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

Exercises

Exercise *

Determine which of the two rational numbers ${}p$ and ${}q$ is larger:

$p={\frac {573}{-1234}}{\text{ and }}q={\frac {-2007}{4322}}.$

Exercise

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?

Exercise

A Bahncard ${}25$ costs ${}62$ Euros and allows for one year to save ${}25$ percentage of the standard price for the travels. A Bahncard ${}50$ costs ${}255$ Euros and allows for one year to save ${}50$ percentage of the standard price for the travels. Determine for which standard price no Bahncard, the Bahncard ${}25$ or the Bahncard ${}50$ is the best option.

Exercise

Two bicyclists, ${}A$ and ${}B$, drive on their bikes along a street. ${}A$ makes ${}40$ pedal turnings per minute, has a gear ratio of pedal to back wheel of ${}1$ to ${}6$ and tyres with a radius of ${}39$ centimeter. ${}B$ needs ${}2$ seconds for one pedal turning, has a gear ratio of pedal to back wheel of ${}1$ to ${}7$ and tyres with a radius of ${}45$ centimeter.

Who is driving faster?

Exercise

Show that in an ordered field the following properties hold.

1. ${}1\geq 0$.
2. ${}a\geq 0$ holds if and only if ${}-a\leq 0$ holds.
3. ${}a\geq b$ holds if and only if ${}a-b\geq 0$ holds.
4. ${}a\geq b$ holds if and only if ${}-a\leq -b$ holds.
5. ${}a\geq b$ and ${}c\geq d$ imply ${}a+c\geq b+d$.
6. ${}a\geq b$ and ${}c\geq 0$ imply ${}ac\geq bc$.
7. ${}a\geq b$ and ${}c\leq 0$ imply ${}ac\leq bc$.
8. ${}a\geq b\geq 0$ and ${}c\geq d\geq 0$ imply ${}ac\geq bd$.
9. ${}a\geq 0$ and ${}b\leq 0$ imply ${}ab\leq 0$.
10. ${}a\leq 0$ and ${}b\leq 0$ imply ${}ab\geq 0$.

Exercise

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 ${}\succeq$. This trigenometric ordering coincides with our ordering as long as both numbers are${}\neq 0$. However, they put

${}0\succeq x\,$

for every rational number ${}x$. The well-known Ethnomathematician Dr. Eisenbeis thinks that this is related to the fact that they worship the number ${}0$.

Show that ${}\succeq$ fulfils the following properties.

1. For any two elements ${}a,b\in \mathbb {Q}$ either ${}a\succ b$ or ${}a=b$ or ${}b\succ a$ holds.
2. From ${}a\succeq b$ and ${}b\succeq c$ we get ${}a\succeq c$ (for arbitrary ${}a,b,c\in \mathbb {Q}$).
3. From ${}a\succeq 0$ and ${}b\succeq 0$ we get ${}a+b\succeq 0$.
4. From ${}a\succeq 0$ and ${}b\succeq 0$ we get ${}ab\succeq 0$.

Which property of an ordered field does ${}(\mathbb {Q} ,\succeq )$ not fulfil?

Exercise

Show that in an ordered field the following properties hold.

1. We have ${}a^{2}\geq 0$.
2. If ${}a\geq b\geq 0$ holds, then also ${}a^{n}\geq b^{n}$ holds for all ${}n\in \mathbb {N}$.
3. From ${}a\geq 1$ we get ${}a^{n}\geq a^{m}$ for integer numbers ${}n\geq m$.

Exercise

Let ${}K$ be an ordered field and ${}x>0$. Show that also the inverse element ${}x^{-1}$ is positive.

Exercise

Let ${}K$ be an ordered field and ${}x\geq 1$. Show that for the inverse element ${}x^{-1}\leq 1$ holds.

Exercise

Let ${}K$ be an ordered field and ${}x>y>0$. Show that for the inverse elements ${}x^{-1}<y^{-1}$ holds.

Exercise

Let ${}K$ be an ordered field and let ${}x,y$ be positive elements. Show that ${}x\geq y$ is equivalent with ${}{\frac {x}{y}}\geq 1$.

Exercise

Let ${}K$ be an ordered field and ${}b\in K$, ${}b>1$. Show that there exists elements ${}c,d>1$ such that ${}b=cd$.

