ORCCA Complex Number Operations

Section 11.4 Complex Number Operations

Complex numbers ¹ are used in many math, science, and engineering applications. In this section, we will learn the basics of complex number operations.

en.wikipedia.org/wiki/Complex_number#Applications

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Subsection 11.4.1 Adding and Subtracting Complex Numbers

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Adding and subtracting complex numbers is just like combining like terms. We combine the terms that are real and the terms that are imaginary. Here are some examples

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Example 11.4.1.

Simplify the expression $(1-7i)+(5+4i)\text{.}$

\begin{aligned} (1 - 7 i) + (5 + 4 i) & = 1 + 5 - 7 i + 4 i \\ = 6 - 3 i \end{aligned}

$\begin{align*} (1-7i)+(5+4i)\amp=1+5-7\highlight{i}+4\highlight{i}\\ \amp=6-3\highlight{i} \end{align*}$

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Example 11.4.2.

Simplify the expression $(3-10i)-(4-6i)\text{.}$

\begin{aligned} (3 - 10 i) - (4 - 6 i) & = 3 - 10 i - 4 + 6 i \\ = - 1 - 4 i \end{aligned}

$\begin{align*} (3-10i)-(4-6i)\amp=3-10\highlight{i}-4+6\highlight{i}\\ \amp=-1-4\highlight{i} \end{align*}$

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Checkpoint 11.4.3.

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Subsection 11.4.2 Multiplying Complex Numbers

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Now let's learn how to multiply complex numbers. It is very similar to multiplying polynomials, with the exception that

i^{2}

$i^2$ must be replaced with

- 1 .

$-1\text{.}$

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Example 11.4.4.

Simplify the expression $2i(3-2i)\text{.}$

We distribute the $2i$ to both terms, then we simplify any powers of $i\text{.}$

\begin{aligned} 2 i (3 - 2 i) & = 2 i \cdot 3 - 2 i \cdot 2 i \\ = 6 i - 4 i^{2} \\ = 6 i - 4 (- 1) \\ = 6 i + 4 \\ = 4 + 6 i \end{aligned}

$\begin{align*} 2i(3-2i)\amp=2i\cdot3-2i\cdot2i\\ \amp=6i-4i^2\\ \amp=6i-4(-1)\\ \amp=6i+4\\ \amp=4+6i \end{align*}$

Note that we always write a complex number in standard form, which is $a+bi\text{.}$

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When we multiply two complex numbers we can use the distributive method, FOIL method, or generic rectangles. Here is an example of each method.

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Example 11.4.5.

Multiply $(1+5i)(2-7i)\text{.}$

We will use the distributive method to multiply the two binomials.

\begin{aligned} (1 + 5 i) (2 - 7 i) & = 2 (1 + 5 i) - 7 i (1 + 5 i) \\ = 2 + 10 i - 7 i - 35 i^{2} \\ = 2 + 10 i - 7 i - 35 (- 1) \\ = 2 + 3 i + 35 \\ = 37 + 3 i \end{aligned}

$\begin{align*} (1+5i)(2-7i)\amp=2(1+5i)-7i(1+5i)\\ \amp=2+10i-7i-35i^2\\ \amp=2+10i-7i-35(-1)\\ \amp=2+3i+35\\ \amp=37+3i \end{align*}$

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Example 11.4.6.

Expand and simplify the expression $(3-4i)^2\text{.}$

Explanation

We will use the FOIL method to expand this perfect square.

A diagram that shows 3*3=9, 3*-4i=-12i, -4i*3=-12i and (-4i)^2=16i^2 — Figure 11.4.7. Using the FOIL method to expand $(3-4i)^2\text{.}$

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Example 11.4.8.

Multiply $(3+4i)(3-4i)\text{.}$

Explanation

We will use the Generic Rectangle Method to multiply those two binomials.

A generic rectangle diagram that shows 3*3=9, 3*-4i=-12i, -4i*3=-12i and (-4i)^2=16i^2 — Figure 11.4.9. Using the Generic Rectangle Method to multiply $(3+4i)(3-4i)\text{.}$

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As the last example shows, it is possible to multiply two complex numbers and get a real number result. Notice that the middle terms,

12 i

$12i$ and

- 12 i,

$-12i\text{,}$ are opposites, which makes the result a real number. This happens when we multiply a sum and difference of the same real and imaginary parts, called complex conjugates. This pair of factors results in the difference of squares:

(a + b) (a - b) = a^{2} - b^{2} .

$\begin{equation*} (a+b)(a-b)=a^2-b^2\text{.} \end{equation*}$

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Example 11.4.10.

Here is an example using the sum and difference formula to multiply $(5+2i)(5-2i)\text{:}$

\begin{aligned} (5 + 2 i) (5 - 2 i) & = 5^{2} - (2 i)^{2} \\ = 25 - 4 i^{2} \\ = 25 - 4 (- 1) \\ = 25 + 4 \\ = 29 \end{aligned}

$\begin{align*} (5+2i)(5-2i)\amp=5^2-(2i)^2\\ \amp=25-4i^2\\ \amp=25-4(-1)\\ \amp=25+4\\ \amp=29 \end{align*}$

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Checkpoint 11.4.11.

