- Warm-up-exercises
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.
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
![{\displaystyle {}a,b\in {]0,1[}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7a8d5f76730252b7fbe1daec643ed47ad3102f98)
and a rational number
![{\displaystyle {}c\in {]0,1[}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/01fea4cb646de38fb17e72f70c364d1e8b8d4ec2)
such that
-
The following exercises should only be made with reference to the ordering axioms of the real numbers.
Show that for a
real number
the inequality
-
holds.
Let
be two real numbers. Show that for the
arithmetic mean
the inequalities
-

hold.
Sketch the following subsets of
.
,
,
,
,
,
,
,
,
,
.
- Hand-in-exercises
Let
be real numbers. Show by
induction
the following inequality
-
Prove the general distributive property for a
field.
Sketch the following subsets of
.
,
,
,
,
,
.
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.