Concept explainers
Fill in the blanks for the following proof that the difference of any rational number and any irralional number is irrational.
Proof (by contradiction]: Suppose not. That is, suppose that there exist (a) x and (b) y such that
Adding y and subtracting
Now both
Trending nowThis is a popular solution!
Chapter 4 Solutions
Discrete Mathematics With Applications
- 13. Prove that if and are rational numbers such that then there exists a rational number such that . (This means that between any two distinct rational numbers there is another rational number.)arrow_forwardUse the fact that 3 is a prime to prove that there do not exist nonzero integers a and b such that a2=3b2. Explain how this proves that 3 is not a rational number.arrow_forwardIf x and y are positive rational numbers, prove that there exists a positive integer n such that nxy. This property is called the Archimedean Property of the rational numbers. (Hint: Write x=a/b and y=c/d with each of a,b,c,d+.)arrow_forward
- Prove that if is an irrational number, then is an irrational number.arrow_forwardProve that 23 is not a rational number.arrow_forwardLabel each of the following statements as either true or false. Every decimal representation of a real number that terminates represents a rational number.arrow_forward
- Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:CengageAlgebra: Structure And Method, Book 1AlgebraISBN:9780395977224Author:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. ColePublisher:McDougal Littell
- Holt Mcdougal Larson Pre-algebra: Student Edition...AlgebraISBN:9780547587776Author:HOLT MCDOUGALPublisher:HOLT MCDOUGALCollege Algebra (MindTap Course List)AlgebraISBN:9781305652231Author:R. David Gustafson, Jeff HughesPublisher:Cengage Learning