complex number(f.4)

(a)
(3 + i)/(2 - 3i) + (5 - 7i)/(2 + 3i)
= (3 + i)(2 + 3i)/(2 - 3i)(2 + 3i) + (5 - 7i)(2 - 3i)/(2 + 3i)(2 - 3i)
= (6 + 9i + 2i - 3)/(4 + 9) + (10 - 15i - 14i -21)/(4 + 9)
= (3 + 11i)/13 + (-11 - 29i)/13
= (3 + 11i - 11 - 29i)/13
= (-8 - 18i)/13
or -(8/13) - (18/13)i

(b)
(1 + 2i)/(3 - i) - (3 + i)/(2 - i)
= (1 + 2i)(3 + i)/(3 - i)(3 + i) - (3 + i)(2 + i)/(2 - i)(2 + i)
= (3 + i + 6i - 2)/(9 + 1) - (6 + 3i + 2i - 1)/(4 + 1)
= (1 + 7i)/10 - (5 + 5i)/5
= (1 + 7i)/10 - (10 + 10i)/10
= (1 + 7i - 10 - 10i)/10
= (-9 - 3i)/10
or -(9/10) - (3i/10)i


(c)
(2 - i)/(i + 3) - (i - 3)/(2 + i)
= (2 - i)(i - 3)/(i + 3)(i - 3) - (i - 3)(2 - i)/(2 + i)(2 - i)
= (2i - 6 + 1 + 3i)/(-1 - 9) - (2i + 1 - 6 + 3i)/(4 + 1)
= (-5 + 5i)/(-10) - (-5 + 5i)/5
= (1 - i)/2 - (-2 + 2i)/2
= (1 - i + 2 - 2i)/2
= (3 - 3i)/2
or (3/2) - (3/2)i


(d)
(-5 + 12i)/(2 + 3i) + (35 + 12i)/(6 + i)
= (-5 + 12i)(2 - 3i)/(2 + 3i)(2 - 3i) + (35 + 12i)(6 - i)/(6 + i)(6 - i)
= (-10 + 15i + 24i + 36)/(4 + 9) + (210 - 35i + 72i + 12)/(36 + 1)
= (26 + 39i)/13 + (222 + 37i)/37
= (2 + 3i) + (6 + i)
= 8 + 4i

2009-09-07 16:50:42 補充：
六呎將軍：
題 (b) 的第一個分母是 3 - i，不是 3 + i。