College Algebra 9th txtbk

4 CHAPTER R BASIC ALGEBRAIC OPERATIONS
Addition and multiplication of rational numbers are commutative; changing the order in
which two numbers are added or multiplied does not change the result.
3
2 5 7 5 7 3 2
3
2 5 7 5 7 3 2
Addition is commutative.
Multiplication is commutative.
Addition and multiplication of rational numbers is also associative; changing the grouping
of three numbers that are added or multiplied does not change the result:
3
2 a5 7 9 4 b a3 2 5 7 b 9 4
3
2 a5 7 9 4 b a3 2 5 7 b 9 4
Addition is associative.
Multiplication is associative.
Furthermore, the operations of addition and multiplication are related in that multiplication
distributes over addition:
3
2 a5 7 9 4 b 3 2 5 7 3 2 9 4
Left distributive law
a 5 7 9 4 b 3 2 5 7 3 2 9 4 3 2
Right distributive law
The rational number 0 is an additive identity; adding 0 to a number does not change
it. The rational number 1 is a multiplicative identity; multiplying a number by 1 does not
change it. Every rational number r has an additive inverse, denoted r; the additive inverse
of 45 is 45, and the additive inverse of 32 is 32. The sum of a number and its additive
inverse is 0. Every nonzero rational number r has a multiplicative inverse, denoted
r 1 ; the multiplicative inverse of 45 is 54, and the multiplicative inverse of 32 is 23.
The product of a number and its multiplicative inverse is 1. The rational number 0 has no
multiplicative inverse.
EXAMPLE 1 Arithmetic of Rational Numbers
Perform the indicated operations.
1
(A)
3 6 5
8
(B)
3 5 4
(C) (179) 1 (D) (6 92) 1
SOLUTIONS
(A)
1
3 6 5 5 18 23
15 15
(B)
8
3 5 4 40
12 10
3
40
12 10
3
because 40 3 12 10
(C) (179) 1 917
(D)
(6 92) 1 a 6
1 9 2 b 1
a 12 9 1
b a 3 1
2
2 b 2 3