Approximating Irrational Numbers 1. Rational Approximation Math 105 History of Mathematics Prof. D. Joyce, Clark University Early in the study of the history of mathematics, we come across various rational approx-imations to irrational numbers. rounding off: Changing a number to the nearest place value as specified; a method of approximating. reducing: Changing a fraction into its lowest terms. x lies between the square root of 75.69 and the square root of 77.44. continued fractions increase. Kyle was more than likely trying to find the best approximation for 77. 30, 7 8, 16, 1 4, 8i, 42, 3.692692, 4S, 20 1. b) When it works. Identify two rational numbers from the list of numbers. Approximating irrational numbers by rational ones 5 2.1. The height of a painting is 90 ft. Identify an integer from the list of numbers. Newton’s Method gives To graph yx= , let us first make a chart with a few x and y values: If we plot the above points and then draw a smooth curve through them, we obtain the A Proof. === [0.0.2] Corollary: (Liouville) Numbers well approximable by rational numbers, in the sense that, for S Identify three irrational numbers form the list of numbers. real number: Any rational or irrational number. 2. 9 and 10 C. 6 and 7 D. 8 and 9 2. 8. true. We will first use the graph of yx= to approximate square roots. c) When it fails Background: A zero of a function is a value of where the graph of = ( ) crosses the -axis. A. What I mean by that is that when what we know to be an irrational number was used in a computation, a rational number was used instead. Both C+ and C− are convex. rational approximation p=qto , jf0(˘)j 1 2 jf 0( )j. For example, 2/4 is reduced to 1/2. A) 12 B) 14 C) 7 Approximating Irrational Numbers with Newton’s Method Objective: Explore Newton’s Method for determining zeros of functions a) How it works. 8 and 9 C. 10 and 11 D. 9 and 10 3. Between which two consecutive whole numbers is 90? 7 and 8 B. The first lemma says that the denominators of conver gents of . 11 and 12 B. Define Then for . Let , , , ... be a sequence of integers, where for . The approximation of irrational numbers by rationals, up to such results as the best possible approximation of Hurwitz, is also given with elementary technique. Similarly for a point of ∂−. The book is wholly self-contained. Now that we know what perfect squares √14 falls between, we can find their square roots to find √14 approximate location on the number line. On a number line, between which two consecutive whole numbers would 61 be located? A. √9 = 3 or -3 and √16 = 4 or -4 So now we know that 3 and 4, and -3 and -4! 27) The approximation of which irrational number is shown by point R on the number line? Approximation by Rational Numbers Continued fractions give the best rational approximations to an irrational number. If P = (q,p) is a point of ∂+, then in the regionx ≥ q the liney = λx lies below the line throughP parallel to it. Concept # _____ reciprocal: The multiplicative inverse of a number. Proposition 1.1 implies that each of them approaches arbitrarily closely D. Approximating Square Roots 1. Lemma. 3. In this section, we'll see in what sense this is . Plotting and comparing IRRATIONAL numbers The two perfect squares that √14 falls between are √9 and √16. NS.2 Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., π 2). Use the following list of numbers to answer each question below. The last third of the monograph treats normal and transcendental numbers, including the Lindemann theorem, and the Gelfond-Schneider theorem. Thus, there is q osuch that for q q o p q 2=jf0( )j qd Replace the constant 2=jf0( )jby a smaller constant C, if necessary, so that the same inequality holds for the nitely-many 1 q

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