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Unformatted text preview: Math 250, Fall 2003 Selected Solutions, Assignment # 1 HH 1.5.5 d) Is the set Q rational numbers open or closed in R , the real numbers? Response; The set of rational numbers Q is neither open nor closed in the real numbers R . Rather, both Q and R- Q are dense in R- any number can be approximated arbitrarily closely by rational numbers, and by irrational numbers. To see that the rational numbers are dense, fix a denominator n . The fractions with denominator n are of the form k n for any integer k . They are evenly spaced, with a separation of distance 1 n between adjacent numbers. For a fixed denominator n , these numbers are not dense. However, every number is within distance 1 n (actually, 1 2 n- why?) of a fraction with denominator 1 n .* Since we may let n get arbitrarily large, we can approximate any number arbitrarily closely by rational numbers; that is, the rational numbers are dense in R . (A common way to approximate irrational numbers by rational numbers is by decimal expansions - every number has an infinite decimal expansion, and the finite parts of this are rational numbers. A given number lies within 10- k of its decimal expansion truncated at the k-th decimal place. The decimal fractions with k places to the right of the decimal point are exactly those numbers which can be written as a fraction with denominator 10- k .) To see that the irrational numbers are dense, observe that, if φ is an irrational number, then rφ is irrational for all rational numbers r . (Why?) Therefore, the same argument as above shows that any number can be approximated as closely as one wishes by numbers rφ . A fortiori , irrational numbers are dense. HH 1.5.19 : For a given angle θ , consider the sequence A n = cos nθ sin nθ- sin nθ cos nθ . For what values of θ does this converge? For what values of θ can you find a convergent subsequence? Response: The matrix A n produces a rotation of the plane through an angle- nθ . Thus, we may think of A n as being a point a n on the unit circle in R 2 , such that the angle between a n and the first standard basis vector e 1 is- nθ . (This point a n is given by the first column of A n , and the second column is just the rotation of a n through -90 ◦ . Clearly, a sequence of rotation matrices converges if and only if their first columns converge in the unit circle. Thus, we will discuss the behavior of the points a n . Of course, in all these considerations, the value of θ only matters modulo 2 π . That is, if we add a multiple of 2 π to θ , the matrix A n and the point a n do not change....
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