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PART I. THE REAL NUMBERS - UH

PART I. THE REAL NUMBERSThis material assumes that you are already familiar with the real number system and the represen-tation of the real NUMBERS as points on the real THE NATURAL NUMBERS AND INDUCTIONLetNdenote the set of natural NUMBERS (positive integers).Axiom:IfSis a nonempty subset ofN, thenShas a least element. That is, there is anelementm Ssuch thatm nfor alln :A set which has the property that each non-empty subset has a least element is said to bewell-ordered. Thus, the axiom tells us that the natural NUMBERS are a subset the following properties:1. 1 S, Simpliesk+1 S,thenS= :SupposeS6=N.

Consequently, q2 =2k2 and so q is even. Thus p and q have the common factor 2, a contradiction. Other examples of irrational numbers are √ m where m is any rational number which is not a perfect square, 3 √ m where m is any rational number which is not a perfect cube, etc. Also, the numbers π and e are irrational. Definition 2. Let S be a ...

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