# What is a base in mathematics?

The term “base” has many meanings in the English language, including several that apply to the field of mathematics. When talking about sets, bases are the open sets whose union forms an abstract entity called a topological space. In geometry, the base represents the side of a polygon or polyhedron that is perceived as its bottom; when referring to an isosceles triangle, the base is the side that differs in length from the other two (thus, the base angles include the side that is thought of as the base). Algebraists also use the term base to describe either the number used with an exponent to create a power, such as 34 = 81; or to write the same number as a subscript to a logarithm, such as log3 81 = 4. (For more information about logarithms, see “Algebra.”)

One of the more familiar uses of the term base in mathematics deals with our numbering system, in which a base is a natural number whose powers are added to produce a specific number. For example, using 10 as a base, the number 2583.789 is actually (2 × 103) + (5 × 102) + (8 × 101) + (3 × 100) + (7 × 10-1) + (8 × 10-2) + (9 × 10-3).

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