Exercises Analysis I

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Winter Semester 1994/95
German Version 

The following exercises were performed at the LMU Munich. They represent only a little part of the whole lessons.
For a complete representation I don't have enough time. I have extended some parts of the exercises for giving background.

Exercise 1
Calculate  exact for 3 decimals
1 < 7 < 8 => 
1,93 = 6.859 < 7
1,913 = 6.967871 < 7 < 1,923 = 7.077888
=> 1,91 <  < 1,92
1,9113 = 6.978821031 < 7, 1,9123 = 6.989782528 < 7,
1,9133 = 7.000755497 > 7 => 1,912 <  < 1,913
=> 1,912 <  < 1,913

Exercise 2
proof
(a)  is irrational
(b) is irrational, if p is a prime number
(a) assume:  is rational  =>


stands for natural numbers

p, q teilerfremd: p, q can not be divided by the same
natural number
e.g. 17 and 23 are "teilerfremd", also 2 and 25, but 4 and 12 can be divided by the same number 2 (4 and 12 are not "teilerfremd"), 4 and 12 can also be divided by the same number 4
p3  ist durch 7 teilbar: p3 can be divided by 7
mit einem : with 
d.h. : that means
7 teilt q3 : q3 can be divided by 7
7 teilt q: 7 divides q
im Widersrich dazu: this contradicts
dass p und q teilerfremd sind: that p and q can not be divided by the same natural number

(b) assume:  is rational =>
a, b teilerfremd: a, b can not be divided by the same
natural number
ist durch p teilbar: can be divided by p
im Widersrich dazu: this contradicts
dass a und b teilerfremd sind: that a and b can not be divided by the same natural number

The foregoing proof uses the following properties of natural numbers: if the product x * y of two natural numbers can be divided by a prime number p, then x or y can be divided by p.

Setting x = y it follows, if x2 = x * x can be divided by p then x can be divided by p.