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(d) It is possible for L to have no largest number and for R to have no smallest number. What type of number does the cut de ne in this case
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(a) Let a be the largest rational number in L, and b the smallest rational number in R. Then either a b or a < b. We cannot have a b since by de nition of the cut every number in L is less than every number in R. We cannot have a < b since by Problem 1.9, 1 a b is a rational number which would be greater 2 than a (and so would have to be in R) but less than b (and so would have to be in L), and by de nition a rational number cannot belong to both L and R. (b) As an indication of the possibility, let L contain the number 2 and all rational numbers less than 2, while 3 3 R contains all rational numbers greater than 2. In this case the cut de nes the rational number 2. A 3 3 similar argument replacing 2 by any other rational number shows that in such case the cut de nes a 3 rational number. (c) As an indication of the possibility, let L contain all rational numbers less than 2, while R contains all 3 rational numbers greaters than 2. This cut also de nes the rational number 2. A similar argument 3 3 shows that this cut always de nes a rational number.
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(d) As an indication of the possibility let L consist of all negative rational numbers and all positive rational numbers whose squares are less than 2, while R consists of all positive numbers whose squares are greater than 2. We can show that if a is any number of the L class, there is always a larger number of the L class, while if b is any number of the R class, there is always a smaller number of the R class (see Problem 1.106). A cut of this type de nes an irrational number. From (b), (c), (d) it follows that every cut in the rational number system, called a Dedekind cut, de nes either a rational or an irrational number. By use of Dedekind cuts we can de ne operations (such as addition, multiplication, etc.) with irrational numbers.
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OPERATIONS WITH NUMBERS 1.35. Given x 3, y 2, z 5, a 3, and b 1, evaluate: 2 4 a Ans. 1.36. 2x y 3y z 5x 2z ; (a) 2200, (b) 32, b xy 2z2 ; 2ab 1 (d) 1 c 3a2 b ab2 ; 2a2 2b2 1 d ax by 2 ay bx 2 : ay bx 2 ax by 2
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(c) 51=41,
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Find the set of values of x for which the following equations are true. Justify all steps in each case. p p a 4f x 2 3 2x 1 g 2 2x 1 12 x 2 2 c x2 8x 7 2x 2 x 1 b 1 1 1 8 x x 2 4 (a) 2, (b) 6; 4 (c) 1; 1 (d) 1 2 d 1 x 3 p 2 2x 5 5 x
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Ans. 1.37.
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Prove that
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x y z 0 giving restrictions if any. z x x y x y y z y z z x
RATIONAL AND IRRATIONAL NUMBERS 1.38. Find decimal expansions for (a) 3, (b) 7 ______ Ans. (a) 0:428571, (b) 2.2360679 . . .
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Show that a fraction with denominator 17 and with numerator 1; 2; 3; . . . ; 16 has 16 digits in the repeating portion of its decimal expansion. Is there any relation between the orders of the digits in these expansions Prove that Prove that (a) (a) p 3, (b) p 3 2 are irrational numbers. (b) p p p 2 3 5 are irrational numbers.
1.40. 1.41. 1.42. 1.43. 1.44.
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Determine a positive rational number whose square di ers from 7 by less than .000001. Prove that every rational number can be expressed as a repeating decimal. Find the values of x for which (a) 2x3 5x2 9x 18 0, (b) 3x3 4x2 35x 8 0, (c) x4 21x2 4 0. p p p Ans. (a) 3; 2; 3=2 (b) 8=3; 2 5 (c) 1 5 17 ; 1 5 17 2 2 p p If a, b, c, d are rational and m is not a perfect square, prove that a b m c d m if and only if a c and b d. p p p p p 1 3 5 12 5 2 15 14 3 7 p p : Prove that 11 1 3 5
INEQUALITIES 1.47. Find the set of values of x for which each of the following inequalities holds: a 1 3 A 5; x 2x (a) 0 < x @ 1, 2 b x x 2 @ 24; c jx 2j < jx 5j; d x x 3 > : x 2 3x 1
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