Complex Numbers

The study of mathematics continuously builds upon itself. Negative integers, for example, fill a void left by the set of positive integers. The set of rational numbers, in turn, fills a void left by the set of integers. The set of real numbers fills a void left by the set of rational numbers. Not surprisingly, the set of real numbers has voids as well. For example, we still have no solution to equations such as

Our best guesses might be +2 or –2. But if we test +2 in this equation, it does not work. If we test –2, it does not work. If we want to have a solution for this equation, we will have to go farther than we have so far. After all, to this point we have described the square root of a negative number as undefined. Fortunately, there is another system of numbers that provides solutions to problems such as these. In this section, we will explore this number system and how to work within it.

Expressing Square Roots of Negative Numbers as Multiples of i

We know how to find the square root of any positive real number. In a similar way, we can find the square root of a negative number. The difference is that the root is not real. If the value in the radicand is negative, the root is said to be an imaginary number. The imaginary number

i

A complex number is the sum of a real number and an imaginary number. A complex number is expressed in standard form when written

a + b i

Showing the real and imaginary parts of 5 + 2i. In this complex number, 5 is the real part and 2i is the complex part.

Imaginary numbers are distinguished from real numbers because a squared imaginary number produces a negative real number. Recall, when a positive real number is squared, the result is a positive real number and when a negative real number is squared, again, the result is a positive real number. Complex numbers are a combination of real and imaginary numbers.

Plotting a Complex Number on the Complex Plane

We cannot plot complex numbers on a number line as we might real numbers. However, we can still represent them graphically. To represent a complex number we need to address the two components of the number. We use the complex plane, which is a coordinate system in which the horizontal axis represents the real component and the vertical axis represents the imaginary component. Complex numbers are the points on the plane, expressed as ordered pairs

(a, b),

Adding and Subtracting Complex Numbers

Just as with real numbers, we can perform arithmetic operations on complex numbers. To add or subtract complex numbers, we combine the real parts and combine the imaginary parts.

Multiplying Complex Numbers

Multiplying complex numbers is much like multiplying binomials. The major difference is that we work with the real and imaginary parts separately.

Multiplying a Complex Numbers by a Real Number

Let’s begin by multiplying a complex number by a real number. We distribute the real number just as we would with a binomial. So, for example,

Showing how distribution works for complex numbers. For 3(6+2i), 3 is multiplied to both the real and imaginary parts. So we have (3)(6)+(3)(2i) = 18 + 6i.

<div data-type="note" data-has-label="true" class="precalculus howto" data-label="How To Feature" markdown="1"> Given a complex number and a real number, multiply to find the product.

Multiplying Complex Numbers Together

Now, let’s multiply two complex numbers. We can use either the distributive property or the FOIL method. Recall that FOIL is an acronym for multiplying First, Outer, Inner, and Last terms together. Using either the distributive property or the FOIL method, we get

Dividing Complex Numbers

Division of two complex numbers is more complicated than addition, subtraction, and multiplication because we cannot divide by an imaginary number, meaning that any fraction must have a real-number denominator. We need to find a term by which we can multiply the numerator and the denominator that will eliminate the imaginary portion of the denominator so that we end up with a real number as the denominator. This term is called the complex conjugate of the denominator, which is found by changing the sign of the imaginary part of the complex number. In other words, the complex conjugate of

a + b i

Note that complex conjugates have a reciprocal relationship: The complex conjugate of

a + b i

Further, when a quadratic equation with real coefficients has complex solutions, the solutions are always complex conjugates of one another.

equals zero. We first write the division as a fraction, then find the complex conjugate of the denominator, and multiply.

Multiply the numerator and denominator by the complex conjugate of the denominator.

Finding Complex Conjugates

Find the complex conjugate of each number.