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Subsection 11.4.3 Dividing Complex Numbers

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When we divide by

i

$i$ we use a process that is similar to rationalizing the denominator. We use the property

\sqrt{x} \cdot \sqrt{x} = x

$\sqrt{x}\cdot\sqrt{x}=x$ when we rationalize the denominator, and we use the property

i \cdot i = - 1

$i\cdot i=-1$ when we have complex numbers. Let's compare these two problems

\frac{2}{\sqrt{2}}

$\frac{2}{\sqrt{2}}$ and

\frac{2}{i} :

$\frac{2}{i}\text{:}$

\begin{aligned} \frac{2}{\sqrt{2}} & = \frac{2 \cdot \sqrt{2}}{\sqrt{2} \cdot \sqrt{2}} \\ = \frac{2 \sqrt{2}}{2} \\ = \sqrt{2} \end{aligned}

$\begin{align*} \frac{2}{\sqrt{2}}\amp=\frac{2\multiplyright{\sqrt{2}}}{\sqrt{2}\multiplyright{\sqrt{2}}}\\ \amp=\frac{2\sqrt{2}}{2}\\ \amp=\sqrt{2} \end{align*}$

\begin{aligned} \frac{2}{i} & = \frac{2 \cdot i}{i \cdot i} \\ = \frac{2 i}{- 1} \\ = - 2 i \end{aligned}

$\begin{align*} \frac{2}{i}\amp=\frac{2\multiplyright{i}}{i\multiplyright{i}}\\ \amp=\frac{2i}{-1}\\ \amp=-2i \end{align*}$

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Example 11.4.12.

Rationalize the denominator in the expression $-\frac{7}{4i}\text{.}$

\begin{aligned} - \frac{7}{4 i} & = - \frac{7 \cdot i}{4 i \cdot i} \\ = - \frac{7 i}{4 (- 1)} \\ = \frac{7 i}{4} \\ = \frac{7}{4} i \end{aligned}

$\begin{align*} -\frac{7}{4i}\amp=-\frac{7\multiplyright{i}}{4i\multiplyright{i}}\\ \amp=-\frac{7i}{4(-1)}\\ \amp=\frac{7i}{4}\\ \amp=\frac{7}{4}i \end{align*}$

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Checkpoint 11.4.13.

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When the denominator is in the form

a + b i,

$a+bi\text{,}$ we need to use the complex conjugate to remove the imaginary terms from the denominator. Here is an example.

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Example 11.4.14.

Simplify the expression $\frac{1}{4+3i}\text{.}$

Explanation

To get a real result in the denominator we multiply the numerator and denominator by $4-3i\text{,}$ and we have:

\begin{align*} \frac{1}{4+3i}\amp=\frac{1}{4+3i}\multiplyright{\frac{(4-3i)}{(4-3i)}}\\ \amp=\frac{4-3i}{16-12i+12i-9i^2}\\ \amp=\frac{4-3i}{16-9(-1)}\\ \amp=\frac{4-3i}{16+9}\\ \amp=\frac{4-3i}{25}\\ \amp=\frac{4}{25}-\frac{3}{25}i \end{align*}

Note that we always write complex numbers in standard form, which is $a+bi\text{.}$

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Now we can divide two complex numbers as in the next example.

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Example 11.4.15.

Simplify the expression $\frac{1+2i}{2-4i}\text{.}$

Explanation

To divide complex numbers, we rationalize the denominator using the conjugate $2+4i\text{:}$

\begin{align*} \frac{1+2i}{2-4i}\amp=\frac{(1+2i)}{(2-4i)}\multiplyright{\frac{(2+4i)}{(2+4i)}}\\ \amp=\frac{2+4i+4i+8i^2}{4+8i-8i-16i^2}\\ \amp=\frac{2+8i+8(-1)}{4-16(-1)}\\ \amp=\frac{2+8i-8}{4+16}\\ \amp=\frac{-6+8i}{20}\\ \amp=\frac{-6}{20}+\frac{8i}{20}\\ \amp=-\frac{3}{10}+\frac{2}{5}i \end{align*}

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1.

Add up the following complex numbers:

$\displaystyle{ ({-8+11i})+({-10+8i}) = }$

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2.

Add up the following complex numbers:

$\displaystyle{ ({-5+2i})+({1-7i}) = }$

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3.

Subtract the following complex numbers:

$\displaystyle{ ({-3-6i})-({11-6i}) = }$

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4.

Subtract the following complex numbers:

$\displaystyle{ ({10-4i})-({12-8i}) = }$

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5.

Write the complex number in standard form.

(2 + i) + (5 + 4 i)

$\begin{equation*} (2+i)+(5+4i) \end{equation*}$

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6.

Write the complex number in standard form.

(5 - 6 i) + (- 7 - 2 i)

$\begin{equation*} (5-6i)+(-7-2i) \end{equation*}$

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7.

Write the complex number in standard form.

