Page 25 - Schaum's Outline of Theory and Problems of Advanced Calculus

16 NUMBERS [CHAP. 1

n
1.31. Prove Bernoulli’s inequality ð1 þ xÞ > 1 þ nx for n ¼ 2; 3; ... if x > 1, x 6¼ 0.
2
2
The statement is true for n ¼ 2 since ð1 þ xÞ ¼ 1 þ 2x þ x > 1 þ 2x.
k
Assume the statement true for n ¼ k, i.e., ð1 þ xÞ > 1 þ kx.
Multiply both sides by 1 þ x (which is positive since x > 1). Then we have
kþ1 2
ð1 þ xÞ > ð1 þ xÞð1 þ kxÞ¼ 1 þðk þ 1Þx þ kx > 1 þðk þ 1Þx
Thus the statement is true for n ¼ k þ 1ifitistruefor n ¼ k.
But since the statement is true for n ¼ 2, it must be true for n ¼ 2 þ 1 ¼ 3; ... and is thus true for all
integers greater than or equal to 2.
n
Note that the result is not true for n ¼ 1. However, the modified result ð1 þ xÞ A 1 þ nx is true for
n ¼ 1; 2; 3; ... .

MISCELLANEOUS PROBLEMS
n
1.32. Prove that every positive integer P can be expressed uniquely in the form P ¼ a 0 2 þ a 1 2 n 1 þ
a 2 2 n 2 þ þ a n where the a’s are 0’s or 1’s.
Dividing P by 2, we have P=2 ¼ a 0 2 n 1 þ a 1 2 n 2 þ þ a n 1 þ a n =2.
Then a n is the remainder, 0 or 1, obtained when P is divided by 2 and is unique.
Let P 1 be the integer part of P=2. Then P 1 ¼ a 0 2 n 1 þ a 1 2 n 2 þ þ a n 1 .
Dividing P 1 by 2 we see that a n 1 is the remainder, 0 or 1, obtained when P 1 is divided by 2 and is
unique.
By continuing in this manner, all the a’s can be determined as 0’s or 1’s and are unique.

1.33. Express the number 23 in the form of Problem 1.32.
The determination of the coefficients can be arranged as follows:
2Þ23
2Þ11 Remainder 1
2Þ5 Remainder 1
2Þ2 Remainder 1
2Þ1 Remainder 0
0 Remainder 1
3
2
4
The coefficients are101 11. Check:23 ¼ 1 2 þ 0 2 þ 1 2 þ 1 2 þ 1.
The number 10111 is said to represent 23 in the scale of two or binary scale.
1.34. Dedekind defined a cut, section,or partition in the rational number system as a separation of all
rational numbers into two classes or sets called L (the left-hand class) and R (the right-hand class)
having the following properties:

I. The classes are non-empty (i.e. at least one number belongs to each class).
II. Every rational number is in one class or the other.
III. Every number in L is less than every number in R.
Prove each of the following statements:
(a) There cannot be a largest number in L and a smallest number in R.
(b)Itis possible for L to have a largest number and for R to have no smallest number. What
type of number does the cut define in this case?
(c)Itis possible for L to have no largest number and for R to have a smallest number. What
type of number does the cut define in this case?