Mathematics for Applied Sciences (Osnabrück 2011-2012)/Part I/Exercise sheet 2



Warm-up-exercises

Let   be elements in a field, and suppose that   and   are not zero. Prove the following fraction rules.

  1.  
  2.  
  3.  
  4.  
  5.  
  6.  
  7.  
  8.  

Does there exist an analogue of formula (8) that arises when one exchanges addition with multiplication (and division with subtraction), that is

 

Show that the popular formula

 

does not hold.


Determine which of the two rational numbers   and   is larger:

 


a) Give an example of rational numbers   such that

 

b) Give an example of rational numbers   such that

 

c) Give an example of irrational numbers   and a rational number   such that

 


The following exercises should only be made with reference to the ordering axioms of the real numbers.

Prove the following properties of real numbers.

  1.  .
  2. From   and   follows  .
  3. From   and   follows  .
  4.   holds.
  5.   implies   for all  .
  6. From   follows   for integers  .
  7. From   follows  .
  8. From   follows  .


Show that for real numbers   the estimate

 

holds.


Let   be two real numbers. Show that for the arithmetic mean   the inequalities

 

hold.


Prove the following properties for the absolute value function

 

(here let   be arbitrary real numbers).

  1.  .
  2.   if and only if  .
  3.   if and only if   or  .
  4.  .
  5.  .
  6. For   we have  .
  7. We have   (triangle inequality for modulus).
  8.  .


Sketch the following subsets of  .

  1.  ,
  2.  ,
  3.  ,
  4.  ,
  5.  ,
  6.  ,
  7.  ,
  8.  ,
  9.  ,
  10.  .




Hand-in-exercises

Exercise (2 marks)

Let   be real numbers. Show by induction the following inequality

 


Exercise (5 marks)

Prove the general distributive law for a field.


Exercise (3 marks)

Sketch the following subsets of  .

  1.  ,
  2.  ,
  3.  ,
  4.  ,
  5.  ,
  6.  .


Exercise (5 marks)

A page has been ripped off from a book. The sum of the numbers of the remaining pages is  . How many pages did the book have?


Hint: Show that it cannot be the last page. From the two statements A page is missing and The last page is not missing two inequalities can be set up to deliver the (reasonable) upper and lower bound for the number of pages.