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12.2 Basic Operations with Complex Numbers

Addition, Subtraction, Multiplication, and Division of Complex Numbers • Complex Numbers on Calculator

The basic operations of addition, subtraction, multiplication, and division of complex numbers are based on the operations for binomials with real coefficients. In performing these operations, we treat j as we would any other literal number, although we must properly handle any powers of j that might occur.

NOTE

[However, we must be careful to express all complex numbers in terms of j before performing these operations.] Therefore, we now have the definitions for these operations.

Basic Operations on Complex Numbers

Addition:

(a + b j) + (c + d j) = (a + c) + (b + d) j

$(a + b j) + (c + d j) = (a + c) + (b + d) j$ (12.3)

Subtraction:

(a + b j) - (c + d j) = (a - c) + (b - d) j

$(a + b j) - (c + d j) = (a - c) + (b - d) j$ (12.4)

Multiplication:

(a + b j) (c + d j) = a c + a d j + b c j + b d j^{2} = (a c - b d) + (a d + b c) j

$(a + b j) (c + d j) = a c + a d j + b c j + b d j^{2} = (a c - b d) + (a d + b c) j$ (12.5)

Division:

\frac{a + b j}{c + d j} = \frac{(a + b j) (c - d j)}{(c + d j) (c - d j)} = \frac{(a c + b d) + (b c - a d) j}{c^{2} + d^{2}}

$\frac{a + b j}{c + d j} = \frac{(a + b j) (c - d j)}{(c + d j) (c - d j)} = \frac{(a c + b d) + (b c - a d) j}{c^{2} + d^{2}}$ (12.6)

Equations (12.3) and (12.4) show that we add and subtract complex numbers by combining the real parts and combining the imaginary parts.

EXAMPLE 1 Adding, subtracting complex numbers

$\begin{array}{rcl} (3 - 2 j) + (- 5 + 7 j) & = & (3 - 5) + (- 2 + 7) j \\ = & - 2 + 5 j \end{array}$ $\begin{array}{rcl} (3 - 2 j) + (- 5 + 7 j) & = & (3 - 5) + (- 2 + 7) j \\ = & - 2 + 5 j \end{array}$
$\begin{array}{rcl} (7 + 9 j) - (6 - 4 j) & = & (7 - 6) + (9 - (- 4)) j \\ = & 1 + 13 j \end{array}$ $\begin{array}{rcl} (7 + 9 j) - (6 - 4 j) & = & (7 - 6) + (9 - (- 4)) j \\ = & 1 + 13 j \end{array}$

When complex numbers are multiplied, Eq. (12.5) indicates that we express numbers in terms of j, proceed as with any algebraic multiplication, and note that $j^{2} = - 1.$ $j^{2} = - 1.$

EXAMPLE 2 Multiplying complex numbers

$\begin{array}{rcll} (6 - \sqrt{- 4}) (\sqrt{- 9}) & = & (6 - 2 j) (3 j) & write in terms of j \\ = & 18 j - 6 j^{2} = 18 j - & 6 (- 1) \\ = & 6 + 18 j \end{array}$ $\begin{array}{rcll} (6 - \sqrt{- 4}) (\sqrt{- 9}) & = & (6 - 2 j) (3 j) & write in terms of j \\ = & 18 j - 6 j^{2} = 18 j - & 6 (- 1) \\ = & 6 + 18 j \end{array}$
$\begin{array}{rcl} (- 9.4 - 6.2 j) (2.5 + 1.5 j) & = & (- 9.4) (2.5) + (- 9.4) (1.5 j) \\ + (- 6.2 j) (2.5) + (- 6.2 j) (1.5 j) \\ = & - 23.5 - 14.1 j - 15.5 j - 9.3 j^{2} \\ = & - 23.5 - 29.6 j - 9.3 (- 1) \\ = & - 14.2 - 29.6 j \end{array}$ $\begin{array}{rcl} (- 9.4 - 6.2 j) (2.5 + 1.5 j) & = & (- 9.4) (2.5) + (- 9.4) (1.5 j) \\ + (- 6.2 j) (2.5) + (- 6.2 j) (1.5 j) \\ = & - 23.5 - 14.1 j - 15.5 j - 9.3 j^{2} \\ = & - 23.5 - 29.6 j - 9.3 (- 1) \\ = & - 14.2 - 29.6 j \end{array}$

The procedure shown in Eq. (12.6) for dividing by a complex number is the same as that used for rationalizing the denominator of a fraction. The result is in the proper form of a complex number. Therefore, to divide by a complex number, multiply the numerator and the denominator by the conjugate of the denominator.

