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

Warm-up-exercises

### Exercise

Let ${\displaystyle {}x,y,z,w}$ be elements in a field and suppose that ${\displaystyle {}z}$ and ${\displaystyle {}w}$ are not zero. Prove the following fraction rules.

1. ${\displaystyle {}{\frac {x}{1}}=x\,,}$
2. ${\displaystyle {}{\frac {1}{z}}=z^{-1}\,,}$
3. ${\displaystyle {}{\frac {1}{-1}}=-1\,,}$
4. ${\displaystyle {}{\frac {0}{z}}=0\,,}$
5. ${\displaystyle {}{\frac {z}{z}}=1\,,}$
6. ${\displaystyle {}{\frac {x}{z}}={\frac {xw}{zw}}\,}$
7. ${\displaystyle {}{\frac {x}{z}}\cdot {\frac {y}{w}}={\frac {xy}{zw}}\,,}$
8. ${\displaystyle {}{\frac {x}{z}}+{\frac {y}{w}}={\frac {xw+yz}{zw}}\,.}$

Does there exist an analogue of formula (8), which arises when one replaces addition by multiplication (and subtraction by division), that is

${\displaystyle {}(x-z)\cdot (y-w)=(x+w)\cdot (y+z)-(z+w)\,?}$

Show that the popular formula

${\displaystyle {}{\frac {x}{z}}+{\frac {y}{w}}={\frac {x+y}{z+w}}\,}$

does not hold.

### Exercise *

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

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

### Exercise *

a) Give an example of rational numbers ${\displaystyle {}a,b,c\in {]0,1[}}$ such that

${\displaystyle {}a^{2}+b^{2}=c^{2}\,.}$

b) Give an example of rational numbers ${\displaystyle {}a,b,c\in {]0,1[}}$ such that

${\displaystyle {}a^{2}+b^{2}\neq c^{2}\,.}$

c) Give an example of irrational numbers ${\displaystyle {}a,b\in {]0,1[}}$ and a rational number ${\displaystyle {}c\in {]0,1[}}$ such that

${\displaystyle {}a^{2}+b^{2}=c^{2}\,.}$

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. ${\displaystyle {}1\geq 0}$.
2. From ${\displaystyle {}a\geq b}$ and ${\displaystyle {}c\geq 0}$ follows ${\displaystyle {}ac\geq bc}$.
3. From ${\displaystyle {}a\geq b}$ and ${\displaystyle {}c\leq 0}$ follows ${\displaystyle {}ac\leq bc}$.
4. ${\displaystyle {}a^{2}\geq 0}$ holds.
5. ${\displaystyle {}a\geq b\geq 0}$ implies ${\displaystyle {}a^{n}\geq b^{n}}$ for all ${\displaystyle {}n\in \mathbb {N} }$.
6. From ${\displaystyle {}a\geq 1}$ follows ${\displaystyle {}a^{n}\geq a^{m}}$ for integers ${\displaystyle {}n\geq m}$.
7. From ${\displaystyle {}a>0}$ follows ${\displaystyle {}{\frac {1}{a}}>0}$.
8. From ${\displaystyle {}a>b>0}$ follows ${\displaystyle {}{\frac {1}{a}}>{\frac {1}{b}}}$.

### Exercise *

Show that for real numbers ${\displaystyle {}x\geq 3}$ the estimate

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

holds.

### Exercise

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

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

hold.

### Exercise

Prove the following properties for the absolute value function

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

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

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

### Exercise

Sketch the following subsets of ${\displaystyle {}\mathbb {R} ^{2}}$.

1. ${\displaystyle {}{\left\{(x,y)\mid x=5\right\}}}$,
2. ${\displaystyle {}{\left\{(x,y)\mid x\geq 4{\text{ and }}y=3\right\}}}$,
3. ${\displaystyle {}{\left\{(x,y)\mid y^{2}\geq 2\right\}}}$,
4. ${\displaystyle {}{\left\{(x,y)\mid \vert {x}\vert =3{\text{ and }}\vert {y}\vert \leq 2\right\}}}$,
5. ${\displaystyle {}{\left\{(x,y)\mid 3x\geq y{\text{ and }}5x\leq 2y\right\}}}$,
6. ${\displaystyle {}{\left\{(x,y)\mid xy=0\right\}}}$,
7. ${\displaystyle {}{\left\{(x,y)\mid xy=1\right\}}}$,
8. ${\displaystyle {}{\left\{(x,y)\mid xy\geq 1{\text{ and }}y\geq x^{3}\right\}}}$,
9. ${\displaystyle {}{\left\{(x,y)\mid 0=0\right\}}}$,
10. ${\displaystyle {}{\left\{(x,y)\mid 0=1\right\}}}$.

Hand-in-exercises

### Exercise (p points)

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

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

### Exercise (p points)

Prove the general distributive property for a field.

### Exercise (p points)

Sketch the following subsets of ${\displaystyle {}\mathbb {R} ^{2}}$.

1. ${\displaystyle {}{\left\{(x,y)\mid x+y=3\right\}}}$,
2. ${\displaystyle {}{\left\{(x,y)\mid x+y\leq 3\right\}}}$,
3. ${\displaystyle {}{\left\{(x,y)\mid (x+y)^{2}\geq 4\right\}}}$,
4. ${\displaystyle {}{\left\{(x,y)\mid \vert {x+2}\vert \geq 5{\text{ and }}\vert {y-2}\vert \leq 3\right\}}}$,
5. ${\displaystyle {}{\left\{(x,y)\mid \vert {x}\vert =0{\text{ and }}\vert {y^{4}-2y^{3}+7y-5}\vert \geq -1\right\}}}$,
6. ${\displaystyle {}{\left\{(x,y)\mid -1\leq x\leq 3{\text{ and }}0\leq y\leq x^{3}\right\}}}$.

### Exercise (p points)

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