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Unformatted text preview: Introduction to Mathematical Reasoning Math 300 Fall 2009 The Real Numbers and the Integers UNDEFINED TERMS To avoid circularity, we cannot give every term a rigorous mathematical definition; we have to accept some things as undefined terms. For this course, we will take the following fundamental notions as undefined terms. You already know what these terms mean; but the only facts about them that can be used in proofs are the ones expressed in the axioms listed below (and any theorems that can be proved from the axioms). Real number: Intuitively, a real number represents a point on the number line, or a (signed) distance left or right from the origin, or any quantity that has a finite or infinite decimal representation. Real numbers include integers, positive and negative fractions, and irrational numbers like , e , and 2. Integer: Intuitively, an integer is a whole number (positive, negative, or zero). Zero: The number zero is denoted by . One: The number one is denoted by 1 . Sum: The sum of two real numbers a and b is denoted by a + b . Product: The product of two real numbers a and b is denoted by ab or a b or a b . Less than: To say that a is less than b , denoted by a < b , means intuitively that a is to the left of b on the number line. DEFINITIONS In all the definitions below, a and b represent arbitrary real numbers. The set of all real numbers is denoted by R , and the set of all integers is denoted by Z . Real numbers a and b are said to be equal , denoted by a = b , if they are the same number. The numbers 2 through 9 are defined by 2 = 1 + 1, 3 = 2 + 1, etc. The decimal representations for other numbers are defined by the usual rules of decimal notation: For example, 23 is defined to be 2 10 + 3, etc. The additive inverse or negative of a is the number- a that satisfies a + (- a ) = 0, and whose existence and uniqueness are guaranteed by Axiom 9. The difference between a and b , denoted by a- b , is the real number defined by a- b = a + (- b ). If a negationslash = 0, the multiplicative inverse or reciprocal of a is the number a- 1 that satisfies a a- 1 = 1, and whose existence and uniqueness are guaranteed by Axiom 10....
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