A number r ∈ R is called algebraic provided for some n ∈ N there exists numbers a0, a1, … , an ∈ Z such that r satisfiesp( r ) = 0where p ( x ) = a0 + a1x + a2x2 + ⋯ + anxn. A number r ∈ R is called transcendental provided it is NOT algebraicprove the set of ALL algebraic numbers is countable and the set of ALL transcendental numbers is uncountable.

Note : In the definition of algebraic numbers , coefficients a0,a1,......,an should be rational…

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prove the set of ALL algebraic numbers is countable and the set of ALL transcendental numbers is uncountable.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter7: Real And Complex Numbers

Section7.1: The Field Of Real Numbers

Problem 26E: Prove that if and are real numbers such that , then there exist a rational number such that ....

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Question

A number r ∈ R is called algebraic provided for some n ∈ N there exists numbers a₀, a₁, … , a_n ∈ Z such that r satisfies

p( r ) = 0

where p ( x ) = a₀ + a₁x + a₂x² + ⋯ + a_nxⁿ.

A number r ∈ R is called transcendental provided it is NOT algebraic

prove the set of ALL algebraic numbers is countable and the set of ALL transcendental numbers is uncountable.

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