What is base in mathematics?

In common terms it is the whole unit at which we add a one in the next available digit place to the left and reset the numbers to the right of it to zero.

Bases are a convenience, and by historical convention and perhaps convenience of having 10 easily accessible digits we use the base 10 when counting. Computers use the base 2. The ancient Babylonians used the base 60.

Mathematically a base is the radix that is raised to a particular exponent to generate any number. E.g. 10^2 = 100 and 10^1.5 = 31.622776. Here 10 is the radix, 2 and 1.5 are the exponents.

Mathematically all bases are equivalent.

Mathematics often makes use of the natural base because of the convenience its properties.