lecnotes2-2_1 - Lecture Notes, Concepts of Mathematics...

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Unformatted text preview: Lecture Notes, Concepts of Mathematics (21-127) Lecture 1, Recitation A–D, Spring 2008 2 Integers and Diophantine Equations READ: pgs. 41–49 EXERCISES: pgs. 50–55, #52–72, 89–108. 2.4 Integers in Different Bases Theorem 2.1 (Theorem 2.41, page 42) Let b be a fixed integer greater than 1. Then any positive integer x can be expressed uniquely as x = r n b n + r n- 1 b n- 1 + ··· + r 2 b 2 + r 1 b + r where ≤ r i < b for i = 0 , 1 , 2 , ··· , n and r n 6 = 0 . This expression for x is called the representation of x in base b and is written as x = ( r n r n- 1 . . . r 2 r 1 r ) b . The r i are obtained through the Division Algorithm, first dividing x by b and repeatedly dividing the quotients by b : x = q b + r , where 0 ≤ r < b q = q 1 b + r 1 , where 0 ≤ r 1 < b q 1 = q 2 b + r 2 , where 0 ≤ r 2 < b . . . q n- 2 = q n- 1 b + r n- 1 , where 0 ≤ r n- 1 < b q n- 1 = 0 · b + r n , where 0 ≤ r n < b. Example 2.1 (#60, page 51) Convert 5766 to base 12, writing A for ten and B for eleven. We have 5766 = 480 · 12 + 6 , 480 = 40 · 12 + 0 , 40 = 3 · 12 + 4 , 3 = 0 · 12 + 3 . Thus 5766 = (3406) 12 . We check the result by computing 3 · 12 3 + 4 · 12 2 + 6 = 3 · 1728 + 4 · 144 + 6 = 5766. 1 2.5 Prime Numbers Definition 2.1 An integer p > 1 is prime if and only if its only positive divisors are 1 and p ; otherwise it is composite . Proposition 2.1 (Proposition 2.51) Every integer larger than 1 can be expressed as a prod- uct of primes. Proof: Suppose not, and let N be the smallest integer (greater than 1) that cannot be written as the product of primes. Since N is not prime itself, we can write N = r · s where 1 < r ≤ s < N...
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This note was uploaded on 01/25/2010 for the course MATH 21127 taught by Professor Howard during the Spring '08 term at Carnegie Mellon.

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lecnotes2-2_1 - Lecture Notes, Concepts of Mathematics...

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