$2 + i \sqrt{5}$
$- \frac{1}{2} i$

The number is already in the form $a + b i .$
The complex conjugate is
$a - b i,$
or
$2 - i \sqrt{5} .$
We can rewrite this number in the form $a + b i$
as
$0 - \frac{1}{2} i .$
The complex conjugate is
$a - b i,$
or
$0 + \frac{1}{2} i .$
This can be written simply as
$\frac{1}{2} i .$

Analysis

Although we have seen that we can find the complex conjugate of an imaginary number, in practice we generally find the complex conjugates of only complex numbers with both a real and an imaginary component. To obtain a real number from an imaginary number, we can simply multiply by $i .$

Dividing Complex Numbers

Divide $(2 + 5 i)$

by $(4 - i) .$

We begin by writing the problem as a fraction.

\frac{(2 + 5 i)}{(4 - i)}

Then we multiply the numerator and denominator by the complex conjugate of the denominator.

\frac{(2 + 5 i)}{(4 - i)} \cdot \frac{(4 + i)}{(4 + i)}

To multiply two complex numbers, we expand the product as we would with polynomials (the process commonly called FOIL).

\begin{array}{l} \frac{(2 + 5 i)}{(4 - i)} \cdot \frac{(4 + i)}{(4 + i)} = \frac{8 + 2 i + 20 i + 5 i^{2}}{16 + 4 i - 4 i - i^{2}} \\ = \frac{8 + 2 i + 20 i + 5 (- 1)}{16 + 4 i - 4 i - (- 1)} & Because i^{2} = - 1 \\ = \frac{3 + 22 i}{17} \\ = \frac{3}{17} + \frac{22}{17} i & Separate real and imaginary parts . \end{array}

Note that this expresses the quotient in standard form.

Simplifying Powers of i

by itself for increasing powers, we will see a cycle of 4. Let’s examine the next 4 powers of

i .

**Can we write $i^{35}$

in other helpful ways?**

*As we saw in [link], we reduced $i^{35}$

to $i^{3}$

by dividing the exponent by 4 and using the remainder to find the simplified form. But perhaps another factorization of $i^{35}$

may be more useful. [link] shows some other possible factorizations.*

| Factorization of

i 35

</math></strong> | $i^{34} \cdot i$

i^{33} \cdot i^{2}

i^{31} \cdot i^{4}

i^{19} \cdot i^{16}


Reduced form	${(i^{2})}^{17} \cdot i$

i^{33} \cdot (- 1)

i^{31} \cdot 1

i^{19} \cdot {(i^{4})}^{4}


Simplified form	${(- 1)}^{17} \cdot i$

- i^{33}

i^{31}

i^{19}

Each of these will eventually result in the answer we obtained above but may require several more steps than our earlier method.

Key Concepts

Verbal

Explain how to add complex numbers.

Add the real parts together and the imaginary parts together.

What is the basic principle in multiplication of complex numbers?

Give an example to show the product of two imaginary numbers is not always imaginary.

i

times $i$

equals –1, which is not imaginary. (answers vary)

What is a characteristic of the plot of a real number in the complex plane?

Algebraic

For the following exercises, evaluate the algebraic expressions.

If f (x) = x^{2} + x - 4,

evaluate $f (2 i) .$

- 8 + 2 i

If f (x) = x^{3} - 2,

evaluate $f (i) .$

If f (x) = x^{2} + 3 x + 5,

evaluate $f (2 + i) .$

14 + 7 i

If f (x) = 2 x^{2} + x - 3,

evaluate $f (2 - 3 i) .$

If f (x) = \frac{x + 1}{2 - x},

evaluate $f (5 i) .$

- \frac{23}{29} + \frac{15}{29} i

If f (x) = \frac{1 + 2 x}{x + 3},

evaluate $f (4 i) .$

Graphical

For the following exercises, determine the number of real and nonreal solutions for each quadratic function shown.

![Graph of a parabola intersecting the real axis.](/precalculus-book/resources/CNX_Precalc_Figure_03_01_201.jpg)

2 real and 0 nonreal

![Graph of a parabola not intersecting the real axis.](/precalculus-book/resources/CNX_Precalc_Figure_03_01_202.jpg)

For the following exercises, plot the complex numbers on the complex plane.

