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Prove that there exist irrational numbers
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Discrete Mathematics with Graph Theory (Classic Version) (3rd Edition) (Pearson Modern Classics for Advanced Mathematics Series)
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- Prove that if is a nonzero rational number and is irrational, then is irrational.arrow_forward13. 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_forwardProve that is irrational. (That is, prove there is no rational number such that .)arrow_forward
- Prove that if is an irrational number, then is an irrational 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_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
- Prove by induction that if r is a real number where r1, then 1+r+r2++rn=1-rn+11-rarrow_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_forwardLet a and b be integers such that ab and ba. Prove that b=0.arrow_forward
- Prove that if and are integers such that and , then .arrow_forwardProve that the statements in Exercises 116 are true for every positive integer n. a+ar+ar2++arn1=a1rn1rifr1arrow_forwardProve or disprove each of the following statements. The set Q of rational numbers is an ideal of the set R of real numbers. The set Z of integers is an ideal of the set Q of rational numbers.arrow_forward
- Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,College Algebra (MindTap Course List)AlgebraISBN:9781305652231Author:R. David Gustafson, Jeff HughesPublisher:Cengage Learning