[Math Lair] Variables, Parameters, and Constants

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What is the difference between variables, parameters, and constants? All of them are represented in a similar fashion, and all of them can vary (even, on occasion, "constants"), but they vary in different ways.

While the exact difference is not always clear-cut, we can think of variables as being more variable than parameters, which in turn are more variable than constants. Constants might be thought of as universal properties, parameters might be thought of as properties of the system in question, and variables can vary within the system itself, independently of the parameters and constants.

Consider the expression y = ekx. In this expression, x is a variable, k is a parameter, and e is a constant (e is the natural logarithm base). Say that you wanted to investigate this relationship by drawing a graph. In doing so, you would typically select a certain value for k and vary x. If you change k, the result is a different, although similar, graph, while the variation in x is represented within the graph itself. In most circumstances, it wouldn't make a lot of sense to vary k while leaving x constant. It would make even less sense to vary e while leaving k and x constant.

To illustrate the difference between variables, constants, and parameters, consider the following analogy. Say that you're driving a golf ball. There are many different ways you can hit a golf ball. You can hit the ball at different angles, with different club speeds, and so on. These factors are analogous to variables. Another thing that affects your speed is your club selection. Your selection of club is analogous to a parameter. Your selection of club is, in a way, less variable than how you hit the ball; if you're trying to find the best shot, you might consider hitting the ball in a different way while using the same club, but you'd be less likely to consider switching clubs and taking the exact same shot. There are other things that affect your shot as well. These include the wind speed, the course conditions and so on. There isn't much you can do about these (unless perhaps you go to another course, or wait until the wind changes direction). These conditions are analogous to constants.

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