#  1.1无穷小是一个数

 1.1无穷小是一个数
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 袁萌  陈启清  10月1日
 附件？
 Chapter 1
 Generalities（概论）
 1.1 Inﬁnitesimals and other nonstandard numbers: getting acquainted
 An inﬁnitesimal is a number that is smaller than every positive real number and is larger than every negative real number, or, equivalently, in absolute value it is smaller than 1/m for all m ∈ IN = {1,2,3,...}. Zero is the only real number that at the same time is an inﬁnitesimal, so that the nonzero inﬁnitesimals do not occur in classical mathematics. Yet, they can be treated in much the same way as can the classical numbers. For example, each nonzero inﬁnitesimal ε can be inverted and the result is the number ω = 1/ε. It follows that | w |> m for all m ∈ IN, for which reason ω is called (positive or negative) hyperlarge (or inﬁnitely large). Hyperlarge numbers too do not occur in classical mathematics, but nevertheless can be treated like classical numbers. If, for example, ω is positive hyperlarge, we can compute √ω, ω/2, ω −1, ω + 1, 2ω, ω2, etc., and we have (ω−1) + (ω + 1) = 2ω, (ω−1)•(ω + 1) = ω2 −1, etc. Also, for all m ∈ IN, m < √ω < ω/2 < ω−1 < ω < ω + 1 < 2ω < ω2 giving seven diﬀerent hyperlarge numbers. The positive hyperlarge numbers must not be confused with inﬁnity (∞), which should not be regarded a number at all, and which anyway does not satisfy these inequalities, except the ﬁrst one.
 Regrettably, there does not seem to exist a synonym for ‘hyperlarge number’ that would make a nice pair with ‘inﬁnitesimal’, so let us introduce the synonym ‘hypersmall number’ for the latter.
 If ε is hypersmall, if δ too is hypersmall but nonzero, and if ω is positive hyperlarge, so that −ω is negative hyperlarge, we write, ε ' 0, δ ∼ 0, ω ∼∞, −ω ∼−∞ respectively.
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 It would be wrong, of course, to deduce from ω ∼∞ that the diﬀerence between ω and ∞, or that between −ω and −∞ would be hypersmall. Given any x ∈ IR, x 6= 0, and any δ ' 0, let t = x + δ, then, ε <| t |< ω, for all ε ∼ 0 and all ω ∼∞. The number t is called appreciable (as it is not too small and not too large).
 Three nonoverlapping sets of numbers (old or new) can now be presented:
 a) the set of all inﬁnitesimals, to which zero belongs, b) the set of all appreciable numbers, to which all nonzero reals belong, and c) the set of all hyperlarge numbers, containing no classical numbers at all.
 Together these three sets constitute the set of all numbers of ‘real nonstandard analysis’. This set, which clearly is an extension of IR is indicated by,
 ∗IR and is called the ∗-transform of IR. The elements of ∗IR are called hyperreal. The use of the preﬁx ‘hyper’ here is not entirely defendable, as, say, 5, which obviously is an element of ∗IR, is just an ordinary real.
 Abbreviating hypersmall, appreciable, and hyperlarge to s, a and l, respectively, and assuming that x and y are positive numbers, for addition and multiplication the following holds, y\x s a l y\x s a l s s a l s s s ? a a a l a s a l l l l l l ? l l addition multiplication
 where the quotation marks stand for s or a or l. Examples for the lower left quotation mark are x ∼ 0 and y = √x−1, or 1/x, or 1/x2. For x−y the results are the same as for x + y (if still x,y > 0), except that if both x and y are appreciable, then x−y is either hypersmall or appreciable, and that if both x and y are hyperlarge, then x−y is either hyperlarge (positive or negative), or appreciable, or hypersmall, as is shown by the following examples: y = x/2, or 2x, or x−1, or x + ε, with ε ' 0. If a number is not hyperlarge it is called ﬁnite or limited.
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 Remark: Elsewhere in the literature, any element of ∗IR is called ﬁnite. Clearly, t is ﬁnite if and only if t = x + ε for some x ∈ IR and some ε ' 0. Given such a t, both x and ε are unique, for, x + ε = y + δ, x,y ∈ IR, ε,δ ' 0 implies that x−y = δ−ε ' 0, so that (as x−y ∈ IR), x−y = 0, hence x = y and ε = δ. By deﬁnition x is called the standard part of t, and this is written as,
 x = st(t).
 The standard part function st provides an important (mainly one-way) bridge between the ﬁnite numbers of nonstandard analysis and the classical numbers. Trivially, if t is itself a classical number, then st(t) = t.
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