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Unformatted text preview: mplex number 1 z+w (0 + (z + w)) = , 2 2 which also represents the midpoint of the diagonal AC . This proves that the two diagonals of a parallelogram bisect each other. Problems for Chapter 1 1. Suppose that a, b, c, d are positive real numbers satisfying a < b and c < d. Show that ac < bd. [Hint: Use the Field axioms and the Order axioms only.]
Chapter 1 : The Number System page 15 of 19 First Year Calculus c W W L Chen, 1982, 2005 2. Find x, y ∈ R such that x < y and x−1 < y −1 . 3. Suppose that x, y, z ∈ R. Use the Field axioms and the Order axioms only to show that a) if x + z = y + z , then x = y ; b) if z = 0 and xz = yz , then x = y ; c) if xy = 0, then x = 0 or y = 0. 4. Show that if x, y, a ∈ R satisfy x < y and a < 0, then ax > ay . [Hint: Use the Field axioms and the Order axioms only.] 5. Prove that 13 + 23 + 33 + . . . + n3 = 1 n2 (n + 1)2 for every n ∈ N. 4 6. Prove that 2n > n3 for every natural number n > 9. 7. Prove that for every n ∈ N, 52n − 6n + 8 is divisible by 9. 8. Complex numbers are num...
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