1.6 The algebra of real numbers 15
EXAMPLES 1.12Examples of the rules of arithmetic
rule examples
1.a 1 + 1 b 1 = 1 b 1 + 1 a 21 + 131 = 131 + 121 = 15
2.ab 1 = 1 ba 21 × 131 = 131 × 121 = 16
3.a 1 + 1 (b 1 + 1 c) 1 = 1 (a 1 + 1 b) 1 + 1 c
!
21 + 1 (3 1 + 1 4) 1 = 121 + 171 = 1 9, and
@
(2 1 + 1 3) 1 + 141 = 151 + 141 = 19
4.a(bc) 1 = 1 (ab)c
!
21 × 1 (3 1 × 1 4) 1 = 121 × 1121 = 1 24, and
@
(2 1 × 1 3) 1 × 141 = 161 × 141 = 124
5.a(b 1 + 1 c) 1 = 1 ab 1 + 1 ac
!
21 × 1 (3 1 + 1 4) 1 = 121 × 171 = 1 14, and
@
21 × 1 (3 1 + 1 4) 1 = 1 (2 1 × 1 3) 1 + 1 (2 1 × 1 4) 1 = 161 + 181 = 114
−2(3 1 + 1 4) 1 = 1 (− 21 × 1 3) 1 + 1 (− 21 × 1 4) 1 = 1 − 61 − 181 = 1 − 14
−2(3 1 − 1 4) 1 = 1 − 21 × 131 − 121 × 1 (−4) 1 = 1 − 61 + 181 = 12
A corollary to rule 5 is
(a 1 + 1 b)(c 1 + 1 d) 1 = 1 a(c 1 + 1 d) 1 + 1 b(c 1 + 1 d)(2 1 + 1 3)(4 1 + 1 5) 1 = 1 2(4 1 + 1 5) 1 + 1 3(4 1 + 1 5) 1 = 1181 + 1271 = 145
Three rules define the properties of zero and unity:
6.a 1 + 101 = 101 + 1 a 1 = 1 a (addition of zero)
7.a 1 × 101 = 101 × 1 a 1 = 1 0 (multiplication by zero)
8.a 1 × 111 = 111 × 1 a 1 = 1 a (multiplication by unity)
We have already seen that subtraction of a number is the same as addition of its
negative, and that division by a number is the same as multiplication by its inverse.
However, division by zero is not defined; there is no number whose inverse is zero.
For example, the number 12 a, for positive values of a, becomes arbitrarily large as the
value of aapproaches zero; we say that 12 atends to infinityas atends to zero:
Although ‘infinity’ is represented by the symbol ∞, it is not a number. If it were a
number then, by the laws of algebra, the equations 1201 = 1 ∞and 2201 = 1 ∞would imply
11 = 12.
The modulusof a real number ais defined as the positive square root of a
2;
(read as ‘mod a’). It is the ‘magnitude’ of the number, equal to+aif ais
positive, and equal to−aif ais negative:
(1.13)
For example,||33= and||−= 33.
||a
aa
aa
=
+>
−<
if
if
0
0
||aa=+
21
0
a
→→∞asa