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1.74. 1.75. 1.76.
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If z1 and z2 are complex numbers, prove Prove (a) jz1 z2 j @ jz1 j jz2 j,
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(b) jz2 j jz1 j2 giving any restrictions. 1 (c) jz1 z2 j A jz1 j jz2 j.
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(b) jz1 z2 z3 j @ jz1 j jz2 j jz3 j,
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Find all solutions of 2x4 3x3 7x2 8x 6 0. Ans. 3, 1, 1 i 2 Let z1 and z2 be represented by points P1 and P2 in the Argand diagram. Construct lines OP1 and OP2 , where O is the origin. Show that z1 z2 can be represented by the point P3 , where OP3 is the diagonal of a parallelogram having sides OP1 and OP2 . This is called the parallelogram law of addition of complex numbers. Because of this and other properties, complex numbers can be considered as vectors in two dimensions. Interpret geometrically the inequalities of Problem 1.75. p p Express in polar form (a) 3 3i, (b) 2 2i, (c) 1 3i, (d) 5, (e) 5i. p3 Ans. (a) 6 cis =6 b 2 2 cis 5=4 c 2 cis 5=3 d 5 cis 0 e 5 cis 3=2 Evaluate (a) 2 cos 258 i sin 258 5 cos 1108 i sin 1108 , b 2i (b) 12 cis 168 : 3 cis 448 2 cis 628
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1.78. 1.79.
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p p Ans. (a) 5 2 5 2i; 1.81.
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Determine all the indicated roots and represent them graphically: p p p (a) 4 2 4 2i 1=3 ; b 1 1=5 ; c 3 i 1=3 ; d i1=4 . Ans. (a) 2 cis 158; 2 cis 1358; 2 cis 2558 (b) p 368; cis 1088; cis 1808 cis 2528; cis 3248 cis p p 1; (c) 3 2 cis 1108; 3 2 cis 2308; 3 2 cis 3508 (d) cis 22:58; cis 112:58; cis 202:58; cis 292:58 Prove that 1 p 3i is an algebraic number. (a) z1 z2 1 2 cis 1 2 , (b) z1 =z2 1 =2 cis 1 2 .
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1.82. 1.83.
If z1 1 cis 1 and z2 2 cis 2 , prove Interpret geometrically.
MATHEMATICAL INDUCTION Prove each of the following. 1.84. 1.85. 1.86. 1.87. 1 3 5 2n 1 n2 1 1 1 1 n 1 3 3 5 5 7 2n 1 2n 1 2n 1 a a d a 2d a n 1 d 1 n 2a n 1 d 2 1 1 1 1 n n 3 1 2 3 2 3 4 3 4 5 n n 1 n 2 4 n 1 n 2 a ar ar2 arn 1 a rn 1 ; r 6 1 r 1
1.88. 1.89. 1.90. 1.91.
13 23 33 n3 1 n2 n 1 2 4 1 5 2 5 2 3 5 3 n 5 n 1 5 4n 1 5n 1 16
x2n 1 y2n 1 is divisible by x y for n 1; 2; 3; . . . .
CHAP. 1]
NUMBERS
1.92. 1.93.
cos  i sin  n cos n i sin n.
Can this be proved if n is a rational number
cos x cos 2x cos nx
sin n 1 x 2 , x 6 0; 2; 4; . . . 2 sin 1 x 2
1.94. 1.95.
sin x sin 2x sin nx
cos 1 x cos n 1 x 2 2 ; x 6 0; 2; 4; . . . 2 sin 1 x 2
a b n an n C1 an 1 b n C2 an 2 b2 n Cn 1 abn 1 bn n n 1 n 2 . . . n r 1 n! C . Here p! p p 1 . . . 1 and 0! is de ned as r! r! n r ! n n r n n 1 ; . . . ; nCn 1 are 1. This is called the binomial theorem. The coe cients n C0 1, n C1 n, n C2 2! n . called the binomial coe cients. n Cr is also written r where n Cr
MISCELLANEOUS PROBLEMS 1.96. Express each of the following integers (scale of 10) in the scale of notation indicated: (a) 87 (two), (b) 64 (three), (c) 1736 (nine). Check each answer. Ans. (a) 1010111, (b) 2101, (c) 2338 If a number is 144 in the scale of 5, what is the number in the scale of (a) 2, (b) 8
1.97. 1.98.
Prove that every rational number p=q between 0 and 1 can be expressed in the form p a1 a2 a 2 n 2 2 2n q where the a s can be determined uniquely as 0 s or 1 s and where the process may or may not terminate. The representation 0:a1 a2 . . . an . . . is then called the binary form of the rational number. [Hint: Multiply both sides successively by 2 and consider remainders.}
Express 2 in the scale of (a) 2, (b) 3, (c) 8, (d) 10. 3 Ans. (a) 0:1010101 . . . ; (b) 0.2 or 0:2000 . . . ; (c) 0:5252 . . . ; (d) 0:6666 . . .
1.100. A number in the scale of 2 is 11.01001. What is the number in the scale of 10. Ans. 3.28125 1.101. In what scale of notation is 3 4 12 Ans. 5 1.102. In the scale of 12, two additional symbols t and e must be used to designate the digits 10 and 11, respectively. Using these symbols, represent the integer 5110 (scale of 10) in the scale of 12. Ans. 2e5t 1.103. Find a rational number whose decimal expansion is 1:636363 . . . . Ans. 18/11 1.104. A number in the scale of 10 consists of six digits. If the last digit is removed and placed before the rst digit, the new number is one-third as large. Find the original number. Ans. 428571 1.105. Show that the rational numbers form a eld. 1.106. Using as axioms the relations 1 9 on Pages 2 and 3, prove that (a) 3 0 0, (b) 2 3 6, (c) 2 3 6.
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