1 - 2 i

![Graph of the plotted point, 1-2i.](/precalculus-book/resources/CNX_Precalc_Figure_03_01_203.jpg)

- 2 + 3 i

i

![Graph of the plotted point, i.](/precalculus-book/resources/CNX_Precalc_Figure_03_01_205.jpg)

- 3 - 4 i

Numeric

For the following exercises, perform the indicated operation and express the result as a simplified complex number.

(3 + 2 i) + (5 - 3 i)

8 - i

(- 2 - 4 i) + (1 + 6 i)

(- 5 + 3 i) - (6 - i)

- 11 + 4 i

(2 - 3 i) - (3 + 2 i)

(- 4 + 4 i) - (- 6 + 9 i)

2 - 5 i

(2 + 3 i) (4 i)

(5 - 2 i) (3 i)

6 + 15 i

(6 - 2 i) (5)

(- 2 + 4 i) (8)

- 16 + 32 i

(2 + 3 i) (4 - i)

(- 1 + 2 i) (- 2 + 3 i)

- 4 - 7 i

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

(3 + 4 i) (3 - 4 i)

\frac{3 + 4 i}{2}

\frac{6 - 2 i}{3}

2 - \frac{2}{3} i

\frac{- 5 + 3 i}{2 i}

\frac{6 + 4 i}{i}

4 - 6 i

\frac{2 - 3 i}{4 + 3 i}

\frac{3 + 4 i}{2 - i}

\frac{2}{5} + \frac{11}{5} i

\frac{2 + 3 i}{2 - 3 i}

\sqrt{- 9} + 3 \sqrt{- 16}

15 i

- \sqrt{- 4} - 4 \sqrt{- 25}

\frac{2 + \sqrt{- 12}}{2}

1 + i \sqrt{3}

\frac{4 + \sqrt{- 20}}{2}

i^{8}

1

i^{15}

i^{22}

- 1

Technology

For the following exercises, use a calculator to help answer the questions.

Evaluate ${(1 + i)}^{k}$

for $k = 4, 8, and 12 .$

Predict the value if $k = 16.$

Evaluate ${(1 - i)}^{k}$

for $k = 2, 6, and 10 .$

Predict the value if $k = 14.$

128i

Evaluate $(1 + i)^{k} - (1 - i)^{k}$

for $k = 4, 8, and 12$

. Predict the value for $k = 16.$

Show that a solution of $x^{6} + 1 = 0$

is $\frac{\sqrt{3}}{2} + \frac{1}{2} i .$

{(\frac{\sqrt{3}}{2} + \frac{1}{2} i)}^{6} = - 1

Show that a solution of $x^{8} - 1 = 0$

is $\frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} i .$

Extensions

For the following exercises, evaluate the expressions, writing the result as a simplified complex number.

\frac{1}{i} + \frac{4}{i^{3}}

3 i

\frac{1}{i^{11}} - \frac{1}{i^{21}}

i^{7} (1 + i^{2})

i^{−3} + 5 i^{7}

\frac{(2 + i) (4 - 2 i)}{(1 + i)}

5 – 5i

\frac{(1 + 3 i) (2 - 4 i)}{(1 + 2 i)}

\frac{{(3 + i)}^{2}}{{(1 + 2 i)}^{2}}

- 2 i

\frac{3 + 2 i}{2 + i} + (4 + 3 i)

\frac{4 + i}{i} + \frac{3 - 4 i}{1 - i}

\frac{9}{2} - \frac{9}{2} i

\frac{3 + 2 i}{1 + 2 i} - \frac{2 - 3 i}{3 + i}

Expressing Square Roots of Negative Numbers as Multiples of i

Plotting a Complex Number on the Complex Plane

Adding and Subtracting Complex Numbers

Multiplying Complex Numbers

Multiplying a Complex Numbers by a Real Number

Multiplying Complex Numbers Together

Dividing Complex Numbers

Simplifying Powers of i

Key Concepts

Verbal

Algebraic

Graphical

Numeric

Technology

Extensions

Glossary