Unformatted text preview: 3 If every real number was algebraic then, R = A ∩ R would be countable. We have shown in class that R is not countable, so R ±⊂ A and hence there must be a non-algebraic real number; indeed there must be an uncountably inﬁnite set of them. (3) Rudin, Chapter 2, Problem 4 We have shown in class that the set of rational numbers, Q ⊂ R is countable. Since R is uncountable it cannot be equal to Q so there must exist irrational real numbers. 1...
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- Fall '09
- Rational number, Irrational number, Countable set