2 s 3 2 3 2 1 1 2 0 1 2 1 3 2 e s2 2 3 proper real

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Unformatted text preview: tions, positive and negative. The union of the set of r ational numbers and the set of ir r ational numbers is called the set of real irr ir numbers R . −2 S −3 − 2 −3 − 2 − 1 −1 − 2 0 1 − 2 1 3 − 2 π e S2 2 3 proper Real number umbers: COMPLETENESS pr oper ty of the Real n umber s: Corresponding to every point on the number line we have a unique real number number point and vice versa. Between any two real numbers on the number line each and every point corresponds to a real number. The set of real numbers R is COMPLETE. The real number line is smooth and CONTINUOUS. There are no gaps, breaks or bumps. This Real number line we call the x-axis x-axis. We now see the connection between the set R of real numbers in Algebra and a straight line (a continuous set of points) in Geometry. Each point xi on the x-axis continuous contin points corresponds to a Real number and vice versa. And when we say x1 < x2 for x1, x2 ∈ R the picture from the Geometry point of view is : ALGEBRA GEOMETRY set R ≡ x-axis x1 x2 x1 < x2 13 4. TENDS T O and LIMIT In elementary geometry you learned the definition or meaning of a point point. You also know how to name or label a point point. In Calculus a point on the number line or x-axis corresponds to a real number. Sometimes we know its exact value, e.g. 2 . Sometimes we will know only the approximate value. However, we may denote it by a special name or label or symbol, e.g. S2 , e, π . R egardless of knowing the exact value or not we know which point we are talking about. In Calculus we prefer to use the word instant rather than point We speak of point. the instant 2, or the instant S2 , and so on. In Calculus we have another way to define an instant Before we do this we need to know what an instant. infinitesimal is. In order to give a formal definition of an inf initesimal we need to use two infinitesimal concepts: TENDS TO and LIMIT. TENDS TO: Let a = 0.5 and x = 0.4, 0.49, 0.499, 0.4999, . . . progr essively. It is clear th...
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This note was uploaded on 11/29/2012 for the course PHYSICS 105 taught by Professor Tamerdoğan during the Fall '09 term at Middle East Technical University.

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