Beyond Measurement: A Guide to Exact, Counted, and Stipulated Values

The Problem: Why Don’t All Numbers Have Uncertainty?

You have learned that every measurement comes with uncertainty, which we communicate using significant figures. You carefully measure the length of a block as 15.5 inches (3 significant figures). Your instructor tells you to convert this to centimeters. The problem states, “Use the conversion factor 1 inch = 2.54 cm.”

You perform the calculation: 15.5 inches × 2.54 cm/inch = 39.37 cm.

A question immediately arises: Should your final answer be limited by the three significant figures in 2.54? Or the three in 15.5? Or something else entirely?

This page will clarify these common and important questions. The key takeaway is simple: the source of a number determines how we treat its precision. We will explore the different types of numbers you will encounter in science.

The Categories of Numbers

To master significant figures, you must first learn to identify the source of every number you use. Is it from a measurement device, a simple count, a universal definition, or a value given to you in a problem?

1. Measured Numbers (The Baseline)

This is the category you are most familiar with. A measured number is any quantity determined using a measurement device.

Source: A ruler, graduated cylinder, digital balance, thermometer, stopwatch, etc.
Key Property: Every measured number has a degree of uncertainty, which is always in the last reported digit.
Treatment: The significant figure rules you have learned apply directly.

Examples: 12.105 g, 25.3 mL, 37.2 °C

2. Exact Numbers by Counting

This is the simplest type of exact number. When you count discrete, whole items, there is no uncertainty or estimation involved.

Source: Direct enumeration of individual, indivisible items.
Key Property: There is zero uncertainty.
Treatment: Counted numbers have an infinite number of significant figures.

Examples: 35 students, 4 atoms of hydrogen in a methane molecule (CH₄).

3. Exact Numbers by Definition

These are numbers that are part of a formally defined relationship, often for units of conversion. The relationship is a matter of universal agreement and is not subject to measurement uncertainty.

Source: A definition that establishes an exact relationship between two units.
Key Property: The relationship is a perfect, unchanging standard.
Treatment: Defined numbers have an infinite number of significant figures.

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

4. Handling Numbers in Word Problems

This is the most nuanced area and requires careful thought. These are numbers provided to you within a problem’s context, but they must be split into two fundamentally different groups.

The Litmus Test

To tell these two groups apart, ask yourself this key question:

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

A. Given Measured Values

These are values provided in a problem that represent a physical measurement. The problem author has simply “done the measurement for you” and is giving you the result.

Source: A value representing a measurement of mass, volume, density, temperature, concentration, etc.
Key Property: The value carries the uncertainty from the original (hypothetical) measurement.
Treatment: It has a finite number of 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 by Agreement

These are values that are exact by their nature within a specific context. They are not based on physical measurement but are set by a rule or agreement.

Source: A price, a wage, a recipe, a technical specification, etc.
Key Property: Within its context, the value is treated as having no uncertainty.
Treatment: It is treated as having infinite number of significant figures for that specific calculation.

Examples:

Financial Rates: A wage of $8.25 per hour.
Recipes & Specifications: “Take one 500 mg tablet.” or “Add 2 cups of sugar.”
Problem-Specific Assumptions: “For this problem, assume there are exactly 5280 feet in a mile.” (Here, a problem explicitly tells you to treat a number as exact).

Calculations in Practice

The Standard Case: Calculations with Measured Numbers

This follows the standard rules. If a calculation contains at least one measured number (either one you take yourself or a Given Measured Value), the final answer’s precision is limited by the least precise measured number.

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?

Classify the numbers:
- 2.50 cm³: Measured (3 sig figs).
- 7.87 g/cm³: Given Measured Value (3 sig figs).
Calculate: 2.50 cm³ × 7.87 g/cm³ = 19.675 g.
Round: The answer is limited by the 3 sig figs in both inputs.
- Final Answer: 19.7 g.

The Special Case: What Happens When There Are No Measured Numbers?

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?

Here, both numbers are Stipulated by Agreement. $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...), which does not seem reasonable.

The Guiding Principle for Non-Measured Calculations

When a calculation involves only Stipulated by Agreement and/or Exact numbers, the precision of the result is determined by applying the standard significant figure rules to the stated precision of the input values.

For multiplication and division, round the final answer to the same number of significant figures as the stipulated input with the fewest significant figures.
For addition and subtraction, round the final answer to the same place value as the last significant digit of the least precise stipulated input.

This rule honors the fact that while stipulated values are treated as exact for the calculation itself, their stated form still implies a certain level of precision that should be respected in the final result.

Let’s apply this guiding principle to our problem:

Analyze Stated Precision:
- $21.46 is stated with 4 significant figures.
- $8.25 is stated with 3 significant figures.
- This is a division problem, so the least number of significant figures among the stipulated inputs is 3.
Perform the calculation: 2.6012121212... hours
Round the final answer: We round the result to 3 significant figures.
- Final Answer: 2.60 hours.

Summary Table

The Final Takeaway

Always be a critical thinker: question the origin of every number you use. Is it a real-world measurement limited by an instrument, or is it a value that is counted, defined by a universal standard, or stipulated for a specific problem? Answering this question is the key to correctly applying the rules of significant figures and reporting your results with the honesty and precision that science demands.