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
-
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
not fulfil?