Download Analysis with ultrasmall numbers by Hrbacek K., Lessmann O., O'Donovan R. PDF

By Hrbacek K., Lessmann O., O'Donovan R.

ISBN-10: 149870266X

ISBN-13: 9781498702669

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Our first principle merely records this assumption formally. Existence Principle There exist ultrasmall real numbers. Exercise 1 (Answer page 241) (1) If x is such that 0 < |x| < |ε| and ε is ultrasmall, then x is ultrasmall. Basic Concepts 9 (2) If x is such that |M | < |x| and M is ultralarge, then x is ultralarge. 3 First Principles In this section we develop systematic rules for computing with ultrasmall and ultralarge numbers. Before starting on this project, we need some principle that would connect the notion of observability with the traditional mathematical concepts.

Q . It follows that the statement is true in some context where the parameters f and a are observable, if and only if it is true in every context where the parameters f and a are observable. The last example is of great importance, and applies generally. By our convention, if a theorem does not specify the context of the relative concepts used in it, then we understand this context to be that of its parameters. By Stability and Exercises 18, 19, the theorem is then true in every context where the parameters are observable.

Hence 1010 , 5, sin(π/7) and log 35 are observable. More generally, the statement used in a definition may depend on one or more additional parameters x1 , . . , xn ; the name given to the unique object should then indicate the parameters on which it depends; thus C(x1 , . . , xn ) could be used. C can be viewed as an operation, defined for those values of x1 , . . , xn for which such unique object exists. The above argument applies to concepts that depend on parameters: Any mathematical object uniquely defined from parameters x1 , .

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