(7 + 7 i) + (2 - 8 i)

$\begin{equation*} (7+7i)+(2-8i) \end{equation*}$

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8.

Write the complex number in standard form.

(9) + (- 10 + 8 i)

$\begin{equation*} (9)+(-10+8i) \end{equation*}$

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9.

Write the complex number in standard form.

(- 9 - 7 i) - (- 2 + 2 i)

$\begin{equation*} (-9-7i)-(-2+2i) \end{equation*}$

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10.

Write the complex number in standard form.

(- 7 + 7 i) - (7 - 4 i)

$\begin{equation*} (-7+7i)-(7-4i) \end{equation*}$

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11.

Write the complex number in standard form.

(- 5 - i) - (- 5 - 10 i)

$\begin{equation*} (-5-i)-(-5-10i) \end{equation*}$

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12.

Write the complex number in standard form.

(- 2 - 8 i) - (4 + 5 i)

$\begin{equation*} (-2-8i)-(4+5i) \end{equation*}$

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Multiplying Complex Numbers

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13.

Multiply the following complex numbers:

$\displaystyle{ {i}({7-10i}) = }$

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14.

Multiply the following complex numbers:

$\displaystyle{ {i}({3-2i}) = }$

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15.

Multiply the following complex numbers:

$\displaystyle{ ({6-11i})({11-4i}) = }$

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16.

Multiply the following complex numbers:

$\displaystyle{ ({8+6i})({-4+4i}) = }$

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17.

Multiply the following complex numbers:

$\displaystyle{ ({11-3i})^{2} = }$

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18.

Multiply the following complex numbers:

$\displaystyle{ ({-11-12i})^{2} = }$

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19.

Multiply the following complex numbers:

$\displaystyle{ ({-8+5i})({-8-5i}) = }$

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20.

Multiply the following complex numbers:

$\displaystyle{ ({-6-4i})({-6+4i}) = }$

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21.

Write the complex number in standard form.

(- 2 + 10 i) (- 2 - 5 i)

$\begin{equation*} (-2+10i)(-2-5i) \end{equation*}$

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22.

Write the complex number in standard form.

(3 i) (7 + 10 i)

$\begin{equation*} (3i)(7+10i) \end{equation*}$

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23.

Write the complex number in standard form.

(2 - 4 i) (- 5 + 4 i)

$\begin{equation*} (2-4i)(-5+4i) \end{equation*}$

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24.

Write the complex number in standard form.

(5 + 9 i) (4 - 2 i)

$\begin{equation*} (5+9i)(4-2i) \end{equation*}$

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Dividing Complex Numbers

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25.

Rewrite the following expression into the form of a+b $i\text{:}$

$\displaystyle{ \frac{8}{i} = }$

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26.

Rewrite the following expression into the form of a+b $i\text{:}$

$\displaystyle{ \frac{4}{i} = }$

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27.

Rewrite the following expression into the form of a+b $i\text{:}$

$\displaystyle{ \frac{{-9+7i}}{{3+i}} = }$

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28.

Rewrite the following expression into the form of a+b $i\text{:}$

$\displaystyle{ \frac{{-8-4i}}{{1-2i}} = }$

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29.

Rewrite the following expression into the form of a+b $i\text{:}$

$\displaystyle{ \frac{{-8-3i}}{{5-7i}} = }$

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30.

Rewrite the following expression into the form of a+b $i\text{:}$

$\displaystyle{ \frac{{-6-6i}}{{6-5i}} = }$

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31.

Write the complex number in standard form.

\frac{0}{2 - 8 i}

$\begin{equation*} \frac{0}{2-8i} \end{equation*}$

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32.

Write the complex number in standard form.

\frac{2 - 7 i}{- 10 - 6 i}

$\begin{equation*} \frac{2-7i}{-10-6i} \end{equation*}$

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33.

Write the complex number in standard form.

\frac{5 + 7 i}{- 1 + 9 i}

$\begin{equation*} \frac{5+7i}{-1+9i} \end{equation*}$

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34.

Write the complex number in standard form.

\frac{7 - i}{7 + 3 i}

$\begin{equation*} \frac{7-i}{7+3i} \end{equation*}$

Section 11.4 Complex Number Operations

Subsection 11.4.1 Adding and Subtracting Complex Numbers

Example 11.4.1.

Example 11.4.2.

Checkpoint 11.4.3.

Subsection 11.4.2 Multiplying Complex Numbers

Example 11.4.4.

Example 11.4.5.

Example 11.4.6.

Example 11.4.8.

Example 11.4.10.

Checkpoint 11.4.11.

Subsection 11.4.3 Dividing Complex Numbers

Example 11.4.12.

Checkpoint 11.4.13.

Example 11.4.14.

Example 11.4.15.

Checkpoint 11.4.16.

Exercises 11.4.4 Exercises

Adding and Subtracting Complex Numbers

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

Multiplying Complex Numbers

13.

14.

15.

16.

17.

18.

19.

20.

21.

22.

23.

24.

Dividing Complex Numbers

25.

26.

27.

28.

29.

30.

31.

32.

33.

34.