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急 ! F4 Number Systems

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23. (a) (9 - 15i)(9 + 15i) = 92 - (15i)2 = 81 + 225 = 306 It is real. (b) (14 - 5i) + 11i - (9 + 6i) = 14 - 5i + 11i - 9 - 6i = 5 It is real. (c) 6 ÷ (1 + √2i) 6(1 - √2i) / [(1 + √2i)(1 - √2i)] = 6(1 - √2i) / (1 + 2) = 2(1 - √2i) It is non-real. (d) (2 - 3i)(4 - 2i)(8 - i) = (8 - 4i - 12i - 6)(8 - i) = (2 - 16i)(8 - i) = 16 - 32i - 98i - 16 = -100i It is non-real. 26. (a) (2a + 7i) - (12 - 4i) = 2a + 7i - 12 + 4i = (2a - 12) + 11i Since it is a pure imaginary number, then 2a - 12 = 0 a = 6 (b) (a + bi)(-10 + 25i) = -10a + 25ai - 10bi - 25b = -(10a + 25b) + (25a - 10b)i Since it is a real number, then 25a - 10b = 0 25a = 10b a/b = 10/25 a : b = 2 : 5 16. (a) √{[2a - (5/3)] [2a + (5/3)] + (25/9)} = √[(2a)2 - (5/3)2] +(25/9)] = √[(2a)2 - (25/9)+ (25/9)] = √(2a)2 = 2a It is rational. (b) √(3a + 1) x √[(a + 2)(a + 1) - (a2 + 1)] = √(3a + 1) x √(a2 + 3a + 2 - a2 - 1) = √(3a + 1) x √(3a + 1) = 3a + 1 It is rational. (c) √[2a - 1 + (a - 1)2] x √6 = √(2a - 1 + a2 -2a + 1) x √6 = √(a2) x√6 = a√6 It is irrational. 17. P = [2(n + 2) - 2] + 2(n + 2) + [2(n + 2) + 2] = 6(n + 2) Q = (n + 1) + (n + 2) + (n + 3) = 3n + 6 = 3(n + 2) (a) √(P/Q) = √[6(n + 2)]/[3(n + 2)] = √2 It is an irrational number. (b) √(PQ/18) = √{[6(n + 2)] x [2(n + 2)] / 18} = √[18(n + 2)2 /18] = n + 2 It is a rational number.

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