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Notice that, for convenience, we have de ned multiplication of negative numbers just as we did in high school The reason is that the de nition that we use for the product of two positive numbers cannot work when one of the two factors is negative (check this as an exercise) We have said what the additive identity is in this realization of the real numbers Of course the multiplicative identity is the cut corresponding to 1, or 1 {t Q : t < 1} We leave it to the reader to verify that if C is any cut then 1 C = C 1 = C It is now routine to verify that the set of all cuts, with this de nition of multiplication, satis es eld Axioms M1 M5 The proofs follow those for A1 A5 rather closely For the distributive property, one rst checks the case when all the cuts are positive, reducing it to the distributive property for the rationals Then one handles negative cuts on a case-by-case basis The two properties of an ordered eld are also easily checked for the set of all cuts We now know that the collection of all cuts forms an ordered eld Denote this eld by the symbol R and call it the real number system We next verify the crucial property of R that sets it apart from Q Theorem 511 The ordered eld R satis es the least upper bound property Proof: Let S be a subset of R which is bounded above That is, there is a cut such that s < for all s S De ne S =
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Then S is clearly nonempty, and it is therefore a cut since it is a union of cuts It is also clearly an upper bound for S since it contains each element of S It remains to check that S is the least upper bound for S In fact if T < S then T S and there is a rational number q in S \T But, by the de nition of S , it must be that q C for some C S So C > T , and T cannot be an upper bound for S Therefore S is the least upper bound for S, as desired We have shown that R is an ordered eld which satis es the least upper bound property It remains to show that R contains (a copy of) Q in a natural way In fact, if q Q we associate to it the element (q) = Cq {x Q : x < q} Then Cq is
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obviously a cut It is also routine to check that (q + q ) = (q) + (q ) and (q q ) = (q) (q )
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Therefore we see that is a ring homomorphism (see [LAN]) and hence represents Q as a sub eld of R
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56 The Nonstandard Real Number System
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Isaac Newton s calculus was premised on the existence of certain in nitesimal numbers numbers that are positive, smaller than any standard real number, but not zero Since limits were not understood in Newton s time, in nitesimals served in their stead But in fact it was just these in nitesimals that called the theory of calculus into doubt More than a century was expended developing the theory of limits in order to dispel those doubts Nonstandard analysis, due to Abraham Robinson (1918 1974), is a model for the real numbers (that is, it is a number system that satis es the axioms for the real numbers that we enunciated in Sec 55) that also contains in nitesimals In a sense, then, Robinson s nonstandard reals are a perfectly rigorous theory that vindicates Newton s original ideas about in nitesimally small numbers
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