Integer
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The
Integer
s are the whole
number
s, both positive and negative and including
zero
. The usual symbol for the
set
of
integer
s is a "blackboard bold"
Z
- a
Z
with two strokes on the diagonal.
${\bb~Z}=\{\ldots,{\quad}-3,{\quad}-2,{\quad}-1,{\quad}0,{\quad}1,{\quad}2,{\quad}3,{\quad}\ldots}$
The
integer
s have a natural
embedding
into the
rational number
s, and contain the
natural number
s.
More technical stuff ...
The
integer
s can be constructed from the counting
number
s (the
natural number
s not including 0) as follows:
Let P be the collection of all pairs of whole
number
s:
$P=\{(a,b):a,b\in{N}}$
Let
(a,b)
be equivalent to
(c,d)
if
a+d=b+c.
For any pair
(a,b)
we can consider the collection of all pairs equivalent to it.
This collection is called the equivance class of
(a,b),
and we write it as
E(a,b)
We can now define
arithmetic
operation
s on the
equivalence class
es:
The sum is obtained as
E(a,b)+E(c,d) = E(a+c,b+d)
The difference is obtained as
E(a,b)-E(c,d) = E(a+d,b+c)
The product:
E(a,b)*E(c,d) = E(ac+bd,ad+bc)
The
equivalence class
es can be thought of as the
integer
s.
We think of
E(a,b)
as being "the same as"
a-b.
Lots and lots of checking required to see that the
arithmetic
on the
equivalence class
es is "the same as" the
arithmetic
on the
integer
s.
The same technique of
equivalence class
es of pairs can be used to create the
rational number
s from the
integer
s.
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