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# 4.5: Complex Numbers

Difficulty Level: At Grade Created by: CK-12
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Plotting points was something you did back in early Algebra, and it likely seems pretty simple now. For instance, plotting the point (4, 5) meant starting at the origin and moving 4 units to the right - the x direction, and 5 units up - the y direction.

In this lesson, one of the things we will consider is the graphing of complex numbers such as 4+3i\begin{align*}4 +3i\end{align*}.

In essence it doesn't sound that hard, but which direction do you only imagine moving 3 units?

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### Guidance

Sometimes when you solve a quadratic equation, the solution has both a real part and an imaginary part. For example, if you want to solve

(x1)2+4=0\begin{align*}(x - 1)^2 + 4 = 0\end{align*}

then

(x1)2=4\begin{align*}(x - 1)^2 = -4\end{align*}

x1=±4\begin{align*}x - 1 = \pm\sqrt{-4}\end{align*}

x1=±14\begin{align*}x - 1 = \pm\sqrt{-1}\sqrt{4}\end{align*}

x1=±2i\begin{align*}x - 1 = \pm2i\end{align*}

x=1±2i\begin{align*}x = 1 \pm 2i\end{align*}

x=1+2i\begin{align*}x = 1 + 2i\end{align*} or 12i\begin{align*}1 - 2i\end{align*}

Notice that these two solutions involve a real part, 1, and an imaginary part, ±2i

a + bi is the standard or rectangular form of a complex number.

A complex number is a number that has a real part (in this case a) and an imaginary part, that is, the imaginary number i with a coefficient b.

The complex numbers are a superset of the real numbers, meaning that all of the real numbers are part of the set of complex numbers:

Given 'a + bi, if b = 0 (meaning there is no imaginary part to the complex number), then all you have remaining is a real number. Viewed this way, every real number can be written as a complex number (just let it equal a), but there are many more complex numbers than real numbers. Hence the complex numbers are a superset of the real numbers.

Graphing complex numbers

In standard form, a + bi, a complex number can be graphed using rectangular coordinates (a, b). The x - coordinate represents “real number” values, while the y - coordinate represents the “imaginary” values.

#### Example A

Graph the complex number: z = 2 + 2i, in rectangular form.

Solution

The coordinate (2, 2) is graphed as shown below:

#### Example B

Solve each equation and express it as a complex number. (Note: If the imaginary part is 0, express the solution as a + 0i)

a) x2 + 24 = 0
b) 2x2 - 4x + 7 = 0

Solutions

a) x=±26i\begin{align*}x = \pm2\sqrt{6}i\end{align*}
b) x=1±23i\begin{align*}x = 1 \pm 2\sqrt{3}i\end{align*}

#### Example C

Plot each of the following complex numbers in rectangular form:

a. (4 + 2i)

b. (-3 + i)

c. (3 - 4i)

d. 3i

Solution

Concept question wrap up

Of course you don't actually just imagine moving in one direction, but it is kind of a funny thought!

When graphing complex numbers in standard form: a + bi, the real component a is plotted on the horizontal or x axis. The imaginary component bi is plotted on the y axis in just the same way a real number would be.

\begin{align*}\therefore 4 + 3i\end{align*} would be located 4 units to the right and 3 units up from the origin.

### Vocabulary

Complex numbers are the combination (technically the 'sum') of a real number and an imaginary number. They are written in standard form as a + bi, where a is the real number and b is the coefficient of the imaginary number.

The complex plane is the graphical representation of the set of all complex numbers.

A superset is a set that includes other sets within it. The complex number set includes all classic complex numbers a + bi, and also, because bi may equal 0, it also includes all of the real numbers as well. This makes it a superset of the set of real numbers.

