Chap 1 The Number System

# We also say that the set r forms a eld under addition

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Unformatted text preview: of all non-zero real numbers forms an abelian group under multiplication. We also say that the set R forms a ﬁeld under addition and multiplication. The property (D) is called the Distributive law. The set of all real numbers also possesses an ordering relation, so we have the Order axioms. ORDER AXIOMS. (O1) For every a, b ∈ R, exactly one of a &lt; b, a = b, a &gt; b holds. (O2) For every a, b, c ∈ R satisfying a &gt; b and b &gt; c, we have a &gt; c. (O3) For every a, b, c ∈ R satisfying a &gt; b, we have a + c &gt; b + c. (O4) For every a, b, c ∈ R satisfying a &gt; b and c &gt; 0, we have ac &gt; bc. Remark. Clearly the Order axioms as given do not appear to include many other properties of the real numbers. However, these can be deduced from the Field axioms and Order axioms. For example, suppose that x &gt; 0. Then by (A4), we have −x ∈ R and x + (−x) = 0. It follows from (O3) and (A3) that 0 = x + (−x) &gt; 0 + (−x) = −x, giving −x &lt; 0. 1.2. The Natural Numbers An important subset of the set R of all real numbers is the set of all natural...
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## This note was uploaded on 02/01/2009 for the course MATH 2343124 taught by Professor Staff during the Fall '08 term at UCSD.

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