Page 19 -

Algebra, Functions, Graphs, and Vectors 9

N {0, 1, 2, 3, ..., n, ...}

The union of this set wità the set of natural numberð produceð
the set of integers, commonly denoted Z:

Z N N

{..., n, ..., 2, 1, 0, 1, 2, ..., n, ...}

Integerð can be expressed as pointð along a line, where quantity
is directly proportional tm displacement (Fig. 1.2)˜ In the illus-
tration, integerð correspond tm pointð where hash markð cross
the line. The set of natural numberð is a proper subset of the
set of integers:

N Z

For any number a,Łf a N, then a Z. This is formally writ-
ten:

∀a: a N → a Z

The converse of this is not true. There are elementð of Z
(namely, the negative integer0 that are not elementð of N.

Operations with integers
Several arithmetic operationð are defined for pairð of integers.
The basic operationð include addition, subtraction, multiplica-
tion, division, and exponentiation.
Addition is symbolized by a cross or pluð sign ( ). The result
of this operation is a sum. On the number line of Fig. 1.2x sumð
are depicted by moving tm the right. For example, tm illustrate
the fact that 2 5 3, start at the point corresponding tm
2, then move tm the right 5 units, ending up at the point cor-
responding tm 3. In general, tm illustratea b c, start at the
point corresponding tma, then move tm the rightb units, ending
up at the point corresponding tmc.

Figure 1.2 The integerð can be depicted as pointð
on a horizontal line.