# Kurs:Mathematik für Anwender (Osnabrück 2011-2012)/Teil I/Arbeitsblatt 3/en/latex

Zur Navigation springen Zur Suche springen

\setcounter{section}{3}

\zwischenueberschrift{Warm-up-exercises}

\inputexercise
{}
{

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

}
{} {}

\inputexercise
{}
{

Show that the binomial coefficients are natural numbers.

}
{} {}

\inputexercise
{}
{

Prove the formula
\mathdisp {2^n = \sum_{k=0}^n \binom { n } { k}} { . }

}
{} {}

\inputexercise
{}
{

Prove by induction for $n \geq 10$ the inequality
\mathdisp {3^n \geq n^4} { . }

}
{} {}

In the following computing tasks regarding complex numbers, the result must always be in the form $a+bi$, with real numbers $a,b$, and these should be as simple as possible.

\inputexercise
{}
{

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

}
{} {}

\inputexercise
{}
{

Show that the \definitionsverweis {complex numbers}{}{} constitute a \definitionsverweis {field}{}{.}

}
{} {}

\inputexercise
{}
{

Prove the following statements concerning the \definitionsverweis {real}{}{} and \definitionsverweis {imaginary}{}{} parts of a \definitionsverweis {complex number}{}{.} \aufzaehlungfuenf{$z= \operatorname{Re} \, { \left( z \right) } + \operatorname{Im} \, { \left( z \right) } { \mathrm i}$. }{$\operatorname{Re} \, { \left( z+w \right) } = \operatorname{Re} \, { \left( z \right) } + \operatorname{Re} \, { \left( w \right) }$. }{$\operatorname{Im} \, { \left( z+w \right) } = \operatorname{Im} \, { \left( z \right) } + \operatorname{Im} \, { \left( w \right) }$. }{For $r \in \R$ we have
\mathdisp {\operatorname{Re} \, { \left( rz \right) } =r \operatorname{Re} \, { \left( z \right) } \text{ and } \operatorname{Im} \, { \left( rz \right) } =r \operatorname{Im} \, { \left( z \right) }} { . }
}{ $z = \operatorname{Re} \, { \left( z \right) }$ if and only if $z \in \R$, and this is exactly the case when $\operatorname{Im} \, { \left( z \right) }=0$. }

}
{} {}

\inputexercise
{}
{

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

}
{} {}

\inputexercise
{}
{

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

}
{} {}

\inputexercise
{}
{

Check the formula we gave in Example 3.15 for the \definitionsverweis {square root}{}{} of a \definitionsverweis {complex number}{}{} $z=a+bi$, in the case $b <0$.

}
{} {}

\inputexercise
{}
{

Determine the two complex solutions of the equation
\mathdisp {z^2+5iz-3=0} { . }

}
{} {}

\zwischenueberschrift{Hand-in-exercises}

\inputexercise
{3}
{

Prove the following formula
\mathdisp {n 2^{n-1} = \sum_{k=0}^n k \binom { n } { k}} { . }

}
{} {}

\inputexercise
{3}
{

Calculate the \definitionsverweis {complex numbers}{}{}
\mathdisp {(1+i)^n} { }
for $n=1,2,3,4,5$.

}
{} {}

\inputexercise
{3}
{

Prove the following properties of the \definitionsverweis {complex conjugation}{}{.} \aufzaehlungsechs{$\overline{ z+w }= \overline{ z } + \overline{ w }$. }{$\overline{ -z } = - \overline{ z }$. }{$\overline{ z \cdot w }= \overline{ z } \cdot \overline{ w }$. }{For $z \neq 0$ we have $\overline{ 1/z } =1/\overline{ z }$. }{$\overline{ \overline{ z } } =z$. }{$\overline{ z } =z$ if and only if
\mathl{z \in \R}{}. }

}
{} {}

\inputexercise
{2}
{

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

}
{} {}

\inputexercise
{3}
{

Calculate the square root, the fourth root and the eighth root of $i$.

}
{} {}

\inputexercise
{4}
{

Find the three complex numbers $z$ such that
\mathdisp {z^3=1} { . }

}
{} {}