{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

axioms - Math 300 Introduction to Mathematical Reasoning...

Info icon This preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
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 π , e , and 2. Integer: Intuitively, an integer is a whole number (positive, negative, or zero). Zero: The number zero is denoted by 0 . 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 ).
Image of page 1

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern