2400practice1

# 2400practice1 - MATH 2400 PRACTICE PROBLEMS FOR EXAM 1 PETE...

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Unformatted text preview: MATH 2400: PRACTICE PROBLEMS FOR EXAM 1 PETE L. CLARK 1) Find all real numbers x such that x 3 = x . Prove your answer! 2) a) Prove that √ 6 is an irrational number. You may use the fact that if an integer x 2 is divisible by 6, then also x is divisible by 6. (For “extra credit”, prove the fact of the previous sentence using the uniqueness of prime factorizations.) b) Show that if x 2 is an irrational number, so is x . c) Show that √ 2 + √ 3 is irrational. 3) A subset S of the real numbers is dense if for any real numbers a < b , there exists x ∈ S such that a < x < b . a) Show that the set of rational numbers is dense. (Suggestion: make use of the fact that every real number has a decimal expansion.) b) Suppose a, b ∈ Q with b ̸ = 0. Show that a + b √ 2 is irrational. c) Show that the set { a + b √ 2 | a, b ∈ Q } is dense. 1 d) Conclude that the set of irrational numbers is dense....
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2400practice1 - MATH 2400 PRACTICE PROBLEMS FOR EXAM 1 PETE...

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