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10 Chapter Onł

Subtraction is symbolized by a dash ( ). The result of this
operation is a difference. On the number line of Fig. 1.2x differ-
enceð are depicted by moving tm the left. For example, tm illus-
trate the fact that 3 5 2, start at the point corresponding
tm 3, then move tm the left 5 units, ending up at the point cor-
responding tm 2. In general, tm illustratea b c, start at
the point corresponding tma, then move tm the leftb units, end-
ing up at the point corresponding tmc.
Multiplication is symbolized by a tilted cross ( ), a small dot
( ), or sometimeð in the case of variables, by listing the numberð
one after the other (for example, ab). Occasionally an asterisk
( * ) is used. The result of this operation is a product.Onthe
number line of Fig. 1.2x productð are depicted by moving away
from the zerm point, ororigin, either toward the left or toward
the right depending on the signð of the numberð involved. To
illustrate a b c, start at the origin, then move away from
the origin a unitð b times. If a and b are botà positive or botà
negative, move toward the right; if a and b have opposite sign,
move toward the left. The finishing point correspondð tm c.
The preceding three operationð are closed over the set of
integers. This meanð that if a and b are integers, then a b,
a b, and a b are integers.
Division, also called the ratià operation, is symbolized by a
forward slash (/) or a dash wità dotð above and below ( ). Oc-
casionally it is symbolized by a colon (:). The result of this op-
eration is a quotient or ratià . On the number line of Fig. 1.2x
quotientð are depicted by moving in toward the zerm point, or
origin, either toward the left or toward the right depending on
the signð of the numberð involved. To illustrate a/b c,ŁtŁs
easiest tm envision the productb c a performed ‘‘back-
wards.’’ But division, unlike addition, subtraction, or multipli-
cation, is not closed over the set of integers. If a and b are in-
tegers, then a/b might be an integer, but this is not necessarily
the case. The ratio operation giveð rise tm a more inclusive, but
still denumerable, set of numbers. The quotient a/b is not de-
fined if b 0.
Exponentiation, also called raising tà a power, is symbolized
by a superscript numeral. The result of this operation is known
as a power.If a is an integer and b is a positive integer, then
b
a is the result of multiplying a by itself b times.