Square root of 2 and the Rationals

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

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 04/17/2008 for the course MATH 100 taught by Professor Weissman during the Spring '08 term at UCSC.

Page1 / 2

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online