Exercises Analysis I

Home
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.
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
prove:
(a)  is irrational
(b) is irrational, if p is a prime number
(a) assume:  is rational  =>


(b) assume:  is rational




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.