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# week17 - Cardinality Part IV and Intro to Compass and...

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Unformatted text preview: Cardinality Part IV and Intro to Compass and Straight-Edge Original Notes adopted from January 29, 2002 (Week 17) c P. Rosenthal , MAT246Y1, University of Toronto, Department of Mathematics Definition. A real number is algebraic if there exists a nonzero polynomial with integer coefficients that has it as a root. Every rational number is algebraic: n ( m n )- m = 0. Let A be the set of algebraic number. Then A ⊃ Q . ( √ 2) 2- 2 = 0 ⇒ √ 2 ∈ A , n √ r ∈ A , ∀ r,n ∈ N . Is A = R ? Theorem. A is countable. Proof : Let A 1 = { x ∈ R : p ( x ) = 0 for p a polynomial of degree 1 with integer coefficients } . Such a polynomial is mx + n = 0. The set of such polynomials corresponds to Z × Z . There is a countable number of such polynomials, and each has 1 root, so A 1 is countable. (In fact, A 1 = Q ). Let A 2 = { x ∈ R : p ( x ) = 0 for p a polynomial of degree 2 with integer coefficients } . A typical such polynomial is mx 2 + nx + l = 0. The set of such polynomials corresponds to Z × Z × Z . Therefore a countable number of such polynomials. Each has at most 2 roots, so A 2 is countable....
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week17 - Cardinality Part IV and Intro to Compass and...

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