A constant has multiple different meanings in mathematics depending on the context. In its simplest definition, a constant is a fixed value, such as the number 5. It does not change with respect to the variables in an expression or equation.
In algebra, constants are one of the types of terms used in an equation:
-3 and 4 are constants because they do not change with respect to x, the variable. While 12 is a fixed number, it is a coefficient, not a constant, because it multiplies the variable. For clarification:
- Constant - a term with a fixed value in an equation that is not affected by any changes in the variable.
- Coefficient - a coefficient is a factor that multiplies the variable. It is often a number, but can also be an expression. The terms coefficient and constant are sometimes used interchangeably, particularly as parts of a function, as a constant can be considered a coefficient where it multiplies the variable raised to the 0th power (eg. 5x0 = 5)
- Variable - the unknown in the equation that is solved for.
Constants as variables
When written as a variable, a constant is commonly denoted as c to represent a fixed value. This terminology may be confusing because of the use of the term "variable," but it all it means is that c is a variable that can represent any fixed value. For example, the general form of the quadratic equation can be written as:
ax2 + bx + c
We know that c is a constant (more specifically the constant term of the polynomial) in that it does not vary with respect to x, but at the same time, it is a variable because c can take on any number of fixed values. In the above, a and b are coefficients, and the unknown variable in the expression is x. The variables a, b, and c, are more specifically referred to as parameters.
The term constant may also be used to refer to constant functions. A constant function is a function whose output is the same regardless of the input value:
f(x) = 5
This is a constant function because the main variable is x, and the constant 5 is not dependent on x. No matter what value we input for x, the output of the function will be 5, since x does not appear in the function.
The term mathematical constant is used to describe fixed, well-defined numbers. Some of these include:
- π = 3.1415926536...
- i2 = -1
- e = 2.7182818284...
- Pythagoras' constant: √2 = 1.4142135623...
- Theodorus' constant: √3 = 1.7320508075...
- Golden ratio: φ = 1.618033...
There are many other mathematical constants, these are just a few of them that are relatively commonly used.