4 Chapter 1Numbers, variables, and units
so that, for example, the subtraction of a negative number is equivalent to the addition
of the corresponding positive number. The operation of multiplication is made
possible by the rules
(−m) 1 × 1 (−n) 1 = 1 +(m 1 × 1 n)
(−m) 1 × 1 (+n) 1 = 1 −(m 1 × 1 n)
(1.6)
Similarly for division. Note that −m 1 = 1 (−1) 1 × 1 m.
EXAMPLES 1.2Addition and multiplication of negative numbers
21 + 1 (−3) 1 = 121 − 131 = 1 − 121 − 1 (−3) 1 = 121 + 131 = 15
(−2) 1 × 1 (−3) 1 = 121 × 131 = 1 6(2) 1 × 1 (−3) 1 = 1 − 21 × 131 = 1 − 6
(−6) 1 ÷ 1 (−3) 1 = 161 ÷ 131 = 1261 ÷ 1 (−3) 1 = 1 − 61 ÷ 131 = 1 − 2
0 Exercises 1–
In equations (1.5) and (1.6) the letters mand nare symbols used to represent any pair of
integers; they are integer variables, whose values belong to the (infinite) set of integers.
Division of one integer by another does not always give an integer; for example
61 ÷ 131 = 12 , but 61 ÷ 14 is not an integer. A set of numbers for which the operation of
divisionis always valid is the set of rational numbers, consisting of all the numbers
m 2 n 1 = 1 m 1 ÷ 1 nwhere mand nare integers (m 2 n, read as ‘m over n’, is the more
commonly used notation for ‘mdivided by n’). The definition excludes the casen 1 = 10
because division by zero is not defined (see Section 1.6), but integers are included
because an integer mcan be written asm 2 1. The rules for the combination of rational
numbers (and of fractions in general) are
(1.7)
(1.8)
(1.9)
where, for example, mqmeansm 1 × 1 q.
EXAMPLES 1.3Addition of fractions
(1) Add and.
1
4
1
2
m
n
p
q
m
n
q
p
mq
np
÷= ×=
m
n
p
q
mp
nq
×=
m
n
p
q
mq np
nq
+=