Transcendental

Transcendental - MATH 135 Algebraic and Transcendental...

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MATH 135 Fall 2008 Algebraic and Transcendental Numbers Number Systems So far this term, we have looked at a number of different number systems: Z (the integers): the set of all whole numbers, including positive, negative and zero Q (the rational numbers): the set of all fractions of integers R (the real numbers): the set of all decimal numbers, including positive, negative and zero C (the complex numbers): the set of all complex numbers We know that these form a chain: Z Q R C . Each of these number systems includes numbers that the previous one does not. Examples 1 2 Q , but 1 2 6∈ Z . 2 i C , but 2 i 6∈ R . Can you remember a number x with x R but x 6∈ Q ? x = 2 works, and thus 2 is called irrational . We proved this in the first couple of weeks of term, but here is another proof of this fact. Theorem 2 is irrational. Proof Suppose that r = 2 is rational. We know that r is a root of x 2 - 2 = 0. What are the possible rational roots of the polynomial f ( x ) = x 2 - 2? Suppose that x = a b is a rational root with a, b Z and gcd( a, b ) = 1. By the Rational Roots Theorem, b | 1 and a | 2, so b = ± 1 and a = ± 1 or ± 2. This means that a b = ± 1 or ± 2. But none of these square to give 2, so none equals 2. Thus, f ( x ) = x 2 - 2 has no rational roots, so 2 is irrational. ± Note We could use a similar method to prove that 5 and 3 2 are irrational. Numbers as Solutions of Polynomial Equations We can justify the need for each new system by looking at different polynomial equations with integer coefficients that have or do not have solutions in that system: x - 2 = 0 has integer solutions 3 x - 2 = 0 does not have integer solutions but does have rational solutions x 2 - 2 = 0 does not have rational solutions but does have real solutions
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Transcendental - MATH 135 Algebraic and Transcendental...

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