**Constants** can tautologically be defined as those entities which are not variable. More specifically, a constant is an unchanging value.

All numbers are constants. 5 will always be 5, just as 339 will always be 339.

There does exist constants which are not written numerically, but whose numerical value is represented with letters (otherwise confused with variables if it werent for the fact that they arent variable).

When non-numbers are used in place for numbers, these are falsely called "variables" only in that they are substitutions and representations for constant values, but aren't actually representative of (potentially) changing parameters as the typical variable would be.

Some of the more common non-numerical constants are Euler's number (e), Pi (π), and the imaginary unit * i*, all of which represent numbers or abstract mathematical concepts.

Other constants are defined and utilized in the physical sciences, like 'c' for the speed of light, 'G' for the gravitational constant, etcetera.

Constants have specific values that never change, as long as that meaning is implicitly or explicitly understood within the context of a mathematical problem.

Of course 'e' need not always be representative of Euler's constant, 'c' need not always be representative of the speed of light. But these contextual facts must necessarily be plainly obvious or implicitly or explicitly stated.

The alternative, aside from the use of simple numbers, is that letters and other such symbols are variables. Variables are changing, or otherwise changeable, such that their value is not known.

## Special properties of constants

In calculus, the derivative of any constant is 0. In Leibniz notation,

- $ \frac{d}{dx}C=0 $

Which agrees with the requirement that a constant does not change. The derivative of a constant multiple is the constant itself:

- $ \frac{d}{dx}Cx=C $

Special properties of limits also exist for constants:

- $ \lim_{n \to a}C=C $

and

- $ \lim_{n \to a}C\pm f(x)=C\pm\lim_{n \to a}f(x) $
- $ \lim_{n \to a}C\,f(x)=C\lim_{n \to a}f(x) $
- $ \lim_{n \to a}\frac{f(x)}{C}=\frac{\lim_{n \to a}f(x)}{C} $,

Which follow from the more general properties of limits.