To think more clearly about how models attempt to represent measurable quantities, we need to think more precisely about the nature of measurements and the state of a system. The most basic form of a measurement is a simple count. Population measurements are examples of counts, where we are interested in an actual number (integer) of individuals making up the total population. A population is an example of an extensive measurement. This means that if you take two different populations and bring them together, the total population is the sum of the parts. Other measurements quantify physical traits of an object or system, such as physical dimensions (length, area, volume), mass, temperature, or energy. As these physical measurements are not pure counts, they require a reference or standard of measurement.

Length and mass are additional examples of extensive attributes, and this is essential to the ability to measure them with reference to a standard unit of measure. Putting two lengths end-to-end creates a new length that is the sum of the parts. Putting two masses together creates a new mass that is the sum of the parts. So length can be measured according to a standard unit length. Essentially, one counts how many unit lengths (using a ruler) must be added together to obtain the total length represented by the object, including fractions of units. Using a ruler with different unit lengths leads to a different numerical measurement for the same length. Similarly, mass is measured by determining relative to a unit mass by counting how may units are required to obtain a total mass equivalent to the mass of the object, perhaps by using a balance. Using different standard units of mass results in different numerical measurements of the same mass. In a sense, a population can also be thought of extensively in terms of a unit population size. Normally, we think of individuals as the standard unit. But it might be more practical to think in terms of hundreds or millions of individuals. In chemistry, populations consist of enormous numbers of atoms in terms of a standard unit called a mole which represents approximately \(6.022 \times 10^{23}\) (Avagadro's number) individual atoms.

Other quantities are known as intensive attributes. The most practical example for us is the idea of a density, which is the ratio of two extensive measurements. Examples of densities include a mass density (ratio of mass to length, area or volume in a solid), population density (ratio of population count to area), and concentrations (ratio of molecular count to volume in a solution). Because the extensive measurements individually are recorded relative to their individual units of measurement, the intensive measurement must specify the units used. Temperature is also an example of an intensive attribute, although one must study the ideas of thermodynamics to understand what ratio is actually involved.

This leads us to the idea that we must distinguish between a measurement and a physical quantity. For example, the physical quantity might be a length. A measurement of that length provides us with a specific number, namely the number of units required to form the total length. While the quantity itself does not change when different units are used, the measurement does change.

For example, consider a meter stick. When measured with units of meters, the measurement is exactly 1. When measured with units of centimeters, the measurement is exactly 100, a different number. We can not say that \(1=100\) since these are different numbers. To compare the measurements, we must include units, \(1\; \hbox{m} = 100 \; \hbox{cm}\). That is, the quantity of length \(L\) itself is represented as a measurement along with the units, written as an equation, \(L = 1 \; \hbox{m}\) or \(L = 100 \; \hbox{cm}\).

Basic mathematics works with pure numbers, not quantities with units. So as we think about applications of mathematics to physical problems, we must begin thinking about equations in terms of physical quantities that are represented. Basic arithmetic operations of addition and subtraction can only work with quantities of the same basic type. For example, it is not appropriate to add a length to a mass. We can, however, add a length of 1 m to a length of 1 ft to create a new length of a total \(1 \; \hbox{m}+1 \; \hbox{ft}\). Notice that the sum is not 2 since the units are not the same. To get a single measurement, we must express both quantities with the same units.

Unit conversion is the process of expressing a quantity measured with respect to one unit in terms of a different unit. Mathematically, we think of unit conversion as simply multiplying by 1, formed from the ratio of measuring one unit in terms of another unit. For example, 1 inch is exactly 2.54 centimeters. So the ratio \(1 \; \hbox{in}:2.54 \; \hbox{cm}\) is exactly the pure number 1. If we need to convert a measurement from inches to centimeters, we will multiply by this ratio or its inverse. Units cancel much like variables, so we want to use the ratio such that the old units cancel leaving the new units.

The ratios that play a role in unit conversion are called unit conversion factors. Online search engines (like Google and Bing) have built in conversion tools for standard units. However, we will need to consider making conversions between non-standard units. Practicing with basic unit conversion is a practical skill toward understanding the process.

Finally, our last new vocabulary word is dimension, which refers to the type of quantity represented by a measurement, such as length, time, population, mass, etc. Two quantities can be added (or subtracted) or compared (inequalities or equations) only if they are the same dimension. In mathematical models, we often create individual terms that are products or ratios of several quantities. The dimension of the term is the corresponding product or ratio of the dimensions of the individual quantities.

We often represent a measured quantity with a symbol, such as a variable. When we wish to talk about the dimension of the quantity, we put square brackets around the variable. If \(x\) is a variable representing a quantity, then \([x]\) is the symbol representing the dimension of the quantity. It is also convenient to have a symbolic representation of individual dimensions, which necessarily look like new variables but do not have a sense of quantity. Dimensional analysis, unfortunately, is not a frequent topic of discussion. Anytime you use dimensional analysis in your writing, you should explicitly explain all of your notation relating to dimensions.

To illustrate, consider velocity as discussed above. Let \(v\) be the variable representing velocity (an actual quantity). We might use \(L\) as a symbol representing the dimension of length (not an actual quantity of length, just the concept of length) and use \(T\) as a symbol representing the dimension of time (also not an actual quantity of time, but the concept of time), then the dimension of \(v\) is symbolically represented by \([v]\). Our recognition that the dimension of velocity is the ratio of a length to a time is represented by the dimensional equation \begin{equation*}[v] = \frac{L}{T}.\end{equation*} Dimensional equations have no numerical interpretation. So a new velocity that is twice \(v\) is still just a velocity (different value, but same dimension). So we can also say that \begin{equation*}[2v] = \frac{L}{T}.\end{equation*}

A quantity where all dimensions cancel is called dimensionless. Dimensionless quantities will play an important role in our analysis and interpretation of models. This is a rather lengthy discussion that will be broken into smaller pieces in future readings. For now, I'll simply point out that all of the basic mathematical functions (like exponentials, logarithms, sine, cosine, tangent, etc.) are defined only for numbers, so that in a model, they must be used in terms of dimensionless quantities.