### Guided Practice

1) Simplify and express as a complex number \begin{align*}-\sqrt{60} + \sqrt{-121}\end{align*}

2) Solve the equation and express the answer as a simplified complex number \begin{align*}x(4x) + 4 = 0\end{align*}

3) Graph the complex numbers:

a) 3 + 2i
b) 2 - 3i
c) -2 + 2i

Solutions

1) To simplify \begin{align*}-\sqrt{60} + \sqrt{-121}\end{align*}

\begin{align*}-\sqrt{4 \cdot 15} + \sqrt{121 \cdot -1}\end{align*} Factor the numbers under the roots
\begin{align*}-2\sqrt{15} + 11\sqrt{-1}\end{align*} Simplify the perfect square roots
\begin{align*}-2\sqrt{15} + 11i\end{align*}

2) To solve \begin{align*}x(4x) + 4 = 0\end{align*}

\begin{align*}4x^2 + 4 = 0\end{align*} Distribute the x
\begin{align*}4x^2 = -4\end{align*} Subtract 4 from both sides
\begin{align*}x^2 = -1\end{align*} Divide both sides by 4
\begin{align*}x = i\end{align*} Take the square root of both sides

3) To graph the complex numbers, graph the real value on the x axis, and the imaginary value on the y axis. The image below shows the points correctly graphed.

### Practice

1. What are complex numbers technically the sum of?
2. What does the complex plane represent?

Express in simplest form in terms of i

1. \begin{align*}13 - \sqrt{-49}\end{align*}
2. \begin{align*}\sqrt{75} + \sqrt{\frac{-16}{256}}\end{align*}
3. \begin{align*}-3 - \sqrt{\frac{-25}{169}}\end{align*}
4. \begin{align*}-\sqrt{36} + \sqrt{-64}\end{align*}
5. \begin{align*}10 - \sqrt{\frac{-4}{36}}\end{align*}
6. \begin{align*}4 + \sqrt{-250}\end{align*}
7. \begin{align*}3^2 \sqrt{-0.0009}\end{align*}
8. \begin{align*}\sqrt{-0.16} -(- \sqrt{27})\end{align*}
9. \begin{align*}10 + \sqrt{-0.49}\end{align*}
10. \begin{align*}-3 + \sqrt{-0.0196}\end{align*}
11. \begin{align*}9 \sqrt{-8j^9} + 3\sqrt{25}\end{align*}
12. \begin{align*}\sqrt{-676b^3a^1c^8} +\sqrt{112}\end{align*}

Graph Complex Numbers

1. Two complex numbers are graphed below. What are the numbers expressed in standard complex number form?

1. Graph the complex number \begin{align*}3 - 4i\end{align*}
2. Graph the complex number \begin{align*}-2 + 3i\end{align*}
3. Graph the complex point \begin{align*}(3 + i)\end{align*}
4. Graph the complex point \begin{align*}(-1 - 2i)\end{align*}

### Notes/Highlights Having trouble? Report an issue.

Color Highlighted Text Notes

### Vocabulary Language: English

$i$

$i$ is an imaginary number. $i=\sqrt{-1}$.

Absolute Value

The absolute value of a number is the distance the number is from zero. The absolute value of a complex number is the distance from the complex number on the complex plane to the origin.

Complex Conjugate

Complex conjugates are pairs of complex binomials. The complex conjugate of $a+bi$ is $a-bi$. When complex conjugates are multiplied, the result is a single real number.

complex number

A complex number is the sum of a real number and an imaginary number, written in the form $a + bi$.

complex plane

The complex plane is the graphical representation of the set of all complex numbers.

i

$i$ is an imaginary number. $i=\sqrt{-1}$.

Real Number

A real number is a number that can be plotted on a number line. Real numbers include all rational and irrational numbers.

rectangular coordinates

A point is written using rectangular coordinates if it is written in terms of $x$ and $y$ and can be graphed on the Cartesian plane.

rectangular form

The rectangular form of a point or a curve is given in terms of $x$ and $y$ and is graphed on the Cartesian plane.

superset

A superset is a set that includes other sets within it. The complex number set includes all classic complex numbers $a+bi$, and also, because $bi$ may equal 0, it includes all of the real numbers as well. This makes it a '''superset''' of the set of real numbers.

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