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


Exercise

Show that the binomial coefficients are natural numbers.


Exercise

Prove the formula


Exercise

Prove by induction for the inequality


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

Exercise

Calculate the following expressions in the complex numbers.

  1. .
  2. .
  3. .
  4. .
  5. .
  6. .


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. .
  2. .
  3. .
  4. For we have
  5. if and only if , and this is exactly the case when .


Exercise

Prove the following calculating rules for the complex numbers.

  1. .
  2. .
  3. .
  4. .
  5. For we have .


Exercise

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

  1. For a real number its real absolute value and its complex absolute value coincide.
  2. We have if and only if .
  3. .
  4. .
  5. .
  6. For we have .


Exercise

Check the formula we gave in Example 3.15 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 points)

Prove the following formula


Exercise (3 points)

Calculate the complex numbers

for .


Exercise (3 points)

Prove the following properties of the complex conjugation.

  1. .
  2. .
  3. .
  4. For we have .
  5. .
  6. if and only if .


Exercise (2 points)

Let with . Show that the equation

has at least one complex solution .


Exercise (3 points)

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


Exercise (4 points)

Find the three complex numbers such that




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