digits, not measurement certainty. The final answer should reflect the precision of the measured data.
Mistake
“The calculator gives \(0.793191489\), so I should report all of those digits.”
Correction
Report \(0.793\) because the least precise measured value in the division has 3 significant figures.
Practice check
Round the final answer correctly
A student calculates the mass of a product using \(2.50\ \text{mol} \times 18.015\ \text{g/mol}\). The calculator gives \(45.0375\ \text{g}\).
Question
How many significant figures should the final answer have, and what mass should be reported?
Show answer
This is multiplication, so use the factor with the fewest significant figures.
\(2.50\) has 3 significant figures. \(18.015\) has 5 significant figures.
The final answer should have 3 significant figures: \(45.0\ \text{g}\). The trailing zero is significant because it shows the tenths place is measured in the rounded answer.
Reasoning check
Reporting \(45\ \text{g}\) would show only 2 significant figures. Reporting \(45.0\ \text{g}\) preserves the required 3 significant figures.
Apply the topic
Use significant figures whenever measured data enter a calculation
A reliable chemistry answer reports both the correct numerical value and the correct precision. That means significant figures belong at the end of nearly every measurement-based problem.
Count significant figures, round measured values, and check calculation precision.
Practice questions Significant Figures QuestionsPractice digit-counting rules, rounding, operation rules, and lab-data interpretation.
How to apply this topic
First decide whether values are measured or exact. Then count precision, perform the calculation with guard digits, and round only the final answer using the correct operation rule.
Final summary
The essential takeaways
They show which digits are meaningful in a measured value.
A measured value includes certain digits plus one uncertain final digit.
Leading zeros are not significant; trapped zeros and decimal trailing zeros are significant.
Round to the least number of decimal places in the measured values.
Round to the least number of significant figures in the measured values.
Keep guard digits during calculation to avoid rounding error.