A Guide to Exact, Counted, and Stipulated Values

Why Don’t All Numbers Have Uncertainty?

Every measurement comes with uncertainty, which we communicate using significant figures. But not all numbers in chemistry calculations are measurements. When you convert 15.5 inches to centimeters using the conversion factor 1 inch = 2.54 cm, should the final answer be limited by the three significant figures in both 2.54 and 15.5?

The answer depends on the source of each number. The measured length (15.5 inches) has finite precision. The conversion factor (1 inch = 2.54 cm) is a defined relationship and is exact. Knowing which numbers carry uncertainty and which do not is necessary for correctly applying significant figure rules.

The Categories of Numbers

1. Measured Numbers

A measured number is any quantity determined using an instrument: a ruler, graduated cylinder, balance, thermometer, or stopwatch. Measured numbers always have uncertainty in the last reported digit and carry finite significant figures determined by the instrument’s precision.

Examples: 12.105 g, 25.3 mL, 37.2 °C

2. Exact Numbers by Counting

When you count discrete, whole items, there is no uncertainty. You cannot have 3.7 students. Counted numbers have infinite significant figures and never limit your answer.

Examples: 35 students, 4 atoms of hydrogen in CH<sub>4</sub>

3. Exact Numbers by Definition

Some numbers are part of formally defined relationships between units, established by international agreement rather than measurement. These defined values have infinite significant figures and never limit your answer.

Examples: 1 foot = 12 inches, 1 minute = 60 seconds, 1 inch = 2.54 cm

Numbers embedded in mathematical formulas are also exact: the 2 in \(KE = \frac{1}{2}mv^2\), the 4 in \(A = 4\pi r^2\), and mathematical constants like \(\pi\) and \(e\) all have infinite significant figures. Since the 2019 SI redefinition, several physical constants are also exact by definition, including Avogadro’s number (\(6.02214076 \times 10^{23}\) mol\(^{-1}\)).

TipThe Practical Takeaway

Exact and counted numbers never limit the significant figures in your answer. When determining how many sig figs to report, ignore them entirely. Only measured values (and stipulated values, by convention) constrain your result.

4. Numbers Given in Problems

Numbers provided in problem statements fall into two distinct groups. To distinguish between them, ask:

“In the real world, would this quantity be determined by a measurement device, or is it set by a rule, definition, or agreement?”

Compare these two statements:

  • “The sample has a mass of 15.2 g.” → Given Measured (someone weighed it)
  • “Each test tube holds 5.00 mL.” → Stipulated (manufacturer’s specification)

Both numbers appear in problem statements, but they have different origins.

A. Given Measured Values

These are physical measurements reported in a problem statement. Someone weighed, measured, or recorded the value, and it carries the uncertainty from that measurement. Treat with finite significant figures, exactly as written.

Examples:

  • “A rock has a mass of 15.2 g.” (3 sig figs)
  • “The density of the sample is 7.87 g/cm³.” (3 sig figs)
  • “A solution has a concentration of 0.100 M.” (3 sig figs)

B. Stipulated Values

Values set by rule, agreement, or specification rather than measurement. Prices, wages, and manufacturer specifications fall in this category. They are not measurements, but they are not exact like defined conversion factors either. Use the number of significant figures as written to determine answer precision.

Examples:

  • “A bottle costs $0.74.”
  • “Hourly wage is $8.25 per hour.”
  • “The tank contains exactly 15 gal of water.” (Problem specifies this as a fixed starting quantity)

Stipulated values appear more often in economics and engineering than in chemistry, but they do show up in mixed calculations.

NoteA Note on Terminology

The term “stipulated values” is used in this resource for clarity. You are unlikely to encounter it in your textbook or on an exam. Most gen-chem courses classify numbers as either measured (finite sig figs) or exact (infinite sig figs) and leave it at that. The stipulated category fills a gap in that framework by giving you a rule for values that are neither measurements nor defined constants.

Applying the Categories

Calculations with Measured Numbers

If a calculation contains at least one measured number, the final answer’s precision is limited by the least precise measured number. This follows the standard significant figure rules.

Example: Using a Given Density

A sample of iron with a measured volume of 2.50 cm³ is used. The density of iron is given as 7.87 g/cm³. What is the mass of the sample?

  1. Classify the numbers:
    • 2.50 cm³: Measured (3 sig figs)
    • 7.87 g/cm³: Given Measured Value (3 sig figs)
  2. Calculate: 2.50 cm³ × 7.87 g/cm³ = 19.675 g
  3. Round: Limited by 3 sig figs in both inputs
    • Final Answer: 19.7 g

Calculations with Stipulated Values

Consider this problem:

A project requires a total materials cost of $21.46. How long (in hours) would you have to work to afford this if your wage is $8.25 per hour?

Both numbers are stipulated values. $21.46 is a price and $8.25/hour is a wage. Neither is a physical measurement. The raw calculation, 21.46 / 8.25, results in an infinitely repeating decimal (2.601212...).

ImportantHow to Handle Stipulated Values

Stipulated values are not measurements, so sig fig rules do not strictly apply to them. But the result of a calculation involving stipulated values is often a physical quantity (like time), and you need some rule for how many digits to report. A practical convention: treat each stipulated value as having the number of significant figures shown in how it is written.

  • For multiplication and division, round to match the stipulated input with the fewest significant figures.
  • For addition and subtraction, round to match the place value of the least precise stipulated input.

This is a heuristic, not a statement about measurement uncertainty. It gives a reasonable, consistent answer when the standard measured-vs-exact framework offers no guidance.

Applying this rule:

  1. Analyze stated precision:

    • $21.46: 4 significant figures
    • $8.25: 3 significant figures
    • Division problem: answer limited to 3 significant figures

  2. Calculate: 21.46 / 8.25 = 2.6012121212... hours

  3. Round:

    • Final Answer: 2.60 hours

Summary Table

Practice: Classify These Numbers

Classify each number and state whether it limits significant figures.

Exact by definition. This is a defined relationship, so it has infinite sig figs and you can ignore it when determining precision.

Measured. Someone used a balance to get this number. It has 4 significant figures and will limit your answer.

Exact by definition (since 1983). Despite having many digits, this value is defined, not measured. It never constrains sig figs.

Exact by counting. You can count apples without a measuring device. Ignore this number when applying sig fig rules.

Stipulated. Not a measurement, but not a defined constant either. Treat as 4 significant figures by convention.

So, returning to the opening question: does 2.54 limit your answer when converting 15.5 inches to centimeters? No. The conversion factor 1 inch = 2.54 cm is exact by definition. Only the 3 significant figures in 15.5 matter, giving 15.5 × 2.54 = 39.4 cm.