index (mathematics)

A notation which simplifies the writing of products, eg 2 × 2 × 2 × 2 is written 2⁴, where 4 is the index (or exponent); it can be extended to give meaning to fractional, negative, and other indices. When numbers are written in index form, eg 16 = 2⁴, certain laws of indices exist. These are a^m × aⁿ = a^m+n; a^m ÷ aⁿ = a^m?n; (a^m)ⁿ = a^mn; a⁰ = 1; a^?n = 1/aⁿ; a¹/q = ^q?a; aⁿ/q = ^q?aⁿ.

The word index is used in variety of senses in mathematics. The index of a subgroup is the number of its left cosets (which is equal to the number of its right cosets). The index of a Fredholm operator is the dimension of its kernel minus the dimension of its cokernel. The index of a real quadratic form Q is defined (but not always consistently) as p − q where Q can be written as a difference of p squared linear terms and q squared linear terms. The index of a vector field v at an isolated zero is the degree of the map taking points near the zero into the unit sphere. This index is used in the statement of the Poincaré–Hopf theorem which relates the sum of the indices of a vector field to the Euler characteristic of the manifold. Confer fixed point index. "An index relates the value of a variable or group of variables) to a base level, which is often the value on a particular date. The base level is set so that the index produces numbers that are easy to understand and compare. Indices are used to report on a wide variety of variables, including prices and wages, ultraviolet levels in sunlight, and even the readability of textbooks."