Difficulty: Introductory | Prerequisites: Basic algebra, scientific notation
This material is the foundation of every quantitative exercise in general chemistry. Before you can calculate a molar mass or an enthalpy change, you need to know how many digits to trust in a measurement, how to express very large or very small numbers cleanly, and how to judge whether your experimental result is close to the accepted value. These skills come back in every lab report and most exam calculations, so getting comfortable with them early pays off throughout the course.
Significant figures tell you how precise a measurement is. Percent error tells you how accurate it is. Density links mass and volume, and shows up in displacement problems constantly.
Significant figures (sig figs)
The digits in a measured quantity that carry meaning about the precision of that measurement. Leading zeros are never significant; trailing zeros after a decimal point are always significant; trailing zeros before a decimal point are ambiguous unless a decimal point is explicitly shown.
In simple terms, sig figs are the digits you can trust based on your measuring instrument.
Percent error
A measure of accuracy calculated as |(experimental value - accepted value) / accepted value| x 100%. It tells you how far off your result is from the known correct value.
In simple terms, percent error is "how wrong were you, as a percentage?"
Density
Mass per unit volume, typically expressed as g/mL or g/cm³. The formula is d = m / V.
In simple terms, density tells you how much stuff is packed into a given space.
Precision
How close repeated measurements are to each other. A set of measurements can be precise (tightly clustered) without being accurate (close to the true value).
Accuracy
How close a measurement is to the accepted or true value. High accuracy with low precision means you got lucky once; high precision with low accuracy means your instrument has a systematic error.
Scientific notation
A way of writing numbers as a coefficient between 1 and 10 multiplied by a power of 10. All digits in the coefficient are significant.
In simple terms, it is a compact way to write very large or very small numbers while making the number of sig figs unambiguous.
All non-zero digits are significant: 345 has 3 sig figs.
Zeros between non-zero digits (captive zeros) are significant: 4007 has 4 sig figs.
Leading zeros are never significant: 0.00340 has 3 sig figs (the 3, 4, and trailing 0).
Trailing zeros after a decimal point are significant: 2.500 has 4 sig figs.
Trailing zeros in a whole number with no decimal point are ambiguous: 120 could be 2 or 3 sig figs. Adding an explicit decimal point (120.) makes all three digits significant.
In scientific notation, every digit in the coefficient counts: 7.00 x 10² has 3 sig figs; 1.20 x 10⁴ has 3 sig figs.
Consider these five measurements:
0.00340 g: 3 sig figs (3, 4, trailing 0)
120 mL: 2 sig figs (ambiguous trailing zero, no decimal point)
0.078 kg: 2 sig figs (7, 8; leading zeros do not count)
7.00 x 10² m: 3 sig figs (7, 0, 0 in the coefficient)
1.20 x 10⁴ s: 3 sig figs (1, 2, 0 in the coefficient)
Three of these measurements (0.00340 g, 7.00 x 10² m, and 1.20 x 10⁴ s) contain exactly three significant figures.
If 120 mL were instead written as 120. mL (with an explicit decimal point), the trailing zero becomes significant, raising the count from 2 to 3.
Multiplication and division: the result has the same number of sig figs as the input with the fewest sig figs.
Addition and subtraction: the result has the same number of decimal places as the input with the fewest decimal places.
The formula:
% error = |(experimental value - accepted value) / accepted value| x 100%
A student measures 9.5 mL of water and finds its mass to be 8.1 g. The accepted density of water is 1.00 g/mL.
Experimental density = mass / volume = 8.1 g / 9.5 mL = 0.853 g/mL
Accepted density = 1.00 g/mL
% error = |0.853 - 1.00| / 1.00 x 100% = 14.7%
A metal sample has a density of 0.0997 g/mL. It is dropped into a graduated cylinder where the water level rises from 15.5 mL to 95.2 mL.
Volume of sample = 95.2 - 15.5 = 79.7 mL
Mass = density x volume = 0.0997 g/mL x 79.7 mL = 7.95 g
If the accepted mass is 8.00 g: % error = |7.95 - 8.00| / 8.00 x 100% = 0.625%
Density: d = m / V
Percent error: % error = |(experimental - accepted) / accepted| x 100%
Percent error is how analytical chemists evaluate the quality of a new assay. If a pharmaceutical lab measures the concentration of a drug in a tablet and gets 2% error, that is good; 15% error means the method needs work. Density measurements by displacement are used to identify unknown metals in forensic and materials science laboratories.
Students often think leading zeros are significant. They are not. In 0.0042, only the 4 and 2 count.
Students forget that a trailing zero after a decimal point is significant. 2.0 has two sig figs, not one.
Students sometimes confuse precision with accuracy. A set of very close but consistently wrong measurements is precise but inaccurate.
The sign of the error in percent error does not matter; the absolute value is taken. A negative result just means the experimental value was below the accepted value.
⚠️ Sig fig counting appears in nearly every calculation-based question. If the exam says "report to the correct number of significant figures," losing a mark for rounding incorrectly is common.
⚠️ The 120 vs 120. distinction (effect of an explicit decimal point on trailing zeros) is a classic multiple-choice trap.
⚠️ Percent error calculations require the absolute value. Forgetting the absolute value will give a negative answer, which is technically still correct in magnitude but may cost marks if the question asks for percent error specifically.
True or false: 0.0050 has 2 significant figures. True.
Fill in the blank: The number of sig figs in 3.040 is ___. 4.
True or false: If your experimental value is higher than the accepted value, percent error is negative. False (the absolute value is taken).
Fill in the blank: Density equals mass divided by ___. Volume.
True or false: In addition/subtraction, the answer is limited by the fewest decimal places among the inputs. True.
Q: A student records a mass of 4.520 g. How many significant figures does this measurement have?
A: Four significant figures. The trailing zero after the decimal point is significant.
Q: Calculate the percent error if a student measures a boiling point of 99.1°C and the accepted value is 100.0°C.
A: % error = |99.1 - 100.0| / 100.0 x 100% = 0.9%.
Q: A sample has a volume of 25.0 mL and a mass of 67.5 g. What is its density, reported to the correct number of significant figures?
A: d = 67.5 g / 25.0 mL = 2.70 g/mL (3 sig figs, matching the least precise input).
Q: If Measurement B is written as 120 mL, how many significant figures does it have? How does writing it as 120. mL change the answer?
A: 120 mL has 2 significant figures (the trailing zero is ambiguous). Writing 120. mL (with the decimal point) makes the trailing zero significant, giving 3 significant figures.
Significant figures carry through to every quantitative topic in the course, from molar mass calculations to enthalpy changes to pH. Density connects to the gas laws (molar volume, ideal gas calculations) and to solution concentration (mass/volume relationships). Percent error appears again in calorimetry experiments and any lab where you compare a measured value to a literature value.
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