MATH 221 Numbers and Functions Assessment

1. What is the 2007th digit after the period in the expansion of 1/7 ?
2. Which of the following fractions have finite decimal expansions?
   a = 2/3 , b = 3 /25 , c = 276937 /15625 .
3. Draw the following sets of real numbers. Each of these sets is the union of one or more intervals. Find those intervals. Which of thee sets are finite?

   A = {x | x^2 ? 3x + 2 ? 0}
   B = {x | x^2 ? 3x + 2 ? 0}
   C =  {x | x2 ? 3x > 3 }
   D = { x | x^2 ? 5 > 2x}
   E = { t | t^2 ? 3t + 2 ? 0}
   F = {? | ?2 ? 3? + 2 ? 0}
   G= (0, 1) ? (5, 7]
   H = {1} ? {2, 3}? (0, 2? 2)
   Q = ? | sin ? = 1/2
   R =  ? | cos ? > 0

4. Suppose A and B are intervals. Is it always true that A ? B is an interval? How about A ?B?
5. Consider the sets M =  x | x > 0 and N =  y | y > 0 . Are these sets the same?
6. Group Problem.
   Write the numbers
   x = 0.3131313131 . . . , y = 0.273273273273 . . . and z = 0.21541541541541541 . . . as fractions (i.e. write them as m/n , specifying m and n.)
   (Hint: show that 100x = x + 31. A similar trick works for y, but z is a little harder.)
7. Group Problem.
   Is the number whose decimal expansion after the period consists only of nines, i.e.
   x = 0.99999999999999999 . . . an integer?

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