Page 30 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 1] NUMBERS 21

n
1.92. ðcos þ i sin Þ ¼ cos n þ i sin n . Can this be proved if n is a rational number?
1
sinðn þ Þx
1.93. 1 2 , x 6¼ 0; 2 ; 4 ; ...
2 þ cos x þ cos 2x þ þ cos nx ¼ 1
2 sin x
2
1
1
cos x cosðn þ Þx
1.94. sin x þ sin 2x þ þ sin nx ¼ 2 2 ; x 6¼ 0; 2 ; 4 ; ...
1
2 sin x
2
n
n
b þ þ n C n 1 ab
1.95. ða þ bÞ ¼ a þ n C 1 a n 1 b þ n C 2 a n 2 2 n 1 þ b n
n!
nðn 1Þðn 2Þ ... ðn r þ 1Þ
¼ n C n r .Here p! ¼ pðp 1Þ .. . 1 and 0! is defined as
r! r!ðn rÞ!
where n C r ¼ ¼
nðn 1Þ
; ... ; n C n ¼ 1are
2!
1. This is called the binomial theorem. The coefficients n C 0 ¼ 1, n C 1 ¼ n, n C 2 ¼
n

called the binomial coefficients. n C r is also written .
r
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
1.97. If a number is 144 in the scale of 5, what is the number in the scale of (a)2, (b)8?
1.98. Prove that every rational number p=q between 0 and 1 can be expressed in the form
p a 1 a 2 a n
q ¼ 2 þ 2 2 þ þ 2 n þ
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:a 1 a 2 ... a n ... is then called the binary form of the rational number. [Hint: Multiply both
sides successively by 2 and consider remainders.}
1.99. 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 þ 4 ¼ 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 first digit,
the new number is one-third as large. Find the original number.
Ans. 428571

1.105. Show that the rational numbers form a field.

1.106. Using as axioms the relations 1–9 on Pages 2 and 3, prove that
(a) ð 3Þð0Þ¼ 0, (b) ð 2Þðþ3Þ¼ 6, (c) ð 2Þð 3Þ¼ 6.