Exercise

Let ${}K$ denote an ordered field. We consider the mapping ${}\mathbb {Z} \rightarrow K$ constructed in exercise *****.

a) Show that this mapping is injective.

b) Show that this mapping can be extended to an injective mapping ${}\mathbb {Q} \rightarrow K$ such that the addition and multiplication in ${}\mathbb {Q}$ and in ${}K$ coincide, and such that the ordering of ${}\mathbb {Q}$ coincides with the ordering of ${}K$.

Exercise

Let ${}K$ denote an ordered field. Show that for ${}x\geq 3$ the relation

${}x^{2}+(x+1)^{2}\geq (x+2)^{2}\,$

holds.

Exercise

Let ${}x<y$ be two real numbers. Show that for the arithmetic mean ${}{\frac {x+y}{2}}$ the inequalities

${}x<{\frac {x+y}{2}}<y\,$

hold.

Exercise

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

$(a,b,c,d,0,0,0,\ldots )$

with ${}b,d\neq 0$. Here ${}a/b$ and ${}c/d$ 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.

Exercise

Discuss the operation

$\mathbb {R} _{\geq 0}\times \mathbb {R} _{\geq 0}\longrightarrow \mathbb {R} _{\geq 0},(x,y)\longmapsto \operatorname {max} \,(x,y),$

looking at associative law, commutative law, existence of a neutral element and existence of inverse element.

Exercise

Some bacterium wants to walk around the earth along the equator. It is quite small and during a day it makes exactly ${}2$ millimeter. How many days does it take for it to orbit the earth once?

Exercise

How many trillionths does it take to reach one billionth?

Exercise

In the forest, a giant is living, which height is ${}8$ meter and ${}37$ cm. There is also a colony of dwarfs, their height at the shoulder is ${}3$ cm and their height including the head is ${}4$. The neck and the head of the giant is ${}1,23$ 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?

Exercise

Show that in ${}\mathbb {R}$ the following properties hold.

1. For ${}x>0$ there exists a natural number ${}n$ such that ${}{\frac {1}{n}}<x$.
2. For two real numbers ${}x<y$ there exists a rational number ${}n/k$ (with ${}n\in \mathbb {Z}$, ${}k\in \mathbb {N} _{+}$) such that

   ${}x<{\frac {n}{k}}<y\,.$

Exercise

Compute the floor

$\left\lfloor {\frac {513}{21}}\right\rfloor .$

Exercise

Prove the following properties for the absolute value function

$\mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto \vert {x}\vert ,$

(here let ${}x,y$ be arbitrary real numbers).

1. ${}\vert {x}\vert \geq 0$.
2. ${}\vert {x}\vert =0$ if and only if ${}x=0$.
3. ${}\vert {x}\vert =\vert {y}\vert$ if and only if ${}x=y$ or ${}x=-y$.
4. ${}\vert {y-x}\vert =\vert {x-y}\vert$.
5. ${}\vert {xy}\vert =\vert {x}\vert \vert {y}\vert$.
6. For ${}x\neq 0$ we have ${}\vert {x^{-1}}\vert =\vert {x}\vert ^{-1}$.
7. We have ${}\vert {x+y}\vert \leq \vert {x}\vert +\vert {y}\vert$ (triangle inequality for modulus).
8. ${}\vert {x+y}\vert \geq \vert {x}\vert -\vert {y}\vert$.

Exercise

Let ${}x_{1},\ldots ,x_{n}$ be real numbers. Show by induction the following inequality

${}\vert {\sum _{i=1}^{n}x_{i}}\vert \leq \sum _{i=1}^{n}\vert {x_{i}}\vert \,.$

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 .

Exercise

We consider the mapping

$\Psi \colon \mathbb {N} ^{4}\longrightarrow \mathbb {N} ^{4}$

that assigns to a four tuple ${}(a,b,c,d)$ the four-tuple

$(\vert {b-a}\vert ,\vert {c-b}\vert ,\vert {d-c}\vert ,\vert {a-d}\vert ).$

We denote by ${}\Psi ^{n}$ the ${}n$-th fold composition of ${}\Psi$ with itself.

