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Erschienen in:
2010 | OriginalPaper | Buchkapitel
Embedding Z in R
verfasst von : Matthias Beck, Ross Geoghegan
Erschienen in: The Art of Proof
Verlag: Springer New York
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We have now defined two number systems,
Z
and
R
. Intuitively, we think of the integers as a subset of the real numbers; however, nothing in our axioms tells us explicitly that
Z
can be viewed as a subset of
R
. In fact, at the moment we have no axiomatic reason to think that the integers we named 0 and 1 are the same as the real numbers we named 0 and 1. Just for now, we will be more careful and write 0
Z
and 1
Z
for these special members of
Z
, and 0
R
and 1
R
for the corresponding special members of
R
. Informally we are accustomed to identifying 0
Z
with 0
R
R and identifying 1
Z
with 1
R
. We will justify this here by giving an embedding of
Z
into
R
, that is, a function that maps each integer to the corresponding number in R.
Before You Get Started
. How could such an
embedding
function of
Z
into
R
be constructed? From what you know about functions, what properties will such a function have?