Difficulty: Introductory | Prerequisites: Basic algebra, comfort with scientific notation.
Significant figures and unit conversions are foundational skills for every calculation you will do in general chemistry. Sig figs govern how precisely you can report a numerical answer, and conversions let you move between measurement systems (metric, imperial, SI). If you cannot handle these confidently, every quantitative problem downstream (stoichiometry, thermodynamics, equilibrium) will carry avoidable errors. You should already be comfortable with basic algebra and scientific notation before working through this material.
Significant figures tell you how many digits in a number are meaningful. For addition and subtraction, your answer keeps the fewest decimal places among the inputs. For multiplication and division, your answer keeps the fewest significant figures among the inputs. Unit conversions use conversion factors (fractions equal to 1) so that unwanted units cancel and the desired units remain.
Significant figures (sig figs)
The digits in a measured or calculated number that carry meaning about its precision. Leading zeros are never significant; trailing zeros after a decimal point are always significant.
In simple terms, sig figs tell you how "trustworthy" the digits in a number are.
Decimal places
The number of digits to the right of the decimal point. This is what matters when you add or subtract.
Think of it as: when adding, your answer can only be as precise as the least precise number you started with, measured by where its last decimal digit falls.
Conversion factor
A fraction that equals 1, built from a known relationship between two units (e.g. 2.54 cm / 1 in). Multiplying by it changes units without changing the quantity.
In simple terms, you are multiplying by a fancy version of 1 so the maths stays valid but the units shift.
Dimensional analysis
The method of tracking units through a calculation by arranging conversion factors so that unwanted units cancel. Also called the "factor-label method" or "unit-factor method."
Think of it as: a bookkeeping system for units. If the units on your final answer are wrong, you know a conversion factor was flipped.
Addition and subtraction: line up the decimal points. Your answer is rounded to the same decimal place as the input with the fewest decimal places.
Example: 9.67 + 2.01 = 11.68 (the result keeps two decimal places because both inputs have two).
Multiplication and division: count the sig figs in each input. Your answer may only have as many sig figs as the input with the fewest.
Example: 11.68 x 1.020 = 11.9136 on a calculator, but 1.020 has 4 sig figs and 11.68 has 4 sig figs, so the answer is reported as 11.91 (4 sig figs).
Combined operations: handle addition/subtraction first (using the decimal-place rule internally), then apply the sig-fig rule to any multiplication or division that follows.
The method: multiply by conversion factors arranged so that the unit you want to remove appears in the denominator and the unit you want to keep appears in the numerator.
Length example: convert 14 inches to centimetres.
14 in x (2.54 cm / 1 in) = 35 cm (the "in" cancels top and bottom).
Volume example: convert 8.0 mm³ to m³.
Because the unit is cubed, you need the conversion factor three times: 8.0 mm³ x (1 m / 1000 mm)³ = 8.0 / 10⁹ m³ = 8.0 x 10⁻⁹ m³.
A common trap: students apply the linear conversion (divide by 1000) once instead of cubing it. For cubic units you must cube the factor.
Speed example: convert 60 km/hr to m/s.
60 km/hr x (1000 m / 1 km) x (1 hr / 3600 s) = 17 m/s (two sig figs, matching the "60" input).
Addition / subtraction rule: round the result to the fewest decimal places of any input.
Multiplication / division rule: round the result to the fewest significant figures of any input.
Conversion factor form: (desired unit / given unit) = 1. Multiply so the given unit cancels.
Cubed / squared units: raise the entire conversion factor to the same power as the unit. Converting mm³ to m³ means cubing the factor (1 m / 1000 mm)³, not just dividing once.
Sig figs matter whenever you report a lab measurement. If your balance reads to 0.01 g, reporting a mass to 0.0001 g is false precision and will cost you marks. In engineering and pharmacy, incorrect sig figs can mean a bridge designed with insufficient tolerances or a medication dosed incorrectly.
Unit conversions are how you move between metric and imperial systems, which you will do constantly in chemistry (e.g. converting a volume in mL to L for molarity calculations) and in everyday life (km/hr to m/s in physics problems, inches to centimetres in lab work).
Students often think leading zeros (e.g. the zeros in 0.0045) are significant. They are not. Only the 4 and the 5 count, giving 2 sig figs.
Students frequently apply the addition/subtraction rule (decimal places) when they should be using the multiplication/division rule (sig figs), and vice versa. The two rules are different and apply to different operations.
When converting cubed or squared units, students often forget to cube or square the conversion factor. Converting mm³ to m³ requires dividing by 10⁹, not by 1000.
Students sometimes flip the conversion factor, putting the desired unit in the denominator instead of the numerator. If your units do not cancel correctly, the factor is upside down.
⚠️ Expect at least one question where you must combine addition and multiplication in one problem and apply the correct rounding rule at each step (the lecture example, (9.67 + 2.01) x 1.020, is a classic template).
⚠️ Volume conversions with cubed units (mm³ to m³, cm³ to L) are a favourite exam trap. Always cube the linear conversion factor.
⚠️ Know the conversion 1 in = 2.54 cm. This exact relationship appears frequently.
⚠️ Questions may ask you to identify how many sig figs a number has before performing any calculation. Trailing zeros without a decimal point (e.g. 1500) are ambiguous, and this ambiguity is testable.
True or false: 0.00320 has 3 significant figures.
Fill in the blank: when multiplying, you round to the fewest ______ among the inputs.
True or false: to convert cm² to m², you divide by 100.
Fill in the blank: 14 in x (2.54 cm / 1 in) = ______ cm.
True or false: in the expression (5.1 + 2.34) x 3.0, you should apply the sig-fig rule to the addition step.
Answers: 1. True (the trailing zero after the 2 counts). 2. Significant figures. 3. False (you divide by 10,000, because you must square the factor of 100). 4. 35 cm. 5. False (you apply the decimal-place rule to the addition, then the sig-fig rule to the multiplication).
Q: How many significant figures are in the number 1.020?
A: Four. The trailing zero after the 2 is significant because it comes after a decimal point.
Q: Calculate (9.67 + 2.01) x 1.020 and report the answer to the correct number of significant figures.
A: 9.67 + 2.01 = 11.68 (2 decimal places). 11.68 x 1.020 = 11.9136, rounded to 4 sig figs = 11.91.
Q: Convert 8.0 mm³ to m³.
A: 8.0 mm³ x (1 m / 1000 mm)³ = 8.0 x 10⁻⁹ m³. You must cube the conversion factor because the unit is cubed.
Q: Convert 60 km/hr to m/s.
A: 60 km/hr x (1000 m / 1 km) x (1 hr / 3600 s) = 16.667 m/s, rounded to 17 m/s (2 sig figs).
Q: A student reports 35.56 cm after converting 14 inches. What did they do wrong?
A: They kept too many significant figures. 14 has 2 sig figs, so the answer should be 35 cm, not 35.56 cm.
Sig figs will reappear in every quantitative chapter of CHEM 101. When you reach stoichiometry, your mole calculations must be reported to the correct number of sig figs. In thermodynamics, heat and enthalpy values follow the same rules. Unit conversions are the backbone of dimensional analysis, which you will use heavily in mole-to-gram and concentration calculations.
Significant figures, sig figs, significant digits, decimal places, rounding rules, unit conversion, dimensional analysis, factor-label method, conversion factor, metric conversion, cubed unit conversion, CHEM 101, general chemistry, Michigan State University