- Exercises
Determine which of the two rational numbers
and
is larger:
-
There are two glasses on the table, in one there is red wine, in the other there is white wine, the same amount. Now a small empty glass is immersed into the red wine glass completely and the content is put into the white wine glass and mixed with its content
(in particular, there is enough space there).
After this, the small glass is immersed into the white wine glass completely and the content is put into the red wine glass. In the end, is there more red wine in the red wine glass than white wine in the white whine glass?
A Bahncard
costs
Euros and allows for one year to save
percentage of the standard price for the travels. A Bahncard
costs
Euros and allows for one year to save
percentage of the standard price for the travels. Determine for which standard price no Bahncard, the Bahncard
or the Bahncard
is the best option.
Two bicyclists,
and
,
drive on their bikes along a street.
makes
pedal turnings per minute, has a gear ratio of pedal to back wheel of
to
and tyres with a radius of
centimeter.
needs
seconds for one pedal turning, has a gear ratio of pedal to back wheel of
to
and tyres with a radius of
centimeter.
Who is driving faster?
Show that in an
ordered field
the following properties hold.
.
holds if and only if
holds.
holds if and only if
holds.
holds if and only if
holds.
and
imply
.
and
imply
.
and
imply
.
and
imply
.
and
imply
.
and
imply
.
On the recently discovered planet Trigeno there lives a species which has some ability to calculate. They use like us the rational numbers with "our“ addition and multiplication. They also use a kind of "ordering“ on the rational numbers, denoted by
. This trigenometric ordering coincides with our ordering as long as both numbers are
. However, they put
-
![{\displaystyle {}0\succeq x\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c1903d82fa8112b6f521e4304e2f0b9f2528843f)
for every rational number
. The well-known Ethnomathematician Dr. Eisenbeis thinks that this is related to the fact that they worship the number
.
Show that
fulfils the following properties.
- For any two elements
either
or
or
holds.
- From
and
we get
(for arbitrary
).
- From
and
we get
.
- From
and
we get
.
Which property of an ordered field does
not fulfil?
Show that in an
ordered field
the following properties hold.
- We have
.
- If
holds, then also
holds for all
.
- From
we get
for integer numbers
.
Let
be an
ordered field
and
.
Show that also the inverse element
is positive.
Let
be an
ordered field
and
.
Show that for the inverse element
holds.
Let
be an
ordered field
and
.
Show that for the
inverse elements
holds.
Let
be an
ordered field
and let
be positive elements. Show that
is equivalent with
.
Let
be an
ordered field
and
,
.
Show that there exists elements
such that
.
Let
denote an ordered field. We consider the mapping
constructed in
exercise *****.
a) Show that this mapping is injective.
b) Show that this mapping can be extended to an injective mapping
such that the addition and multiplication in
and in
coincide, and such that the ordering of
coincides with the ordering of
.
Let
denote an
ordered field.
Show that for
the relation
-
![{\displaystyle {}x^{2}+(x+1)^{2}\geq (x+2)^{2}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/429ef3c2ec21af639240d08ac5a77fdceb0e95b8)
holds.
Let
be two real numbers. Show that for the
arithmetic mean
the inequalities
-
![{\displaystyle {}x<{\frac {x+y}{2}}<y\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2198ff9519a5a37882fa0b94d9cbbef7310d2b14)
hold.
Write a computer-program
(pseudocode)
which computes the
arithmetic mean
of two given non negative rational numbers.
- The computer has as many memory units as needed, which can contain natural numbers.
- It can add the content of two memory units and write the result into another memory unit.
- It can multiply the content of two memory units and write the result into another memory unit.
- It can print contents of memory units and it can print given texts.
The initial configuration is
-
with
.
Here
and
represent the rational numbers from which we want to compute the arithmetic mean. The result should be printed
(in the form numerator denominator)
and then the program shall stop.
Discuss the
operation
-
looking at associativity, commutativity, existence of a neutral element and existence of inverse element.
Some bacterium wants to walk around the earth along the equator. It is quite small and during a day it makes exactly
millimeter. How many days does it take for it to orbit the earth once?
How many trillionths does it take to reach one billionth?
In the forest, a giant is living, which height is
meter and
cm. There is also a colony of dwarfs, their height at the shoulder is
cm and their height including the head is
. The neck and the head of the giant is
meter high. On the shoulder of the giant there stands a dwarf. How many dwarfs have to stand above each other
(on their shoulders)
such that the dwarf on top is at least on the level of the dwarf on the giant?
Show that in
the following properties hold.
