Mathematics for Applied Sciences (Osnabrück 2011-2012)/Part I/Exercise sheet 3
- Warm-up-exercises
Show that the binomial coefficients satisfy the following recursive relation
Show that the binomial coefficients are natural numbers.
Prove the formula
Show by induction that for the estimate
holds.
In the following computing tasks regarding complex numbers, the result must always be given in the form , with real numbers , and these should be as simple as possible.
Calculate the following expressions in the complex numbers.
- .
- .
- .
- .
- .
- .
Show that the complex numbers constitute a field.
Prove the following statements concerning the real and imaginary parts of a complex number.
- .
- .
- .
- For
we have
- The equation holds if and only if and this holds if and only if .
Prove the following calculating rules for the complex numbers.
- .
- .
- .
- .
- For we have .
Prove the following properties of the absolute value of a complex number.
- For a real number its real absolute value and its complex absolute value coincide.
- We have if and only if .
- For we have .
Check the formula we gave in example for the square root of a complex number
in the case .
Determine the two complex solutions of the equation
- Hand-in-exercises
Exercise (3 marks)
Prove the following formula
Exercise (3 marks)
Calculate the complex numbers
for .
Exercise (3 marks)
Prove the following properties of the complex conjugation.
- .
- .
- .
- For we have .
- .
- if and only if .
Exercise (2 marks)
Let with . Show that the equation
has at least one complex solution .
Exercise * (5 marks)
Calculate the square roots, the fourth roots and the eighth roots of .
Exercise (3 marks)
Find the three complex numbers such that