This number is equal to 1. In other words, "0.999..." and "1" represent the same number. (In other systems, 0.999... can have the same meaning, a different definition, or be undefined.) More generally, every nonzero terminating decimal has two equal representations (for example, 8.32 and 8.31999...), which is a property of all base representations. The utilitarian preference for the terminating decimal representation contributes to the misconception that it is the only representation. For this and other reasons?such as rigorous proofs relying on non-elementary techniques, properties, or disciplines?some people can find the equality sufficiently counterintuitive that they question or reject it. This has been the subject of several studies in mathematics education.
Infinitesimals The standard definition of the number 0.999... is the limit of the sequence 0.9, 0.99, 0.999, ... A different definition involves what Terry Tao refers to as ultralimit, i.e., the equivalence class [(0.9, 0.99, 0.999, ...)] of this sequence in the ultrapower construction, which is a number that falls short of 1 by an infinitesimal amount.
