An exact number is a value that is known with complete certainty. In other words, an exact number has zero uncertainty and an infinite number of significant figures. An exact number cannot be simplified or reduced.
Examples of Exact Numbers
Examples of exact numbers included counted numbers, defined units, and defined unit conversions. Many exact numbers are integers, but some are decimals. Here are specific examples.
- The number of people in a family
- The number of apples in a bag
- Speed of light in a vacuum (exactly 299792458 m/s)
- The number of feet in one mile (exactly 5280)
- The number of centimeters in a meter (exactly 100)
- Number of protons in a carbon atom (6)
- Minutes in one hour (60)
- Avogadro’s number (6.02214076 × 1023)
- Zero
- The number of centimeters in one inch (exactly 2.54)
- Number of pages in a book
Not all units and their conversions are defined, so determining whether or not a specific one is an exact number can be tricky. As of 2019, all of the SI base units are defined, so they are all exact numbers. Some imperial units are also defined according to the SI base units, so they are also exact numbers. In the past, many of these values depended on empirical measurements and were inexact. For example, the speed of light and the mass of a kilogram were measured values until relatively recently.
Examples of Inexact Numbers
Other numbers contain uncertainty. Measured values, estimates, rounded numbers, and some unit conversions are inexact numbers. Here are some examples:
- Your weight on a scale
- Distance between the North and South pole
- Using 6.022 x 1023 for Avogadro’s number
- Approximating the numbers of pounds per kilogram as 1 kg = 2.2 lb
- Number of hours you sleep (because it’s a measured value)
Exact Numbers in Chemistry
Several relationships in chemistry use exact numbers.
- The molecular formula lists the exact number and type of atoms in a single molecule.
- The empirical formula is the simplest whole number ratio of atoms in a molecule.
- An atom contains an exact number of protons, neutrons, and electrons.
- The mole ratio between any reactants and products in a balanced equation is an exact number.
- The oxidation state and number of valence electrons of an atom are exact numbers.
- Usually, the number of atoms in a molecule is an exact number. Exceptions include some polymers.
Exact Numbers, Uncertainty, and Significant Figures
Exact numbers are important in scientific calculations because they contain no uncertainty and have infinite significant figures. So, even though an exact number may be a single digit, it doesn’t limit the significant figures in a calculate to one digit. Be careful identifying exact numbers. They do not always include decimal points.
For example, you report the mass of two moles of carbon atoms. In this case, the “2” is an exact number. It’s written as “2” and not “2.0” or “2.00”. The mass of one mole of carbon atoms is a rounded number (12.01 g/mol) that contains four significant figures. Your answer, in turn, has four significant figures and not one.
The mass of two moles is 2 x 12.01 = 24.01 grams.
Sometimes the exact number has more digits than other numbers in a calculation. When this occurs, remember the value with the least number of significant digits determines the number of significant figures in your answer.
Is Pi an Exact Number?
Pi (π) is an exact number. However, it’s a tricky example because it’s also an irrational number. You can’t write pi as a common fraction or a decimal value without introducing uncertainty.
References
- Bureau International des Poids et Mesures (2019). The International System of Units (SI) (9th ed.).
- Myers, R. Thomas; Oldham, Keith B.; Tocci, Salvatore (2000). Chemistry. Austin, Texas: Holt Rinehart Winston. ISBN 0-03-052002-9.