# Mathematics for Applied Sciences (Osnabrück 2011-2012)/Part I/Exercise sheet 3

    Warm-up-exercises

### Exercise

Show that the binomial coefficients satisfy the following recursive relation
    ${\binom {n+1}{k}}={\binom {n}{k}}+{\binom {n}{k-1}}\,.$

### Exercise

Show that the binomial coefficients are natural numbers.

### Exercise

Prove the formula
    $2^{n}=\sum _{k=0}^{n}{\binom {n}{k}}\,.$

### Exercise *

Show by induction that for $n\geq 10$ the estimate
    $3^{n}\geq n^{4}\,$
holds.
In the following computing tasks regarding complex numbers, the result must always be given in the form $a+b{\mathrm {i} }$, with real numbers $a,b$, and these should be as simple as possible.

### Exercise

Calculate the following expressions in the complex numbers.
* $(5+4{\mathrm {i} })(3-2{\mathrm {i} })$.
* $(2+3{\mathrm {i} })(2-4{\mathrm {i} })+3(1-{\mathrm {i} })$.
* $(2{\mathrm {i} }+3)^{2}$.
* ${\mathrm {i} }^{1011}$.
* $(-2+5{\mathrm {i} })^{-1}$.
* ${\frac {4-3{\mathrm {i} }}{2+{\mathrm {i} }}}$.

### Exercise

Show that the complex numbers constitute a field.

### Exercise

Prove the following statements concerning the real and imaginary parts of a complex number.
* $z=\operatorname {Re} \,{\left(z\right)}+\operatorname {Im} \,{\left(z\right)}{\mathrm {i} }$.
* $\operatorname {Re} \,{\left(z+w\right)}=\operatorname {Re} \,{\left(z\right)}+\operatorname {Re} \,{\left(w\right)}$.
* $\operatorname {Im} \,{\left(z+w\right)}=\operatorname {Im} \,{\left(z\right)}+\operatorname {Im} \,{\left(w\right)}$.
* For $r\in \mathbb {R}$ we have
    $\operatorname {Re} \,{\left(rz\right)}=r\operatorname {Re} \,{\left(z\right)}{\text{ und }}\operatorname {Im} \,{\left(rz\right)}=r\operatorname {Im} \,{\left(z\right)}.$
* 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

Prove the following calculating rules for the complex numbers.
* $\vert {z}\vert ={\sqrt {z\ {\overline {z}}}}$.
* $\operatorname {Re} \,{\left(z\right)}={\frac {z+{\overline {z}}}{2}}$.
* $\operatorname {Im} \,{\left(z\right)}={\frac {z-{\overline {z}}}{2{\mathrm {i} }}}$.
* ${\overline {z}}=\operatorname {Re} \,{\left(z\right)}-{\mathrm {i} }\operatorname {Im} \,{\left(z\right)}$.
* For $z\neq 0$ we have $z^{-1}={\frac {\overline {z}}{\vert {z}\vert ^{2}}}$.

### Exercise

Prove the following properties of the absolute value of a complex number.
    $\vert {z}\vert ={\sqrt {z\ {\overline {z}}}}\,.$
* For a real number $z$ its real absolute value and its complex absolute value coincide.
* We have $\vert {z}\vert =0$ if and only if $z=0$.
    $\vert {z}\vert =\vert {\overline {z}}\vert \,.$
    $\vert {zw}\vert =\vert {z}\vert \vert {w}\vert \,.$
* For $z\neq 0$ we have $\vert {1/z}\vert =1/\vert {z}\vert$.
    $\vert {\operatorname {Re} \,{\left(z\right)}}\vert ,\vert {\operatorname {Im} \,{\left(z\right)}}\vert \leq \vert {z}\vert \,.$

### Exercise

Check the formula we gave in example for the square root of a complex number
    $z=a+b{\mathrm {i} }\,$
in the case $b<0$.

### Exercise

Determine the two complex solutions of the equation
    $z^{2}+5{\mathrm {i} }z-3=0\,.$
    Hand-in-exercises

### Exercise (3 marks)

Prove the following formula
    $n2^{n-1}=\sum _{k=0}^{n}k{\binom {n}{k}}\,.$

### 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.
* ${\overline {z+w}}={\overline {z}}+{\overline {w}}$.
* ${\overline {-z}}=-{\overline {z}}$.
* ${\overline {z\cdot w}}={\overline {z}}\cdot {\overline {w}}$.
* For $z\neq 0$ we have ${\overline {1/z}}=1/{\overline {z}}$.
* ${\overline {\overline {z}}}=z$.
* ${\overline {z}}=z$ if and only if $z\in \mathbb {R}$.

### Exercise (2 marks)

Let $a,b,c\in \mathbb {C}$ with $a\neq 0$. Show that the equation
    $az^{2}+bz+c=0\,$
has at least one complex solution $z$.

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