We define a real number to be algebraic if it is a solution of some polynomial...

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We define a real number to be algebraic if it is a solution of some polynomial equation Anx" + an−1x”−¹+...+ ª₁x + ao = 0, where all the coefficients are integers. Thus √2 is algebraic because it is a solution of r² - 2 = 0. The number is not algebraic because no such polynomial equation can ever be found (although this is hard to prove). Show that the set of algebraic numbers is countable. A real number that is not algebraic is said to be transcendental. For example, it is known that e and are transcendental. What can you say about the existence of other transcendental numbers? We define a real number to be algebraic if it is a solution of some polynomial equation Anx" + an−1x”−¹+...+ ª₁x + ao = 0, where all the coefficients are integers. Thus √2 is algebraic because it is a solution of r² - 2 = 0. The number is not algebraic because no such polynomial equation can ever be found (although this is hard to prove). Show that the set of algebraic numbers is countable. A real number that is not algebraic is said to be transcendental. For example, it is known that e and are transcendental. What can you say about the existence of other transcendental numbers?

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