EXAMPLE 3 Dividing complex numbers

$The process of dividing the complex number, start fraction 7 minus 2 j over 3 + 4 j end fraction.$

.2-2 Full Alternative Text

This could be written in the form $a + b j$ $a + b j$ as $\frac{13}{25} - \frac{34}{25} j,$ $\frac{13}{25} - \frac{34}{25} j,$ but this type of result is generally left as a single fraction. In decimal form, the result would be expressed as $0.52 - 1.36 j .$ $0.52 - 1.36 j .$
$\begin{array}{rcl} \frac{1}{j} + \frac{2}{3 + j} & = & \frac{(3 + j) + 2 j}{j (3 + j)} = \frac{3 + 3 j}{3 j - 1} = \frac{3 + 3 j}{- 1 + 3 j} \cdot \frac{- 1 - 3 j}{- 1 - 3 j} \\ = & \frac{- 3 - 12 j - 9 j^{2}}{1 - 9 j^{2}} = \frac{6 - 12 j}{10} \\ = & \frac{3 - 6 j}{5} \end{array}$ $\begin{array}{rcl} \frac{1}{j} + \frac{2}{3 + j} & = & \frac{(3 + j) + 2 j}{j (3 + j)} = \frac{3 + 3 j}{3 j - 1} = \frac{3 + 3 j}{- 1 + 3 j} \cdot \frac{- 1 - 3 j}{- 1 - 3 j} \\ = & \frac{- 3 - 12 j - 9 j^{2}}{1 - 9 j^{2}} = \frac{6 - 12 j}{10} \\ = & \frac{3 - 6 j}{5} \end{array}$

Most calculators are programmed for complex numbers. The solutions for (a) and (b) are shown in Fig. 12.2 with the results in decimal form.

A calculator screen with input 7 minus 2 i, divided by, 3 plus 4 i, and output 0.52 minus 1.36 i; input 1 divided by i plus 2 divided by, 3 plus i, and output 0.6 minus 1.2 i. — Fig. 12.2

EXAMPLE 4 Dividing complex numbers—alternating current

In an alternating-current circuit, the voltage E is given by $E = I Z,$ $E = I Z,$ where I is the current (in A) and Z is the impedance (in $Ω$ $Ω$ ). Each of these can be represented by complex numbers. Find the complex number representation for I if $E = 4.20 - 3.00 j$ $E = 4.20 - 3.00 j$ volts and $Z = 5.30 + 2.65 j$ $Z = 5.30 + 2.65 j$ ohms. (This type of circuit is discussed in Section 12.7.)

Because $I = E / Z,$ $I = E / Z,$ we have

The process of finding the complex number representation for I.

.2-2 Full Alternative Text

EXERCISES 12.2

In Exercises 1–4, perform the indicated operations on the resulting expressions if the given changes are made in the indicated examples of this section.

In Example 1(b), change the sign in the first parentheses from $+$ $+$ to $-$ $-$ and then perform the addition.
In Example 2(b), change the sign before 6.2j from $-$ $-$ to $+,$ $+,$ and then perform the multiplication.
In Example 3(a), change the sign in the denominator from $+$ $+$ to $-$ $-$ and then simplify.
In Example 3(b), change the sign in the second denominator from $+$ $+$ to $-$ $-$ and then simplify.

In Exercises 5–38, perform the indicated operations, expressing all answers in the form $a + b j .$ $a + b j .$

