Kurs:Mathematik für Anwender (Osnabrück 2011-2012)/Teil I/Arbeitsblatt 2/en

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Warm-up-exercises

Exercise

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

Does there exist an analogue of formula (7), which arises when one replaces addition by multiplication (and subtraction by division), that is

Show that the “popular formula”

does not hold.


Exercise

Determine which of the two rational numbers and is larger:


Exercise

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.

Exercise

Prove the following properties of real numbers.

  1. .
  2. From and it follows .
  3. From and it follows .
  4. holds.
  5. From it follows for all .
  6. From it follows für ganze Zahlen .
  7. From it follows .
  8. From it follows .


Exercise

Show that for a real number the inequality

holds.


Exercise

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

hold.


Exercise

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 the absolute value).


Exercise

Sketch the following subsets of .

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




Hand-in-exercises

Exercise (2 points)

Let be real numbers. Show by induction the following inequality


Exercise (5 points)

Prove the general distributive property for a field.


Exercise (3 points)

Sketch the following subsets of .

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


Exercise (5 points)

A page has been ripped 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.



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