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re that there are countably many algebraic numbers. (Hint: Use (i) and here you may ental theorem of Algebra, that any polynomial has finitely many roots.) ve that there are uncountably many transcendental numbers. (Remark: e and T are two sr

re that there are countably many algebraic numbers. (Hint: Use (i) and here you may ental theorem of Algebra, that any polynomial has finitely many roots.) ve that there are uncountably many transcendental numbers. (Remark: e and T are two sr

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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5. A real number x is said to be algebraic (over Q) if it satisfies some polynomial equation anx" +an-1x"-1+ ...+ a1x + ao = 0, where each a; E Q, an +0. If x is not algebraic, it is called transcendental. (i) Show that the set of all polynomials over Q (as mentioned above) is countable. (ii) Prove that there are countably many algebraic numbers. (Hint: Use (i) and here you may use the Fundamental theorem of Algebra, that any polynomial has finitely many roots.) (iii) Prove that there are uncountably many transcendental numbers. (Remark: e and T are two such num- bers.)
Transcribed Image Text:5. A real number x is said to be algebraic (over Q) if it satisfies some polynomial equation anx" +an-1x"-1+ ...+ a1x + ao = 0, where each a; E Q, an +0. If x is not algebraic, it is called transcendental. (i) Show that the set of all polynomials over Q (as mentioned above) is countable. (ii) Prove that there are countably many algebraic numbers. (Hint: Use (i) and here you may use the Fundamental theorem of Algebra, that any polynomial has finitely many roots.) (iii) Prove that there are uncountably many transcendental numbers. (Remark: e and T are two such num- bers.)
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