$(3 - 7 j) + (2 - j)$ $(3 - 7 j) + (2 - j)$
$(- 4 - j) + (- 7 - 4 j)$ $(- 4 - j) + (- 7 - 4 j)$
$(7 j - 6) - (19 - 3 j)$ $(7 j - 6) - (19 - 3 j)$
$(5.4 - 3.4 j) - (2.9 j + 5.5)$ $(5.4 - 3.4 j) - (2.9 j + 5.5)$
$0.23 - (0.46 - 0.19 j) + 0.67 j$ $0.23 - (0.46 - 0.19 j) + 0.67 j$
$(7 - j) - (4 - 4 j) - (j - 6)$ $(7 - j) - (4 - 4 j) - (j - 6)$
$(12 j - 21) - (15 - 18 j) - 9 j$ $(12 j - 21) - (15 - 18 j) - 9 j$
$(0.062 j - 0.073) - 0.030 j - (0.121 - 0.051 j)$ $(0.062 j - 0.073) - 0.030 j - (0.121 - 0.051 j)$
$(7 - j) (7 j)$ $(7 - j) (7 j)$
$(- 2.2 j) (1.5 j - 4.0)$ $(- 2.2 j) (1.5 j - 4.0)$
$(4 - j) (5 + 2 j)$ $(4 - j) (5 + 2 j)$
$(8 j - 5) (7 + 4 j)$ $(8 j - 5) (7 + 4 j)$
$(\sqrt{- 18} \sqrt{- 4}) (3 j)$ $(\sqrt{- 18} \sqrt{- 4}) (3 j)$
$\sqrt{- 6} \sqrt{- 12} \sqrt{30}$ $\sqrt{- 6} \sqrt{- 12} \sqrt{30}$
$(2 - 3 j) (5 + 4 j)$ $(2 - 3 j) (5 + 4 j)$
$(9 - 2 j) (6 + j)$ $(9 - 2 j) (6 + j)$
$j \sqrt{- 7} - j^{6} \sqrt{112} + 3 j$ $j \sqrt{- 7} - j^{6} \sqrt{112} + 3 j$
$j^{2} \sqrt{- 7} - \sqrt{- 28} + 8 j^{3}$ $j^{2} \sqrt{- 7} - \sqrt{- 28} + 8 j^{3}$
${(3 - 7 j)}^{2}$ ${(3 - 7 j)}^{2}$
${(8 j + 20)}^{2}$ ${(8 j + 20)}^{2}$
$(8 + 3 j) (8 - 3 j)$ $(8 + 3 j) (8 - 3 j)$
$(6 + 8 j) (6 - 8 j)$ $(6 + 8 j) (6 - 8 j)$
$\frac{6 j}{2 - 5 j}$ $\frac{6 j}{2 - 5 j}$
$\frac{0.25}{3 - \sqrt{- 1}}$ $\frac{0.25}{3 - \sqrt{- 1}}$
$\frac{1 - j}{3 j}$ $\frac{1 - j}{3 j}$
$\frac{12 + 10 j}{6 - 8 j}$ $\frac{12 + 10 j}{6 - 8 j}$
$\frac{j \sqrt{2} - 5}{j \sqrt{2} + 3}$ $\frac{j \sqrt{2} - 5}{j \sqrt{2} + 3}$
$\frac{j^{5} - j^{3}}{3 + j}$ $\frac{j^{5} - j^{3}}{3 + j}$
$\frac{j^{2} - j}{2 j - j^{8}}$ $\frac{j^{2} - j}{2 j - j^{8}}$
$\frac{3}{2 j} - \frac{5}{j - 6}$ $\frac{3}{2 j} - \frac{5}{j - 6}$
$\frac{4 j}{1 - j} - \frac{j + 8}{2 + 3 j}$ $\frac{4 j}{1 - j} - \frac{j + 8}{2 + 3 j}$
$\frac{(6 j + 5) (2 - 4 j)}{(5 - j) (4 j + 1)}$ $\frac{(6 j + 5) (2 - 4 j)}{(5 - j) (4 j + 1)}$
${(4 j^{5} - 5 j^{4} + 2 j^{3} - 3 j^{2})}^{2}$ ${(4 j^{5} - 5 j^{4} + 2 j^{3} - 3 j^{2})}^{2}$
${(2 j^{2} - 3 j^{3} + 2 j^{4} - 2 j^{5})}^{6}$ ${(2 j^{2} - 3 j^{3} + 2 j^{4} - 2 j^{5})}^{6}$

In Exercises 39–42, evaluate each expression on a calculator. Express answers in the form $a + b j .$ $a + b j .$

$(3 j^{9} - 5 j^{3}) (4 j^{6} - 6 j^{8})$ $(3 j^{9} - 5 j^{3}) (4 j^{6} - 6 j^{8})$
$(5 j - 4 j^{2} + 3 j^{7}) (2 j^{12} - j^{13})$ $(5 j - 4 j^{2} + 3 j^{7}) (2 j^{12} - j^{13})$
$\frac{{(2 - j^{3})}^{4}}{{(j^{8} - j^{6})}^{3}} + j$ $\frac{{(2 - j^{3})}^{4}}{{(j^{8} - j^{6})}^{3}} + j$
${(1 + j)}^{- 3} {(2 - j)}^{- 2}$ ${(1 + j)}^{- 3} {(2 - j)}^{- 2}$

In Exercises 43–56, solve the given problems.