- For
there exists a natural number
such that
.
- For two real numbers
there exists a rational number
(with
,
) such that
-
![{\displaystyle {}x<{\frac {n}{k}}<y\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3c17be7669571cd67a0248274a242e13ca11dc6b)
Compute the
floor
-
Prove the following properties for the
absolute value function
-
(here let
be arbitrary real numbers).
.
-
if and only if
.
if and only if
or
.
.
.
- For
we have
.
- We have
(triangle inequality for modulus).
.
Let
be real numbers. Show by
induction
the following inequality
-
![{\displaystyle {}\vert {\sum _{i=1}^{n}x_{i}}\vert \leq \sum _{i=1}^{n}\vert {x_{i}}\vert \,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7102656d72b8e10d14cc6ebf4b3355430f4382d8)
The idea of the following exercises came from http://jwilson.coe.uga.edu/emt725/Challenge/Challenge.html, also have a look at http://www.vier-zahlen.bplaced.net/raetsel.php .
We consider the mapping
-
which assigns to a four tuple
the four tuple
-
We denote by
the
-th fold
composition
of
with itself.
- Compute
-
until the result is the zero tuple
.
- Compute
-
until the result is the zero tuple
.
- Show
for every
.
We consider the mapping
-
which assigns to a four tuple
the four tuple
-
Determine whether
is
injective
and whether
is
surjective.
We consider the mapping
-
which assigns to a four tuple
the four tuple
-
Show that for any initial value
, after finitely many iterations this map reaches the zero tuple.
We consider the mapping
-
which assigns to a four tuple
the four tuple
-
Find an example of a four tuple
with the property that all iterations
for
do not yield the zero tuple. Check your result on http://www.vier-zahlen.bplaced.net/raetsel.php .
We will later deal with the question on how it is with real four tuples, see in particular
Exercise 28.10
.
Let
be an
ordered field
and
.
Show the following statements.
- The mapping
-
is
strictly increasing.
- The mapping
-
is for
odd
strictly increasing.
- The Mapping
-
is for
even
strictly decreasing.
Let
-
be functions, which are increasing or decreasing, and let
be their
composition.
Let
be the number of the decreasing functions among the
's. Show that if
is even, then
is
increasing,
and if
is odd, then
is
decreasing.
The solution to the exercises of complex numbers always has to be written like
with real numbers
whereas those have to be as simple as possible.
Calculate the following expressions in the
complex numbers.
.
.
.
.
.
.
Show that the
complex numbers
constitute a
field.
Show that
with componentwise addition and componentwise multiplication is not a
field.
Prove the following statements concerning the
real
and
imaginary
parts of a
complex number.
.
-
.
.
- For
we have
-
- The equation
holds if and only if
and this holds if and only if
.
Show that for a
complex number
the following relations hold.
.
.
.
Prove the following properties of the
absolute value
of a
complex number.
-
![{\displaystyle {}\vert {z}\vert ={\sqrt {z\ {\overline {z}}}}\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2fd947d4b336af3577b594925c6c80841786a025)
- For a real number
its real absolute value and its complex absolute value coincide.
- We have
if and only if
.
-
![{\displaystyle {}\vert {z}\vert =\vert {\overline {z}}\vert \,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fb5315eea600d0865c7b5c7ab6eda2a5d1a46a49)
-
![{\displaystyle {}\vert {zw}\vert =\vert {z}\vert \vert {w}\vert \,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b94153ca8c04a2287a6c1dafb14af75b2951d3f9)
- For
we have
.
-
![{\displaystyle {}\vert {\operatorname {Re} \,{\left(z\right)}}\vert ,\vert {\operatorname {Im} \,{\left(z\right)}}\vert \leq \vert {z}\vert \,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4d8ee8cd07b249c7d793ed4f199e3874a666f5e1)
- Hand-in-exercises
Let
be an
ordered field
and
.
Show that also the inverse element
is negative.
Prove that a strictly increasing function
-
is injective.
We consider the mapping
-
which assigns to a four tuple of nonnegative rational numbers
the four tuple
-
Show that after finitely many iterations, this mapping yields the zero tuple.
Hint: Use
Exercise 5.27
.
Calculate the
complex numbers
-
for
.
Prove the following properties of the
complex conjugation.
.
.
.
- For
we have
.
.
if and only if
.
Calculate the square roots, the fourth roots and the eighth roots of
.
Find the three complex numbers
such that
-
![{\displaystyle {}z^{3}=1\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/90f0a6f58a7d831807c0c76c4f3e5794e87f86a6)