ࡱ>  " )bjbj .,    $/$ S    giiiiii iQi  F  gg:q, p#  , S0R&!&!&&!4ii@&! : Things to look at: Paper is full-adjusted. Paper is double-spaced. The font is 12-point Times New Roman. There is a title page. There is an abstract page. There is an introduction. There is a conclusion. Sections are numbered. Definitions, theorems, and examples are numbered by section. Theorems are italicized. Definitions, theorems, and examples are marked by a bold-faced word. Words that are being defined are italicized. Definitions and theorems, when taken verbatim, are referenced. References are numbered, and the number of the reference is how they are referred in the paper. Irrational Numbers Daniel Dreibelbis MAS 4932 March 12, 2008 Abstract This paper describes the properties of irrational numbers. We begin by defining what an irrational number is, distinguishing it from rational numbers. Examples of commonly used rational numbers are given. We then give a simple proof that the square root of 2 is an irrational number. If this was a twenty page paper, I would have more things to say here, and my abstract would list all of the major goals of my paper. As is, I only have two goals, so I will end this abstract now. 1) Introduction Irrational numbers have been used throughout mathematics. They are really special. Legend has it that Pythagoras and his followers were the first mathematicians to recognize the existence of irrational numbers. This was quite a problem for Pythagoras, because he and his school believed that all distances would be rational multiples of each other. In fact, the Pythagorean Theorem was used to prove the existence of an irrational number, and it destroyed their entire system of belief. In Section 2, we give the definitions of rational and irrational numbers, and we give some examples of each. In Section 3, we give a proof that  EMBED Equation.3  is irrational. 2) Definitions Throughout this paper, we only consider real numbers, as opposed to complex numbers. We first need to define a rational number: Definition 2.1. [1] A rational number is a number that can be written as p/q, where p and q are integers. Example 2.2. The following numbers are all rational: 2, 0, -1, 2/3, 100/102, 234.45, -4.123423523, etc. As the name implies, irrational numbers will the numbers that are not rational: Definition 2.4. [2] An irrational number is a number that cannot be written as p/q with p and q integers. Example 2.5. [2] The following numbers are all irrational: , e,  EMBED Equation.3 ,  EMBED Equation.3 . 3) Square root of 2 There is a very nice proof showing that  EMBED Equation.3  is irrational. Theorem 3.1 [3] The number  EMBED Equation.3  is irrational. Proof: Assume that  EMBED Equation.3  is not irrational, i.e., assume that  EMBED Equation.3  is rational. The there exists relatively prime integers p and q such that  EMBED Equation.3 . Rearranging and squaring both sides, our equation becomes 2q2 = p2 Since the left-hand side is even, that implies that the right-hand side is even, which implies that p is even. But if p is even, then p = 2k for some integer k. Plugging this into our equation, we get 2q2 = 4k2 So q2 = 2k2, which implies q2 is even, which implies q is even. Hence 2 divides both p and q, which is a contradiction to the assumption that p and q are relatively prime. Hence  EMBED Equation.3  must be irrational. % 4) Conclusion We have barely scratched the surface of all the wonderful applications of irrational numbers. It is truly amazing how often they show up in mathematics. While our proof about  EMBED Equation.3  was relatively simple, proving that other numbers are irrational (like e and ) take quiteO ^ q r  3456IWXr"#$%-.34DQNOPQWX]^jvxмм h6 h5hjh@Vh@VEHUj\0K h@VCJUVaJjh@VUh@Vhh*JCJ aJ h*J5CJaJhh*J5CJ aJ h*Jh*JCJaJ h*J5h*Jh*J5CJ0aJ0h*JhY4,Dj! ; L N O P Q R S T U V W X ^gdY & FgdYX Y Z [ \ ] ^ q r HIX $da$gd*J$a$gd*J$a$gd*JCiprt:j(v((())$a$gd+h $da$gd^U $da$gd^U $da$gd*J$d`a$gdx{ 468:>@fhjlrt<Z\rt ֓~zuz h^U5h^Ujh@Vh@VEHUhh6jh@Vh@VEHUh5CJ aJ hH\tj[h@Vh@VEHUj0K h@VCJUVaJj.h@Vh@VEHUj{0K h@VCJUVaJjh@VU h6h@Vh h5-   12EFGHVWijyz~:ⲥhhH\th^UH* h^Uh^Uj;h@Vh@VEHUj1K h@VCJUVaJ h^U6j h@Vh@VEHUj{0K h@VCJUVaJh@Vh^Ujh@VUj h@Vh@VEHU9:<FH "$DFHJ(k(v(z(((())))))Ž۬ꢹŹ h*Jh*J h+hh+hU h@V6 h+h6jh@Vh@VEHUh+hh+hCJ aJ h+h5CJ aJ jh@Vh@VEHUj{0K h@VCJUVaJh@Vjh@VUh^U h^U6) a bit of ingenuity. Irrational numbers are our friend, and we cannot escape them, even if we wanted to. References (1) Weisstein, Eric W. "Rational Number." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/RationalNumber.html (2) Weisstein, Eric W. "Irrational Number." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/IrrationalNumber.html (3) Smith, D., Eggen, M, and St. Andre, R. A Transition to Advanced Mathematics. Third Edition, Brooks/Cole, 1990. )))$a$gd+h,1h/ =!"#$% .Dd |Tb  c $A? ?3"`?2xԻAo0F6QETD`!LԻAo0F6QE` 0XJx]PJA}3g4xD E:%ll(SZDS!:K,LFA (xn0S+؆=/yD\}jZ}ɋ97VױҒK^z0D8]Q;t`&_PWw)_r.y0wB? [-r9?3u qm:<{0ry :нu6ԍ\ky4~U`gKLzjID-Dd |Tb  c $A? ?3"`?2wQt\9 cISr`!KQt\9 cI` 0XJx]PJA}3&xD "i"! 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