Part IV. ALGEBRAIC REASONING
11. Generalizing Relationships
We have encountered many instances in this book where solving a problem numerically required numbers that we didn’t have. We often don’t know all the numbers needed to solve real problems. In some cases, the simplest way to overcome this issue is to estimate or guess a value. However, in many other cases, the value of an important quantity isn’t constant in space or time – our lack of knowledge is not a reflection of uncertainty in measurement. Instead, there is a systematic variation in the real value of a quantity and we need to allow for those changes. Under these circumstances we need to treat these quantities as variables that have unknown numerical values. Perhaps we have some idea of how large or small the numerical values can get,[1] but within these limits, the variable can take on any value.
In this new world of uncertainty, we have the tools of algebra at our disposal. At least in parts of the problem-solving process, this can be disorienting as we have to carry symbols rather than reducible values through any operations that we find necessary. However as we’ll see shortly, performing symbolic manipulations as a means to solving problems can lead to versatile and reusable solutions. What we have done prior to now can be called specializing, where we seek particular numerical values in every calculation when possible. The alternative, which you’ll recognize as a stepping-stone for algebra, is to generalize.
Writing algebraic relationships can seem to be hocus pocus at first. However, the mystique fades a bit when we remind ourselves that mathematical relationships are little more than formal logical statements. By carefully assembling what we know about quantities of interest, striving to sustain generality, and following a few tips, we can begin to use algebraic reasoning as a powerful tool for creativity and sense-making.
Writing algebraic expressions
- Identify the relevant variables and constants
- Introduce descriptive notation for each quantity
- List what is already known about each variable, using expressions with symbolic notation when possible
- Look for ways to set expressions equal to one another based on what you know; are there two ways to define the same quantity using the variables of interest?
- Guess or infer unknown relationships
- Write and simplify equated expressions as a symbolic equation
- Check for dimensional or unit consistency
When we express and manipulate equations with symbolic variables, we are doing algebra. When we state systematic relationships between symbolic variables, we’re using functions. Functions can describe derived, hypothetical, or observed relationships, depending upon how we arrive at them.
11.1 Families of Functions
In the natural and environmental sciences, a few families of functions can be used to describe relationships between key variables of interest. We will explore the most prevalent of these kinds of functions, examining