Mathematics for Applied Sciences (Osnabrück 2011-2012)/Part I/Exercise sheet 2
- Warm-up-exercises
Exercise
Let be elements in a field and suppose that and are not zero. Prove the following fraction rules.
Does there exist an analogue of formula (8), which arises when one replaces addition by multiplication (and subtraction by division), that is
Show that the popular formula
does not hold.
Exercise *
Determine which of the two rational numbers and is larger:
Exercise *
a) Give an example of rational numbers such that
b) Give an example of rational numbers such that
c) Give an example of irrational numbers and a rational number such that
The following exercises should only be made with reference to the ordering axioms of the real numbers.
Exercise
Prove the following properties of real numbers.
- .
- From and follows .
- From and follows .
- holds.
- implies for all .
- From follows for integers .
- From follows .
- From follows .
Exercise *
Show that for real numbers the estimate
holds.
Exercise
Let be two real numbers. Show that for the arithmetic mean the inequalities
hold.
Exercise
Prove the following properties for the absolute value function
(here let be arbitrary real numbers).
- .
- if and only if .
- if and only if or .
- .
- .
- For we have .
- We have (triangle inequality for modulus).
- .
Exercise
Sketch the following subsets of .
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- ,
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- ,
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- ,
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- Hand-in-exercises
Exercise (2 marks)
Let be real numbers. Show by induction the following inequality
Exercise (5 marks)
Prove the general distributive property for a field.
Exercise (3 marks)
Sketch the following subsets of .
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- ,
- ,
- ,
- ,
- .
Exercise (5 marks)
A page has been ripped off from a book. The sum of the numbers of the remaining pages is . How many pages did the book have?
Hint: Show that it cannot be the last page. From the two statements A page is missing and The last page is not missing two inequalities can be set up to deliver the (reasonable) upper and lower bound for the number of pages.