# Mathematics for Applied Sciences (Osnabrück 2011-2012)/Part I/Exercise sheet 2

    Warm-up-exercises

### Exercise

Let $x,y,z,w$ be elements in a field, and suppose that $z$ and $w$ are not zero. Prove the following fraction rules.
    ${\frac {x}{1}}=x\,,$
    ${\frac {1}{z}}=z^{-1}\,,$
    ${\frac {1}{-1}}=-1\,,$
    ${\frac {0}{z}}=0\,,$
    ${\frac {z}{z}}=1\,,$
    ${\frac {x}{z}}={\frac {xw}{zw}}\,$
    ${\frac {x}{z}}\cdot {\frac {y}{w}}={\frac {xy}{zw}}\,,$
    ${\frac {x}{z}}+{\frac {y}{w}}={\frac {xw+yz}{zw}}\,.$
Does there exist an analogue of formula (8) that arises when one exchanges addition with multiplication (and division with subtraction), that is
    $(x-z)\cdot (y-w)=(x+w)\cdot (y+z)-(z+w)\,?$
Show that the popular formula
    ${\frac {x}{z}}+{\frac {y}{w}}={\frac {x+y}{z+w}}\,$
does not hold.

### Exercise *

Determine which of the two rational numbers $p$ and $q$ is larger:
    $p={\frac {573}{-1234}}{\text{ and }}q={\frac {-2007}{4322}}.$

### Exercise *

a) Give an example of rational numbers $a,b,c\in0,1$ such that
    $a^{2}+b^{2}=c^{2}\,.$
b) Give an example of rational numbers $a,b,c\in0,1$ such that
    $a^{2}+b^{2}\neq c^{2}\,.$
c) Give an example of irrational numbers $a,b\in0,1$ and a rational number $c\in0,1$ such that
    $a^{2}+b^{2}=c^{2}\,.$
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.
* $1\geq 0$.
* From $a\geq b$ and $c\geq 0$ follows $ac\geq bc$.
* From $a\geq b$ and $c\leq 0$ follows $ac\leq bc$.
* $a^{2}\geq 0$ holds.
* $a\geq b\geq 0$ implies $a^{n}\geq b^{n}$ for all $n\in \mathbb {N}$.
* From $a\geq 1$ follows $a^{n}\geq a^{m}$ for integers $n\geq m$.
* From $a>0$ follows ${\frac {1}{a}}>0$.
* From $a>b>0$ follows ${\frac {1}{a}}>{\frac {1}{b}}$.

### Exercise *

Show that for real numbers $x\geq 3$ the estimate
    $x^{2}+(x+1)^{2}\geq (x+2)^{2}\,$
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

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

### Exercise

Sketch the following subsets of $\mathbb {R} ^{2}$.
* ${\left\{(x,y)\mid x=5\right\}}$,
* ${\left\{(x,y)\mid x\geq 4{\text{ and }}y=3\right\}}$,
* ${\left\{(x,y)\mid y^{2}\geq 2\right\}}$,
* ${\left\{(x,y)\mid \vert {x}\vert =3{\text{ and }}\vert {y}\vert \leq 2\right\}}$,
* ${\left\{(x,y)\mid 3x\geq y{\text{ and }}5x\leq 2y\right\}}$,
* ${\left\{(x,y)\mid xy=0\right\}}$,
* ${\left\{(x,y)\mid xy=1\right\}}$,
* ${\left\{(x,y)\mid xy\geq 1{\text{ and }}y\geq x^{3}\right\}}$,
* ${\left\{(x,y)\mid 0=0\right\}}$,
* ${\left\{(x,y)\mid 0=1\right\}}$.
    Hand-in-exercises

### Exercise (2 marks)

Let $x_{1},\ldots ,x_{n}$ be real numbers. Show by induction the following inequality
    $\vert {\sum _{i=1}^{n}x_{i}}\vert \leq \sum _{i=1}^{n}\vert {x_{i}}\vert \,.$

### Exercise (5 marks)

Prove the general distributive law for a field.

### Exercise (3 marks)

Sketch the following subsets of $\mathbb {R} ^{2}$.
* ${\left\{(x,y)\mid x+y=3\right\}}$,
* ${\left\{(x,y)\mid x+y\leq 3\right\}}$,
* ${\left\{(x,y)\mid (x+y)^{2}\geq 4\right\}}$,
* ${\left\{(x,y)\mid \vert {x+2}\vert \geq 5{\text{ and }}\vert {y-2}\vert \leq 3\right\}}$,
* ${\left\{(x,y)\mid \vert {x}\vert =0{\text{ and }}\vert {y^{4}-2y^{3}+7y-5}\vert \geq -1\right\}}$,
* ${\left\{(x,y)\mid -1\leq x\leq 3{\text{ and }}0\leq y\leq x^{3}\right\}}$.

### Exercise (5 marks)

A page has been ripped off from a book. The sum of the numbers of the remaining pages is $65000$. 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.