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.

1. ${}{\frac {x}{1}}=x\,,$

2. ${}{\frac {1}{z}}=z^{-1}\,,$

3. ${}{\frac {1}{-1}}=-1\,,$

4. ${}{\frac {0}{z}}=0\,,$

5. ${}{\frac {z}{z}}=1\,,$

6. ${}{\frac {x}{z}}={\frac {xw}{zw}}\,$

7. ${}{\frac {x}{z}}\cdot {\frac {y}{w}}={\frac {xy}{zw}}\,,$

8. ${}{\frac {x}{z}}+{\frac {y}{w}}={\frac {xw+yz}{zw}}\,.$

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 ${}a,b,c\in {]0,1[}$ such that

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

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

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

c) Give an example of irrational numbers ${}a,b\in {]0,1[}$ and a rational number ${}c\in {]0,1[}$ 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. ${}1\geq 0$.
2. From ${}a\geq b$ and ${}c\geq 0$ follows ${}ac\geq bc$.
3. From ${}a\geq b$ and ${}c\leq 0$ follows ${}ac\leq bc$.
4. ${}a^{2}\geq 0$ holds.
5. ${}a\geq b\geq 0$ implies ${}a^{n}\geq b^{n}$ for all ${}n\in \mathbb {N}$.
6. From ${}a\geq 1$ follows ${}a^{n}\geq a^{m}$ for integers ${}n\geq m$.
7. From ${}a>0$ follows ${}{\frac {1}{a}}>0$.
8. 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).

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

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

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

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 ${}\mathbb {R} ^{2}$.

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

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.