Square root of 2 and the Rationals

Square root of 2 and the Rationals - Math 100 Square Roots...

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Math 100 Square Roots Martin H. Weissman 1. The real square root of two Since we have not thoroughly discussed it, we begin by mentioning the intermediate value theorem for polynomials: Suppose that P ( x ) is a polynomial with real coefficients, with one variable x . Suppose that a, b R , and let u = P ( a ), and v = P ( b ). Suppose that a < b , and u < v . If u < w < v , then there exists c R , such that a < c < b , and P ( c ) = w . Rather than proving such a theorem, we take it as an axiom : a fact about numbers which we assume. This axiom guarantees that every positive real number has a real square root: Theorem 1. If y is a real number, and y > 0 , then there exists a real number x such that x 2 = y . Proof. Suppose that y is a positive real number. Let P be the polynomial P ( x ) = x 2 . Then, 0 < y , and P (0) < P ( y + 1), since P (0) = 0 and P ( y + 1) = ( y + 1)( y + 1) and the product of positive numbers is positive. We claim that 0 < y < P ( y + 1). Note that 0 < y since y is positive. Moreover, we can compare P ( y + 1) and y as follows: P ( y + 1) = y 2 + 2 y + 1, and so P ( y + 1) - y = y 2 + y + 1. Since 1, y , and y 2 are all positive, it follows that P ( y + 1) - y is also positive. Hence P ( y + 1) - y > 0, and so P ( y + 1) > y . We have proven that 0 < y < P ( y + 1). Now, we may apply the intermediate value theorem: since P (0) < y < P ( y + 1), and 0 < y + 1, we can choose x R such that 0 < x < y + 1 and P ( x ) = y . Therefore x 2 = y . / Note that the above proof used many of the basic axioms for real numbers. Moreover, it was not a completely “straightforward” proof – at one point, the expression y + 1 was used, even though y
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