1. Compute

   $\Psi (6,5,2,8),\,\Psi ^{2}(6,5,2,8),\,\Psi ^{3}(6,5,2,8),\,\Psi ^{4}(6,5,2,8)\,...$

   until the result is the zero-tuple ${}(0,0,0,0)$.

2. Compute

   $\Psi (1,10,100,1000),\,\Psi ^{2}(1,10,100,1000),\,\Psi ^{3}(1,10,100,1000),\,\Psi ^{4}(1,10,100,1000)\,...$

   until the result is the zero-tuple ${}(0,0,0,0)$.

3. Show that ${}\Psi ^{4}(0,0,n,0)=(0,0,0,0)$ for every ${}n\in \mathbb {N}$.

Exercise

We consider the mapping

$\Psi \colon \mathbb {N} ^{4}\longrightarrow \mathbb {N} ^{4}$

that assigns to a four-tuple ${}(a,b,c,d)$ the four-tuple

$(\vert {b-a}\vert ,\vert {c-b}\vert ,\vert {d-c}\vert ,\vert {a-d}\vert ).$

Determine whether ${}\Psi$ is injective and whether ${}\Psi$ is surjective.

Exercise

We consider the mapping

$\Psi \colon \mathbb {N} ^{4}\longrightarrow \mathbb {N} ^{4}$

that assigns to a four-tuple ${}(a,b,c,d)$ the four-tuple

$(\vert {b-a}\vert ,\vert {c-b}\vert ,\vert {d-c}\vert ,\vert {a-d}\vert ).$

Show that for any initial value ${}(a,b,c,d)$, after finitely many iterations, this map reaches the zero-tuple.

Exercise

We consider the mapping

$\Psi \colon \mathbb {N} ^{4}\longrightarrow \mathbb {N} ^{4}$

that assigns to a four-tuple ${}(a,b,c,d)$ the four-tuple

$(\vert {b-a}\vert ,\vert {c-b}\vert ,\vert {d-c}\vert ,\vert {a-d}\vert ).$

Find an example of a four-tuple ${}(a,b,c,d)$ with the property that all iterations ${}\Psi ^{n}(a,b,c,d)$ for ${}n\leq 25$ 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 .

Exercise

Let ${}K$ be an ordered field and ${}n\in \mathbb {N} _{+}$. Show the following statements.

1. The mapping

   $K_{\geq 0}\longrightarrow K,x\longmapsto x^{n},$

   is strictly increasing.

2. The mapping

   $K_{\leq 0}\longrightarrow K,x\longmapsto x^{n},$

   is for ${}n$ odd strictly increasing.

3. The Mapping

   $K_{\leq 0}\longrightarrow K,x\longmapsto x^{n},$

   is for ${}n$ even strictly decreasing.

Exercise

Let

$f_{1},\ldots ,f_{n}\colon \mathbb {R} \longrightarrow \mathbb {R}$

be functions, which are increasing or decreasing, and let ${}f=f_{n}\circ \cdots \circ f_{1}$ be their composition. Let ${}k$ be the number of the decreasing functions among the ${}f_{i}$'s. Show that if ${}k$ is even, then ${}f$ is increasing, and if ${}k$ is odd, then ${}f$ is decreasing.

The solution to the exercises of complex numbers always has to be written like ${}a+b{\mathrm {i} }$ with real numbers ${}a,b$ whereas those have to be as simple as possible.

Exercise

Calculate the following expressions in the complex numbers.

1. ${}(5+4{\mathrm {i} })(3-2{\mathrm {i} })$.
2. ${}(2+3{\mathrm {i} })(2-4{\mathrm {i} })+3(1-{\mathrm {i} })$.
3. ${}(2{\mathrm {i} }+3)^{2}$.
4. ${}{\mathrm {i} }^{1011}$.
5. ${}(-2+5{\mathrm {i} })^{-1}$.
6. ${}{\frac {4-3{\mathrm {i} }}{2+{\mathrm {i} }}}$.

Exercise

Show that the complex numbers constitute a field.

Exercise

Show that ${}P=\mathbb {R} ^{2}$ with componentwise addition and componentwise multiplication is not a field.

Exercise

Prove the following statements concerning the real and imaginary parts of a complex number.

