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.

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

Exercise

Show that the complex numbers constitute a field.

Exercise

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

1. ${}z=\operatorname {Re} \,{\left(z\right)}+\operatorname {Im} \,{\left(z\right)}{\mathrm {i} }$.
2. ${}\operatorname {Re} \,{\left(z+w\right)}=\operatorname {Re} \,{\left(z\right)}+\operatorname {Re} \,{\left(w\right)}$.
3. ${}\operatorname {Im} \,{\left(z+w\right)}=\operatorname {Im} \,{\left(z\right)}+\operatorname {Im} \,{\left(w\right)}$.
4. For ${}r\in \mathbb {R}$ we have

   $\operatorname {Re} \,{\left(rz\right)}=r\operatorname {Re} \,{\left(z\right)}{\text{ und }}\operatorname {Im} \,{\left(rz\right)}=r\operatorname {Im} \,{\left(z\right)}.$

5. 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.

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

1. ${}\vert {z}\vert ={\sqrt {z\ {\overline {z}}}}\,.$

2. For a real number ${}z$ its real absolute value and its complex absolute value coincide.
3. We have ${}\vert {z}\vert =0$ if and only if ${}z=0$.

4. ${}\vert {z}\vert =\vert {\overline {z}}\vert \,.$

5. ${}\vert {zw}\vert =\vert {z}\vert \vert {w}\vert \,.$

6. For ${}z\neq 0$ we have ${}\vert {1/z}\vert =1/\vert {z}\vert$.

7. ${}\vert {\operatorname {Re} \,{\left(z\right)}}\vert ,\vert {\operatorname {Im} \,{\left(z\right)}}\vert \leq \vert {z}\vert \,.$

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.

1. ${}{\overline {z+w}}={\overline {z}}+{\overline {w}}$.
2. ${}{\overline {-z}}=-{\overline {z}}$.
3. ${}{\overline {z\cdot w}}={\overline {z}}\cdot {\overline {w}}$.
4. For ${}z\neq 0$ we have ${}{\overline {1/z}}=1/{\overline {z}}$.
5. ${}{\overline {\overline {z}}}=z$.
6. ${}{\overline {z}}=z$ 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

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