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4. (i) Recall that a number is said to be algebraic if it can be expressed as the root of a polynomial expression p(x) = a,a" +an-1"-1+...+ao having integer coefficients ao, a1,.., an; otherwise, it is said to be transcendental. Denote by A the set of algebraic numbers. Show that A = No; (ii) Give three examples of algebraic numbers that are not rational numbers; (iii) Denote by T the set of transcendental numbers. Use (i) above to show that |T| > No, i.e. the transcendental numbers are non-denumerable.

4. (i) Recall that a number is said to be algebraic if it can be expressed as the root of a polynomial expression p(x) = a,a" +an-1"-1+...+ao having integer coefficients ao, a1,.., an; otherwise, it is said to be transcendental. Denote by A the set of algebraic numbers. Show that A = No; (ii) Give three examples of algebraic numbers that are not rational numbers; (iii) Denote by T the set of transcendental numbers. Use (i) above to show that |T| > No, i.e. the transcendental numbers are non-denumerable.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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4. (i) Recall that a number is said to be algebraic if it can be expressed as the root of a polynomial expression p(x) = anx" + an-1x"-1 +...+ao having integer coefficients ao, a1,., an; otherwise, it is said to be transcendental. Denote by A the set of algebraic numbers. Show that A| = No; .... (ii) Give three examples of algebraic numbers that are not rational numbers; (iii) Denote by T the set of transcendental numbers. Use (i) above to show that T| > No, i.e. the transcendental numbers are non-denumerable.
Transcribed Image Text:4. (i) Recall that a number is said to be algebraic if it can be expressed as the root of a polynomial expression p(x) = anx" + an-1x"-1 +...+ao having integer coefficients ao, a1,., an; otherwise, it is said to be transcendental. Denote by A the set of algebraic numbers. Show that A| = No; .... (ii) Give three examples of algebraic numbers that are not rational numbers; (iii) Denote by T the set of transcendental numbers. Use (i) above to show that T| > No, i.e. the transcendental numbers are non-denumerable.
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