Problems in Mathematical Analysis (1) An integer is any whole number, zero, or n

Question

Problems in Mathematical Analysis

(1) An integer is any whole number, zero, or negative of a whole number. A rational number is any

number that can be written as a fraction a b, in which a and b are integers and b is not zero. Show

that if x, y are rational numbers, then x + y and xy are rational numbers.

(2) Find all real numbers x that satisfy 1/x < x.

(3) A terminating decimal number is one that has ?nitely many digits, such as 1/8= 0.125. Binary

numbers are de?ned as decimal numbers, except that the base is 2 rather than 10. For example, the

decimal number 5.5 written in binary is: 101.1. Give a terminating binary representation of the the

number 7 /16.

(4) Find the value of the limit, limn?? sqrt(n^2 + 1 ?n)

.

(5) Let x1 = 8 and de?ne xn+1 = 1/2xn + 2 for all natural numbers n ? 1. Assuming that the sequence

converges, ?nd the limit, limn??xn.

Explanation / Answer

1
if x and y are rational numbers, then by definition they can be expressed as x = a/b and y = c/d for some integers a, b, c and d
x+y = (a/b) + (c/d) = (ad + cb) / bd. Since ad, cd, and bd are all integers, and since adding two integers gives you another integer, then what you have here again is an integer over an integer. So x+y is rational .

2 .
1/x < x
x - 1/x > 0
x^2 - 1 > 0
(x+1)(x-1) > 0
x > 1 & -1 < x < 0

3.
0.0111

4 .
limit if n-> 0 then 0
or if n-> infinity then infinity

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