Unformatted text preview: a/b ). Show that the following operations of “addition” and “multiplication” on Q are well de±ned: [ a, b ] + [ c, d ] = [ ad + bc, bd ] , [ a, b ] [ c, d ] = [ ac, bd ] . 4. Show that every positive integer n has a binary expansion, i.e. can be expressed as a sum of distinct powers of 2. For example, 2010 = 2 10 + 2 9 + 2 8 + 2 7 + 2 6 + 2 4 + 2 3 + 2 1 . (Hint: by the division theorem you can write n = 2 q + r with r ∈ { , 1 } ; use induction.) Extra credit: show that the binary expansion of a given positive integer is unique. 5. Fraleigh, section 2, exercises 31, 32, 34. 6. Fraleigh, section 3, exercises 3, 4, 7. 7. How challenging did you ±nd this assignment? How long did it take?...
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 Spring '08
 OGUS
 Math, positive integer, binary expansion

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