- Warm-up-exercises
Let
be elements in a field and suppose that
and
are not zero. Prove the following fraction rules.
-
-
![{\displaystyle {}{\frac {x}{1}}=x\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7726c5274bfca0a0675f777a2b267f128bdf6773)
-
![{\displaystyle {}{\frac {1}{z}}=z^{-1}\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c0cb6f560ae3533855eff66b2e3634f213d4f5bf)
-
![{\displaystyle {}{\frac {1}{-1}}=-1\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6d33e4048c7b56006e72c27e909d4fb99aa6848a)
-
![{\displaystyle {}{\frac {0}{z}}=0\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/95992602052e4ab8d9593a556d1658d1715de0c4)
-
-
![{\displaystyle {}{\frac {z}{z}}=1\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1214c5baf39a9c3fafe6369643e69a6f1ff1c8d7)
-
![{\displaystyle {}{\frac {x}{z}}={\frac {xw}{zw}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f2266090d42e38af933bd7f534ba1d8e4a0b9f7c)
-
![{\displaystyle {}{\frac {x}{z}}\cdot {\frac {y}{w}}={\frac {xy}{zw}}\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/717e70395683c6d78cbfaa16e8d2ce34d686ad9e)
-
![{\displaystyle {}{\frac {x}{z}}+{\frac {y}{w}}={\frac {xw+yz}{zw}}\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e0b5658c5d20bf6d91447a7437518877886f6991)
Does there exist an analogue of formula (8), which arises when one replaces addition by multiplication (and subtraction by division), that is
-
![{\displaystyle {}(x-z)\cdot (y-w)=(x+w)\cdot (y+z)-(z+w)\,?}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0fc3611de915a95f229d55f61ed528ff123ae3e7)
Show that the popular formula
-
![{\displaystyle {}{\frac {x}{z}}+{\frac {y}{w}}={\frac {x+y}{z+w}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fdd89b9acb2e2fc5ebaab34bf4d750711cee8435)
does not hold.
Determine which of the two rational numbers
and
is larger:
-
a) Give an example of rational numbers
such that
-
![{\displaystyle {}a^{2}+b^{2}=c^{2}\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fdb76114e0284e15753b6b460b03c6babf3287f9)
b) Give an example of rational numbers
such that
-
![{\displaystyle {}a^{2}+b^{2}\neq c^{2}\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2d8ffbd3b3fdcc3cc732f2b33b09d70859d3fbb3)
c) Give an example of irrational numbers
and a rational number
such that
-
![{\displaystyle {}a^{2}+b^{2}=c^{2}\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fdb76114e0284e15753b6b460b03c6babf3287f9)
The following exercises should only be made with reference to the ordering axioms of the real numbers.
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
.
Show that for
real numbers
the estimate
-
![{\displaystyle {}x^{2}+(x+1)^{2}\geq (x+2)^{2}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/429ef3c2ec21af639240d08ac5a77fdceb0e95b8)
holds.
Let
be two real numbers. Show that for the
arithmetic mean
the inequalities
-
![{\displaystyle {}x<{\frac {x+y}{2}}<y\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2198ff9519a5a37882fa0b94d9cbbef7310d2b14)
hold.
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).
.
Sketch the following subsets of
.
,
,
,
,
,
,
,
,
,
.
- Hand-in-exercises
Let
be real numbers. Show by
induction
the following inequality
-
![{\displaystyle {}\vert {\sum _{i=1}^{n}x_{i}}\vert \leq \sum _{i=1}^{n}\vert {x_{i}}\vert \,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7102656d72b8e10d14cc6ebf4b3355430f4382d8)
Prove the general distributive property for a
field.
Sketch the following subsets of
.
,
,
,
,
,
.
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.