# Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I/Exercise sheet 5

*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

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

holds.

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

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.

- There is a stop command.

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

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

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
- 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.

- For a real number its real absolute value and its complex absolute value coincide.
- We have if and only if .
- For we have .

*Hand-in-exercises*

### Exercise (2 marks)

Let be an ordered field and . Show that also the inverse element is negative.

### Exercise (2 marks)

Prove that a strictly increasing function

is injective.

### Exercise (4 marks)

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
.

### Exercise (3 marks)

Calculate the complex numbers

for .

### Exercise (3 marks)

Prove the following properties of the complex conjugation.

- .
- .
- .
- For we have .
- .
- if and only if .

### Exercise * (5 marks)

Calculate the square roots, the fourth roots and the eighth roots of .

### Exercise (3 marks)

Find the three complex numbers such that

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