Show that $- 1 - j$ $- 1 - j$ is a solution to the equation $x^{2} + 2 x + 2 = 0.$ $x^{2} + 2 x + 2 = 0.$
Show that $1 - j \sqrt{3}$ $1 - j \sqrt{3}$ is a solution to the equation $x^{2} + 4 = 2 x .$ $x^{2} + 4 = 2 x .$
What is the sum of the solutions for the equation $x^{2} - 4 x + 13 = 0 ?$ $x^{2} - 4 x + 13 = 0 ?$
What is the product of the solutions to the equation in Exercise 45?
Multiply $- 3 + j$ $- 3 + j$ by its conjugate.
Divide $2 - 3 j$ $2 - 3 j$ by its conjugate.
Write the reciprocal of $3 - j$ $3 - j$ in rectangular form.
Write the reciprocal of $2 + 5 j$ $2 + 5 j$ in rectangular form.
Write $j^{- 2} + j^{- 3}$ $j^{- 2} + j^{- 3}$ in rectangular form.
Solve for x: ${(x + 2 j)}^{2} = 5 + 12 j$ ${(x + 2 j)}^{2} = 5 + 12 j$
Solve for x: ${(x + 3 j)}^{2} = 7 - 24 j$ ${(x + 3 j)}^{2} = 7 - 24 j$
For $\frac{3}{5} + \frac{4}{5} j,$ $\frac{3}{5} + \frac{4}{5} j,$ find: (a) the conjugate; (b) the reciprocal.
If $f (x) = x + \frac{1}{x},$ $f (x) = x + \frac{1}{x},$ find $f (1 + 3 j) .$ $f (1 + 3 j) .$
When finding the current in a certain electric circuit, the expression $(s + 1 + 4 j) (s + 1 - 4 j)$ $(s + 1 + 4 j) (s + 1 - 4 j)$ occurs. Simplify this expression.

In Exercises 57–60, solve the given problems. Refer to Example 4.

If $I = 0.835 - 0.427 j$ $I = 0.835 - 0.427 j$ amperes and $Z = 250 + 170 j$ $Z = 250 + 170 j$ ohms, find the complex-number representation for E.
If $E = 5.70 - 3.65 j$ $E = 5.70 - 3.65 j$ mV and $I = 0.360 - 0.525 j μ A,$ $I = 0.360 - 0.525 j μ A,$ find the complex-number representation for Z.
If $E = 85 + 74 j$ $E = 85 + 74 j$ volts and $Z = 2500 - 1200 j$ $Z = 2500 - 1200 j$ ohms, find the complex-number representation for I.
In an alternating-current circuit, two impedances $Z_{1}$ $Z_{1}$ and $Z_{2}$ $Z_{2}$ have a total impedance $Z_{T}$ $Z_{T}$ of $Z_{T} = \frac{Z_{1} Z_{2}}{Z_{1} + Z_{2}} .$ $Z_{T} = \frac{Z_{1} Z_{2}}{Z_{1} + Z_{2}} .$ Find $Z_{T}$ $Z_{T}$ for $Z_{1} = 3.2 + 4.8 j m Ω$ $Z_{1} = 3.2 + 4.8 j m Ω$ and $Z_{2} = 4.8 - 6.4 j m Ω .$ $Z_{2} = 4.8 - 6.4 j m Ω .$

In Exercises 61–64, answer or explain as indicated.

What type of number is the result of (a) adding a complex number to its conjugate and (b) subtracting a complex number from its conjugate?
If the reciprocal of $a + b j$ $a + b j$ equals $a - b j,$ $a - b j,$ what condition must a and b satisfy?
Explain why the product of a complex number and its conjugate is real and nonnegative.
Explain how to show that the reciprocal of the imaginary unit is the negative of the imaginary unit.

Answers to Practice Exercises

$- 3 - j$ $- 3 - j$
$41 - 57 j$ $41 - 57 j$
$\frac{- 11 - 41 j}{53}$ $\frac{- 11 - 41 j}{53}$

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Table of Contents for 12.2 Basic Operations with Complex Numbers

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Table of Contents for
12.2 Basic Operations with Complex Numbers