Variables, constants, and parameters

In our previous articles (links in the Prerequisites above), we've discussed the ways that the use of mathematical symbols in science differs markedly from what you typically see in introductory math classes. For one, we use many more symbols. For another, the values our symbols stand for are typically built from measurements and have units. 

A challenging difference is how we think about whether symbols are intended to change or not: the difference between variables, constants, and parameters.

Variables, independent and dependent

In introductory math classes, especially up through "the calculus of one variable",  equations typically have one variable or two -- perhaps x or y or t. They often are written with numbers, like this

$$y=f(x)=2x^2 + 4x - 7$$

The two symbols $y$ and $x$ are variables. They are place holders that can take on a range of allowed numerical values (in this case, any real number). Since we can choose the value of $x$ freely, we call $x$ an independent variable. Since the the value of $y$ is determined by (depends on) which value of $x$ we choose, we call $y$ a dependent variable. A graph of our equation is shown at the right.

Constants and parameters

Sometimes in math classes the equations will not involve specific numbers but place holders given by a letter — typically from the beginning of the alphabet like a, b, or c. We might then write our equation this way:

$$y=f(x)=ax^2 + bx +c$$

When a, b, and c have the values 2, 4, and -7, we get the same function and the same graph. The symbols a, b, and c stand for specific numbers: they are constants.

But notice that now, in the program, a, b, and c are specified by sliders. We can change those sliders and see how the function and its graph changes for different values of a, b, and c.

(These graphs were set up with the Desmos Graphing Calculator. It's easy to use. Set these functions up for yourself and see what happens when you change the sliders.) Our constants have become parameters -- symbols that are thought of as constants but that can change so we can compare different situations and conditions.

Clearly, having parameters instead of constants lets you see how an equation changes when the "constants" in the equation change. For most equations in science, it's much better to think of the "constants" in our equations as parameters — constants that we are willing to consider changing. On some equations, we might even take a derivative with respect to a parameter to see how something would change if we moved one of the sliders. (For a nice example of this, see the page Repackaging: Changing physics equations to math (and back).)

Joe Redish 7/12/18