resulting proofs are more self-contained. Dedekind cuts In the Dedekind cut approach, each real number x is defined as the infinite set of all rational numbers less than x. In particular, the real number 1 is the set of all rational numbers that are less than 1. Every positive decimal expansion easily determines a Dedekind cut: the set of rational numbers which are less than some stage of the expansion. So the real number 0.999... is the set of rational numbers r such that r < 0, or r < 0.9, or r < 0.99, or r is less than some other number of the form Every element of 0.999... is less than 1, so it is an element of the real number 1. Conversely, all elements of 1 are rational numbers that can be written as with and . This implies and thus and since by the definition above, every element of 1 is also an element of 0.999..., and, combined with the proof above that every element of 0.999... is also an element of 1, the sets 0.999... and 1 contain the same rational numbers, and are therefore the same set, that is, 0.999... = 1. The definition of real numbers as Dedekind cuts was first published by Richard Dedekind in 1872. The above approach to assigning a real number to each decimal expansion is due to an expository paper titled "Is 0.999 ... = 1?" by Fred Richman in Mathematics Magazine, which is targeted at teachers of collegiate mathematics, especially at the junior/senior level, and their students. Richman notes that taking Dedekind cuts in any dense subset of the rational numbers yields the same results; in particular, he uses decimal fractions, for which the proof is more immediate. He also notes that typically the definitions allow { x : x < 1 } to be a cut but not { x : x ≤ 1 } (or vice versa) "Why do that? Precisely to rule out the existence of distinct numbers 0.9* and 1. [...] So we see that in the traditional definition of the real numbers, the equation 0.9* = 1 is built in at the beginning." A further modification of the procedure leads to a different structure where the two are not equal. Although it is consistent, many of the common rules of decimal arithmetic no longer hold, for example the fraction has no representation;