Exact Numbers and Defined Quantities

Introduction

Every measured number carries some uncertainty, which is why we have spent so much effort on significant figures. But not every number in a calculation is measured. If you count 24 students in a class, there are exactly 24 — not 23.9 or 24.1. If you convert centimetres to metres, the 100 in "1 m = 100 cm" is fixed by definition. These exact numbers behave differently in calculations, and knowing when a number is exact stops you from rounding answers too harshly.

Core explanation

Measured numbers. A reading from a balance, burette or thermometer is limited by the instrument's resolution and by the person reading it. A mass of 12.46 g has four significant figures because the last digit is the first uncertain one. Every measured number has a finite number of significant figures.

Exact numbers. An exact number has no uncertainty at all. You can think of it as having an unlimited number of significant figures: 24 students is really 24.000000… students. There are two main kinds.

1. Counted values. When you count separate, whole objects — beakers on a bench, atoms in a formula, repeats of an experiment — the result is exact, as long as the count is small enough to be done without error. The formula H₂O contains exactly 2 hydrogen atoms and exactly 1 oxygen atom. A titration repeated 3 times has exactly 3 results.

2. Defined quantities. Some relationships are true because scientists have agreed on them. Within the SI system, prefixes are defined: 1 km = 1000 m, 1 mg = 0.001 g, 1 dm³ = 1000 cm³. The conversion between Celsius and kelvin, T(K) = θ(°C) + 273.15, uses a defined offset. Since 2019 several fundamental constants are also exact by definition, including the Avogadro constant (6.02214076 × 10²³ mol⁻¹) and the speed of light (299 792 458 m/s).

Numbers inside formulae. Pure numbers that come from the mathematics of a formula are exact. The 2 in "diameter = 2 × radius" and the ½ in a formula for kinetic energy are exact. So is the divisor when you calculate a mean: dividing a total by 3 readings uses an exact 3.

Why this matters. The significant-figure rules for multiplication and division say the answer should have as many significant figures as the least precise measured value . Exact numbers are never the least precise value, so they are simply ignored when deciding how to round. Likewise, in addition and subtraction the decimal-places rule applies only to measured values.

Not everything that looks exact is exact. "About 30 people attended" is an estimate. A count of millions of blood cells, or a population census, contains counting errors. A conversion such as 1 inch = 2.54 cm is exact by definition, but a rounded constant such as g = 9.8 m/s² is a measured value that has been rounded, so it does limit precision.

Step-by-step reasoning

To decide how to round an answer that mixes exact and measured numbers:

1. List every number used in the calculation. 2. Label each one as measured, counted or defined. 3. Ignore the counted and defined numbers when applying the rounding rule. 4. Apply the significant-figure or decimal-place rule to the measured numbers only. 5. Round the final answer once, at the end.

Visual explanation

Imagine a table with two columns. The left column, labelled "Measured", holds readings such as 25.0 cm³ and 0.462 g, each with a small ± sign beside it. The right column, labelled "Exact", holds 3 repeats, 2 atoms and 1000 mL per litre, with no ± sign at all. Only the left column controls how you round.

Real-world analogy

A recipe says "use 3 eggs and 250 g of flour". The 3 eggs are counted — you cannot have 2.9 eggs by accident. The 250 g of flour is weighed on a kitchen scale and might really be 248 g or 252 g. Any uncertainty in the cake comes from the weighing, not from counting the eggs.

Real-world example

Pharmacists converting a doctor's prescription from milligrams to grams use the defined factor 1 g = 1000 mg. The precision of the dose depends entirely on how accurately the medicine was measured, not on the conversion, because the factor of 1000 is exact and introduces no extra uncertainty.

Why?

Why do exact numbers not limit precision? Uncertainty comes from the limitations of instruments and observers. A defined relationship has no instrument involved, and a small count can be made without any doubt. With nothing uncertain about them, exact numbers cannot make the result less reliable, so they cannot reduce its significant figures.

Common misconception

"The answer to 3 × 12.46 g must be rounded to 1 significant figure because 3 has only 1 significant figure." This is wrong when the 3 is a count. Three identical samples of 12.46 g have a total mass of 37.38 g, keeping four significant figures from the measured value.

Worked example

Question: Three repeat titres are 24.15 cm³, 24.20 cm³ and 24.10 cm³. Calculate the mean titre.

Reasoning: Add the measured values: 24.15 + 24.20 + 24.10 = 72.45 cm³ (two decimal places). Divide by the number of readings, 3. The 3 is a count, so it is exact and does not limit the answer. 72.45 ÷ 3 = 24.15 cm³.

Answer: Mean titre = 24.15 cm³, quoted to two decimal places like the readings.

Quick check

1. In the formula CO₂, is the number 2 measured or exact? Explain. Answer: Exact — it is a count of oxygen atoms in one molecule, so it has no uncertainty.

Exam focus

When an exam asks you to give an answer "to an appropriate number of significant figures", look only at the measured data in the question. Do not round to 1 significant figure because of a count such as "2 moles of product per mole of reactant" or a conversion factor such as 1000.

Advanced insight

The 2019 redefinition of the SI fixed the numerical values of seven constants, including the Planck constant and the Avogadro constant. As a result, the kilogram and the mole are now defined through exact numbers rather than physical objects. Paradoxically, this means some quantities that were once measured with uncertainty — such as the number of entities in one mole — are now exact by agreement.

Summary

Measured numbers always have uncertainty and a limited number of significant figures. Exact numbers — counted values, defined conversion factors, pure numbers in formulae and defined constants — have no uncertainty. When rounding an answer, apply the significant-figure or decimal-place rules only to the measured values and ignore the exact ones.

Practice questions

1. Classify each as exact or measured: (a) 1 L = 1000 mL; (b) a mass of 5.02 g; (c) 6 carbon atoms in C₆H₁₂O₆. Answer: (a) exact, defined; (b) measured; (c) exact, counted. 2. Five identical coins each have a measured mass of 7.50 g. What is their total mass, correctly rounded? Answer: 5 × 7.50 = 37.5 g; the 5 is a count, so the answer keeps the three significant figures of 7.50 g. 3. Convert 0.0356 kg to grams and state how many significant figures the answer should have. Answer: 0.0356 × 1000 = 35.6 g, with three significant figures, because 1000 g per kg is exact. 4. Why is g = 9.8 m/s² not treated as an exact number? Answer: It is a measured quantity that has been rounded to two significant figures, so it carries uncertainty and can limit the precision of an answer.