1. ${}z=\operatorname {Re} \,{\left(z\right)}+\operatorname {Im} \,{\left(z\right)}{\mathrm {i} }$.
2. ${}\operatorname {Re} \,{\left(z+w\right)}=\operatorname {Re} \,{\left(z\right)}+\operatorname {Re} \,{\left(w\right)}$.
3. ${}\operatorname {Im} \,{\left(z+w\right)}=\operatorname {Im} \,{\left(z\right)}+\operatorname {Im} \,{\left(w\right)}$.
4. For ${}r\in \mathbb {R}$ we have

   $\operatorname {Re} \,{\left(rz\right)}=r\operatorname {Re} \,{\left(z\right)}{\text{ und }}\operatorname {Im} \,{\left(rz\right)}=r\operatorname {Im} \,{\left(z\right)}.$

5. The equation ${}z=\operatorname {Re} \,{\left(z\right)}$ holds if and only if ${}z\in \mathbb {R}$ and this holds if and only if ${}\operatorname {Im} \,{\left(z\right)}=0$.

Exercise

Show that for a complex number ${}z$ the following relations hold.

1. ${}{\overline {z}}=\operatorname {Re} \,{\left(z\right)}-{\mathrm {i} }\operatorname {Im} \,{\left(z\right)}$.
2. ${}\operatorname {Re} \,{\left(z\right)}={\frac {z+{\overline {z}}}{2}}$.
3. ${}\operatorname {Im} \,{\left(z\right)}={\frac {z-{\overline {z}}}{2{\mathrm {i} }}}$.

Exercise

Prove the following properties of the absolute value of a complex number.

1. ${}\vert {z}\vert ={\sqrt {z\ {\overline {z}}}}\,.$

2. For a real number ${}z$ its real absolute value and its complex absolute value coincide.
3. We have ${}\vert {z}\vert =0$ if and only if ${}z=0$.

4. ${}\vert {z}\vert =\vert {\overline {z}}\vert \,.$

5. ${}\vert {zw}\vert =\vert {z}\vert \vert {w}\vert \,.$

6. For ${}z\neq 0$ we have ${}\vert {1/z}\vert =1/\vert {z}\vert$.

7. ${}\vert {\operatorname {Re} \,{\left(z\right)}}\vert ,\vert {\operatorname {Im} \,{\left(z\right)}}\vert \leq \vert {z}\vert \,.$

Hand-in-exercises

Exercise (2 marks)

Let ${}K$ be an ordered field and ${}x<0$. Show that also the inverse element ${}x^{-1}$ is negative.

Exercise (2 marks)

Prove that a strictly increasing function

$f\colon \mathbb {R} \longrightarrow \mathbb {R}$

is injective.

Exercise (4 marks)

We consider the mapping

$\Psi \colon \mathbb {Q} _{\geq 0}^{4}\longrightarrow \mathbb {Q} _{\geq 0}^{4}$

that assigns to a four-tuple of nonnegative rational numbers ${}(a,b,c,d)$ the four-tuple

$(\vert {b-a}\vert ,\vert {c-b}\vert ,\vert {d-c}\vert ,\vert {a-d}\vert )$

Show that after finitely many iterations, this mapping yields the zero-tuple.
Hint: Use Exercise 5.27 .

Exercise (3 marks)

Calculate the complex numbers

$(1+{\mathrm {i} })^{n}$

for ${}n=1,2,3,4,5$.

Exercise (3 marks)

Prove the following properties of the complex conjugation.

1. ${}{\overline {z+w}}={\overline {z}}+{\overline {w}}$.
2. ${}{\overline {-z}}=-{\overline {z}}$.
3. ${}{\overline {z\cdot w}}={\overline {z}}\cdot {\overline {w}}$.
4. For ${}z\neq 0$ we have ${}{\overline {1/z}}=1/{\overline {z}}$.
5. ${}{\overline {\overline {z}}}=z$.
6. ${}{\overline {z}}=z$ if and only if ${}z\in \mathbb {R}$.

Exercise * (5 marks)

Calculate the square roots, the fourth roots and the eighth roots of ${}{\mathrm {i} }$.

Exercise (3 marks)

Find the three complex numbers ${}z$ such that

${}z^{3}=1\,.$

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