College Algebra 9th txtbk

SECTION 1–4 Complex Numbers 81
ZZZ EXPLORE-DISCUSS 2
From Theorem 4 in Section R-2, we know that if a and b are positive real numbers,
then
1a 1b 1ab
(1)
So we can evaluate expressions like 19 14 two ways:
19 14 1(9)(4) 136 6 and 19 14 (3)(2) 6
Evaluate each of the following two ways. Is equation (1) a valid property to use
in all cases?
(A) 19 14 (B) 19 14 (C) 19 14
ZZZ CAUTION ZZZ Note that in Example 6, part D, we wrote 1 19 1 3i before proceeding
with the simplification. This is a necessary step because some of the properties of
radicals that are true for real numbers turn out not to be true for complex numbers.
In particular, for positive real numbers a and b,
1a 1b 1ab
but
1a 1b 1(a)(b)
(See Explore-Discuss 2.)
To avoid having to worry about this, always convert expressions of the form 1a to
the equivalent form in terms of i before performing any operations.
Z Solving Equations Involving Complex Numbers
EXAMPLE 7 Equations Involving Complex Numbers
(A) Solve for real numbers x and y:
(3x 2) (2y 4)i 4 6i
(B) Solve for complex number z:
(3 2i)z 3 6i 8 4i
SOLUTIONS
(A) This equation is really a statement that two complex numbers are equal:
(3x 2) (2y 4)i, and 4 6i. In order for these numbers to be equal, the real
parts must be the same, and the imaginary parts must be the same as well.
3x 2 4
and
(B) Solve for z, then write the answer in standard form.
(3 2i)z 3 6i 8 4i
(3 2i)z 11 10i
z
11 10i
3 2i
13 52i
13
1 4i
A check is left to the reader.
3x 6
x 2
(11 10i)(3 2i)
(3 2i)(3 2i)
2y 4 6
2y 10
y 5
Add 3 and subtract 6i from both sides.
Divide both sides by 3 2i.
Multiply numerator and denominator by the
conjugate of the denominator.
Simplify.
Write in standard form.