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
Exercise
Show that the binomial coefficients are natural numbers.
Exercise
Prove the formula
Exercise *
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.
Exercise
Calculate the following expressions in the complex numbers.
- .
- .
- .
- .
- .
- .
Exercise
Show that the complex numbers constitute a field.
Exercise
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 .
Exercise
Prove the following calculating rules for the complex numbers.
- .
- .
- .
- .
- For we have .
Exercise
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 .
Exercise
Check the formula we gave in example for the square root of a complex number
in the case .
Exercise
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