On the recently discovered planet Trigeno there lives a species which has some ability to calculate. They use like us the rational numbers with "our“ addition and multiplication. They also use a kind of "ordering“ on the rational numbers, denoted by
. This trigenometric ordering coincides with our ordering as long as both numbers are
. However, they put
-
![{\displaystyle {}0\succeq x\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c1903d82fa8112b6f521e4304e2f0b9f2528843f)
for every rational number
. The well-known Ethnomathematician Dr. Eisenbeis thinks that this is related to the fact that they worship the number
.
Show that
fulfils the following properties.
- For any two elements
either
or
or
holds.
- From
and
we get
(for arbitrary
).
- From
and
we get
.
- From
and
we get
.
Which property of an ordered field does
![{\displaystyle {}(\mathbb {Q} ,\succeq )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7504e19868a21410e60ae126dd0b6e0524682800)
not fulfil?