Every mathematical problem involves quantities. Some stay fixed throughout a discussion; others change. Recognizing which is which is an essential skill in algebra, physics, and calculus. This section gives precise meanings to the terms constant, variable, and parameter.
Quick Reference
| Term | Meaning | Example |
|---|---|---|
| Quantity | Anything that can be measured | distance, time, temperature |
| Constant | Value that does not change in a given problem | radius $R$ of a fixed circle |
| Variable | Value that changes during the discussion | angle $\theta$ of a moving particle |
| Parameter | Constant that can take different assigned values | slope $m$ and intercept $b$ in $y = mx + b$ |
Constants, Variables and Parameters
What Is a Quantity?
A quantity is anything that can be measured: distance, time, weight, temperature, and so on.
Constants
A quantity whose value remains unchanged throughout a given problem or discussion is called a constant.
For example, if we assume that the temperature does not change during an experiment, then the temperature $T$ is a constant for that experiment.
Variables
A quantity that changes its value during a problem or discussion is called a variable.
Consider a particle moving in a circle of radius $R$. The angle $\theta$ that the ray from the center to the particle makes with the horizontal changes as the particle moves; $\theta$ is a variable that ranges from $0$ to $360°$. But the distance of the particle from the center, $R$, stays fixed throughout, so $R$ is a constant.
Similarly, if we heat water in a container, its temperature $T$ changes over time, so $T$ is a variable in that context. The same symbol can be a constant in one problem and a variable in another, depending on the situation.
Parameters
Parameters (also called arbitrary constants) are constants to which infinitely many values may be assigned, but once assigned, those values are held fixed throughout the investigation.
The equation of a straight line can be written as
$y = mx + b,$where $m$ is the slope and $b$ is the $y$-intercept. Both $m$ and $b$ can take any numerical values, but for a particular line, they are specific fixed numbers. Because they can each represent infinitely many different values, yet remain fixed within any single problem, $m$ and $b$ are parameters of the equation.
The equation $y = mx + b$ has two parameters: $m$ and $b$.
- Setting $m = 2, b = -1$ gives the line $y = 2x - 1$.
- Setting $m = -3, b = 4$ gives the line $y = -3x + 4$.
In each case, $m$ and $b$ are fixed for that particular line, but they vary from one line to another.
The Difference between a Constant and a Parameter
Both constants and parameters are fixed in a given problem. The key distinction is scope:
- A constant refers to a specific number that does not change at all, such as $\pi$ or the speed of light $c$.
- A parameter is a placeholder for a number that could be anything but is held fixed once chosen, such as $m$ and $b$ in $y = mx + b$.
In practice, parameters let us describe an entire family of functions or equations with a single formula, each member of the family corresponding to a particular assignment of the parameter values.
