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. Understanding which numbers have uncertainty and which do not is fundamental to correctly applying significant figure rules.

The source of a number determines how we treat its precision.

The Categories of Numbers

Numbers in chemistry calculations fall into distinct categories based on their source. Each category is treated differently when applying significant figure rules.

1. Measured Numbers

A measured number is any quantity determined using a measurement device.

• Source: Ruler, graduated cylinder, balance, thermometer, stopwatch, or any other instrument.
• Key Property: Has uncertainty in the last reported digit.
• Treatment: Finite significant figures as determined by the instrument 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.

• Source: Direct count of individual, indivisible items.
• Key Property: Zero uncertainty.
• Treatment: Infinite significant figures.

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

3. Exact Numbers by Definition

These numbers are part of formally defined relationships between units. The relationship is established by universal agreement, not measurement.

• Source: Defined conversion factors or relationships.
• Key Property: Perfect, unchanging standard.
• Treatment: Infinite significant figures.

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

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)
• “Add exactly 15.0 mL of water.” → Stipulated (specified by the procedure)

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

A. Given Measured Values

Values that represent physical measurements. The problem provides the result of a measurement.

• Source: Mass, volume, density, temperature, concentration, or other measured quantities.
• Key Property: Carries the uncertainty from the measurement.
• Treatment: 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 that are set by rule, agreement, or specification rather than measurement. These occupy a category distinct from both measured and exact numbers.

• Source: Prices, wages, specifications, or problem-specific fixed quantities.
• Key Property: Not measured, but also not exact like defined conversion factors.
• Treatment: 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 problems. But understanding the category helps you recognize when standard significant figure reasoning requires extra thought.

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 occupy a unique category: - They are not measurements (no instrument uncertainty) - They are not exact like conversion factors (not infinite precision) - They are fixed values with stated precision

The Rule: When your calculation involves stipulated values, determine the precision of your answer by treating each stipulated value as having exactly 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 approach respects that while these values are fixed in their context, their stated form indicates the appropriate precision for reporting your result.

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