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alittlebitofcalculus-pdf-january2011-111112001007-phpapp01

# This is denoted by t2 t1 and each instant on the way

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Unformatted text preview: 2 → t1 . And each instant on the way to continuous contin instant t1 corresponds to a definite real number in R. We may even let t2 coincide with t1 . This is expressed in Analysis terminology as: Analysis Anal LIMIT t2 → t 1 After we develop the concept of the CONTINUOUS INFINITESIMAL δt, the t2 → t1 may be used in the calculation as : t2 = t1 + δt And the concept of t2 coinciding with t1 may be expressed using the word LIMIT with δt in the calculation as: L I M I T { expression involving t1 and t2 } ≡ L I M I T { expression involving t1 and δt } t2 → t1 δt → 0 particular eneral We then extend these concepts from a par ticular instant t1 to a general instant t par simply by dropping the subscript. 7 2. NUMBERS AND THE NUMBER LINE Natur number tural umbers: Natur al number s: N = { 0, 1, 2, 3, . . . } * 1. We note that between any two natural numbers there are FINITELY many natural numbers. e.g. between 1 and 5 there are exactly three natural numbers: 2, 3 and 4. 2. We cannot exhaust counting the natural numbers because there are infinitely many of them. But we can COUNT or ENUMERATE them in an orderly manner without missing any. This property we call COUNTABLE or ENUMERABLE. A set of numbers with these two properties is called DISCRETE. The set of natural numbers N is DISCRETE. We can draw an infinite line and represent the natural numbers on this line in an orderly manner as individual points spaced evenly apar t. 0 1 2 3 Integers: Integer s: Z = { . . . −3, −2, −1, 0, 1, 2, 3, . . . } 1. Between any two integers there are FINITELY many integers. 2. We may COUNT the intergers in an orderly manner without missing any as follows: 0, +1, −1, +2, −2, +3, −3, . . . . The set Z is COUNTABLE. Hence the set Z is DISCRETE. To continue with our representation of numbers on an infinite line we can represent the integers in an orderly manner as: − 3 − 2 − 1 0 1 2 3 with the line extending to infinity,∞, in both directions. * Note: In some contexts zero is not consid...
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