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

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Warm-up-exercises

### Exercise

Show that the binomial coefficients satisfy the following recursive relation

${\displaystyle {}{\binom {n+1}{k}}={\binom {n}{k}}+{\binom {n}{k-1}}\,.}$

### Exercise

Show that the binomial coefficients are natural numbers.

### Exercise

Prove the formula

${\displaystyle 2^{n}=\sum _{k=0}^{n}{\binom {n}{k}}.}$

### Exercise

Prove by induction for ${\displaystyle {}n\geq 10}$ the inequality

${\displaystyle 3^{n}\geq n^{4}.}$

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

### Exercise

Calculate the following expressions in the complex numbers.

1. ${\displaystyle {}(5+4i)(3-2i)}$.
2. ${\displaystyle {}(2+3i)(2-4i)+3(1-i)}$.
3. ${\displaystyle {}(2i+3)^{2}}$.
4. ${\displaystyle {}i^{1011}}$.
5. ${\displaystyle {}(-2+5i)^{-1}}$.
6. ${\displaystyle {}{\frac {4-3i}{2+i}}}$.

### Exercise

Show that the complex numbers constitute a field.

### Exercise

Prove the following statements concerning the real and imaginary parts of a complex number.

1. ${\displaystyle {}z=\operatorname {Re} \,{\left(z\right)}+\operatorname {Im} \,{\left(z\right)}{\mathrm {i} }}$.
2. ${\displaystyle {}\operatorname {Re} \,{\left(z+w\right)}=\operatorname {Re} \,{\left(z\right)}+\operatorname {Re} \,{\left(w\right)}}$.
3. ${\displaystyle {}\operatorname {Im} \,{\left(z+w\right)}=\operatorname {Im} \,{\left(z\right)}+\operatorname {Im} \,{\left(w\right)}}$.
4. For ${\displaystyle {}r\in \mathbb {R} }$ we have
${\displaystyle \operatorname {Re} \,{\left(rz\right)}=r\operatorname {Re} \,{\left(z\right)}{\text{ and }}\operatorname {Im} \,{\left(rz\right)}=r\operatorname {Im} \,{\left(z\right)}.}$
5. ${\displaystyle {}z=\operatorname {Re} \,{\left(z\right)}}$ if and only if ${\displaystyle {}z\in \mathbb {R} }$, and this is exactly the case when ${\displaystyle {}\operatorname {Im} \,{\left(z\right)}=0}$.

### Exercise

Prove the following calculating rules for the complex numbers.

1. ${\displaystyle {}\vert {z}\vert ={\sqrt {z\ {\overline {z}}}}}$.
2. ${\displaystyle {}\operatorname {Re} \,{\left(z\right)}={\frac {z+{\overline {z}}}{2}}}$.
3. ${\displaystyle {}\operatorname {Im} \,{\left(z\right)}={\frac {z-{\overline {z}}}{2{\mathrm {i} }}}}$.
4. ${\displaystyle {}{\overline {z}}=\operatorname {Re} \,{\left(z\right)}-{\mathrm {i} }\operatorname {Im} \,{\left(z\right)}}$.
5. For ${\displaystyle {}z\neq 0}$ we have ${\displaystyle {}z^{-1}={\frac {\overline {z}}{\vert {z}\vert ^{2}}}}$.

### Exercise

Prove the following properties of the absolute value of a complex number.

1. For a real number ${\displaystyle {}z}$ its real absolute value and its complex absolute value coincide.
2. We have ${\displaystyle {}\vert {z}\vert =0}$ if and only if ${\displaystyle {}z=0}$.
3. ${\displaystyle {}\vert {z}\vert =\vert {\overline {z}}\vert }$.
4. ${\displaystyle {}\vert {zw}\vert =\vert {z}\vert \vert {w}\vert }$.
5. ${\displaystyle {}\vert {\operatorname {Re} \,{\left(z\right)}}\vert ,\vert {\operatorname {Im} \,{\left(z\right)}}\vert \leq \vert {z}\vert }$.
6. For ${\displaystyle {}z\neq 0}$ we have ${\displaystyle {}\vert {1/z}\vert =1/\vert {z}\vert }$.

### Exercise

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

### Exercise

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

Hand-in-exercises

### Exercise (3 points)

Prove the following formula

${\displaystyle n2^{n-1}=\sum _{k=0}^{n}k{\binom {n}{k}}.}$

### Exercise (3 points)

Calculate the complex numbers

${\displaystyle (1+i)^{n}}$
for ${\displaystyle {}n=1,2,3,4,5}$.

### Exercise (3 points)

Prove the following properties of the complex conjugation.

1. ${\displaystyle {}{\overline {z+w}}={\overline {z}}+{\overline {w}}}$.
2. ${\displaystyle {}{\overline {-z}}=-{\overline {z}}}$.
3. ${\displaystyle {}{\overline {z\cdot w}}={\overline {z}}\cdot {\overline {w}}}$.
4. For ${\displaystyle {}z\neq 0}$ we have ${\displaystyle {}{\overline {1/z}}=1/{\overline {z}}}$.
5. ${\displaystyle {}{\overline {\overline {z}}}=z}$.
6. ${\displaystyle {}{\overline {z}}=z}$ if and only if ${\displaystyle {}z\in \mathbb {R} }$.

### Exercise (2 points)

Let ${\displaystyle {}a,b,c\in \mathbb {C} }$ with ${\displaystyle {}a\neq 0}$. Show that the equation

${\displaystyle az^{2}+bz+c=0}$
has at least one complex solution ${\displaystyle {}z}$.

### Exercise (3 points)

Calculate the square root, the fourth root and the eighth root of ${\displaystyle {}i}$.

### Exercise (4 points)

Find the three complex numbers ${\displaystyle {}z}$ such that

${\displaystyle z^{3}=1.}$