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4.4: Imaginary and Complex Numbers

Difficulty Level: At Grade Created by: CK-12

Learning Objectives

• Write square roots with negative radicands in terms of i
• Recognize and write complex numbers in standard form
• Describe the relationship between the sets of integers, rational numbers, real numbers and complex numbers
• Plot z = a + bi in the complex number plane

Introduction

While working with quadratic equations, you may have solved an equation such as:

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

No matter which method of solving quadratics you used, the solutions to that equation are not real numbers, and you find that they are 1 + 2i and 1 - 2i. (Recall that 1=i\begin{align*}\sqrt{-1} = i\end{align*}). These solutions combine imaginary and real numbers, and are called complex numbers.

The use of the word imaginary does not mean these numbers are useless. For a long period in the history of mathematics, it was thought that the square root of a negative number was in fact only within the mathematical imagination without real-world significance hence, imaginary. That has changed. Mathematicians now consider the imaginary number as another set of numbers that have real significance, but do not fit on what is called the number line—and engineers, scientists, and others solve real world problems using complex numbers!

Recognize

Recognize i=1,x=ix\begin{align*}i = \sqrt{-1}, \sqrt{-x} = i\sqrt{x}\end{align*}

Where did complex numbers come from? If you solve the equation x2+1=0\begin{align*}x^2 + 1 = 0\end{align*}, you get x=±1\begin{align*}x = \pm\sqrt{-1}\end{align*}. But there is no real number that, when multiplied by itself, yields -1. To fix this problem, mathematicians defined the imaginary constant, i. by definition,

i=1\begin{align*}i = \sqrt{-1}\end{align*}

or squaring both sides,

i2=1\begin{align*}i^2 = -1\end{align*}

Recall that you can simplify radicals by factoring out perfect squares in the radicand. For instance, 8=42=42=22\begin{align*}\sqrt{8} = \sqrt{4 \cdot 2} = \sqrt{4}\sqrt{2} = 2\sqrt{2}\end{align*}. The same procedure works with i. If you have a negative number in the radicand, you can factor out the -1 and use the identity i=1\begin{align*}i = \sqrt{-1}\end{align*} to simplify.

Example: Simplify 5\begin{align*}\sqrt{-5}\end{align*}

Solution:

5=(1)(5)\begin{align*}\sqrt{-5} = \sqrt{(-1)\cdot (5)}\end{align*}

=15\begin{align*}= \sqrt{-1}\sqrt{5}\end{align*}

=i5\begin{align*}= i\sqrt{5}\end{align*}

This also works in combination with the other method of factoring out perfect squares. See the following example.

Example: Simplify 72\begin{align*}\sqrt{-72}\end{align*}

Solution:

72=(1)(72)\begin{align*}\sqrt{-72} = \sqrt{(-1)\cdot (72)}\end{align*}

=172\begin{align*}= \sqrt{-1}\sqrt{72}\end{align*}

=i72\begin{align*}= i\sqrt{72}\end{align*}

But, we’re not done yet. 72 = 36\begin{align*}\cdot\end{align*}2, so

i72=i362\begin{align*}i\sqrt{72} = i\sqrt{36} \sqrt{2}\end{align*}

=i(6)2\begin{align*}= i(6)\sqrt{2}\end{align*}

=6i2\begin{align*}= 6i\sqrt{2}\end{align*}

Standard Form of Complex Numbers (a + bi)

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 \pm 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

z = 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.

Set of Complex Numbers (complex, real, irrational, rational, etc.)

The complex numbers are a superset of the real numbers. Given z = a + bi, if b = 0 then z is a real number. 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.

When you were first introduced to mathematics, you probably used positive whole numbers, that is 1, 2, 3, 4,...

Later, negative whole numbers are investigated. The set of all whole numbers, both positive and negative, including the number zero, is known as integers: ... - 2, - 1, 0, 1, 2,...

Later, students are introduced to fractions. The set of all numbers that CAN be expressed as a quotient of two integers (where the denominator is not zero) is called the set of Rational Numbers. Rational Numbers can also be expressed as a terminating or repeating decimal. Some rational numbers are 1,35,73,1,000,002,0\begin{align*}-1, \frac{3}{5}, - \frac{7}{3}, 1, 000, 002, 0\end{align*}. Of course there are an infinite number of rational numbers between any two whole numbers, so listing all rational numbers neatly is difficult (but it is possible--can you think of a way to do it?).

Notice, that all integers are in the set of rational numbers (for example, 5 CAN be written as the quotient of 10 and 2 since 5=102\begin{align*}5 = \frac{10}{2}\end{align*}), so the integers are a subset of the rational numbers. Finally, when working with circles students encounter a number that can be approximated as a quotient of two integers but cannot be expressed EXACTLY as that quotient, that is the number π. Recall that π was often expressed as APPROXIMATELY 227\begin{align*}\frac{22}{7}\end{align*} or 3.14, BUT NOT EXACTLY THOSE VALUES.

When first exploring using the Pythagorean Theorem to find the length of a diagonal of a square whose side is 1, the number 2\begin{align*}\sqrt{2}\end{align*} was introduced. 2\begin{align*}\sqrt{2}\end{align*} often was approximated as 1.4 or 1.414, but again you can’t possibly write out all of the decimals in 2\begin{align*}\sqrt{2}\end{align*}. These two numbers are examples of IRRATIONAL numbers, that is numbers that cannot be expressed as a quotient of two integers, and therefore CANNOT be expressed as a terminating or repeating decimal. The set of all rational and irrational numbers together is called REAL numbers.

Finally, when working again with the Pythagorean Theorem in the coordinate plane (where “negative distances” are possible), negative values appeared within the square root! But what number times itself can result in a negative number?

Historically, when this occurred, mathematicians thought that this was only an oddity of the theorem and not something that can actually exist. They therefore called such numbers imaginary. But, some real-world problems can be solved with imaginary numbers.

The set of all real numbers AND imaginary numbers is called the set of Complex numbers.

Complex Number Plane

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

For example, given the complex number in standard form: z = 2 + 2i, you can graph this number in the coordinate plane To graph this point, the coordinate (2, 2) is graphed as shown below:

Lesson Summary

When graphing a complex number using rectangular coordinates, the x-axis plots the real number, while the y-axis plots the coefficient of the imaginary number.

Points to Consider

Given a point in a rectangular coordinate system that represents a complex number, multiply that complex number by i and graph this new complex number. If the points that represent the original complex number and the new complex number have a line drawn from the origin to each point- note the angle between the two lines. Multiply the second complex number by i and plot this third point. Draw a line from the origin to this point. Note the angle between the second line and the third line. What appears to happen when a complex number is multiplied by i?

Review Questions

Using rectangular coordinate system, graph

1. Simplify the following radicals a. 9\begin{align*}\sqrt{-9}\end{align*} b. 12\begin{align*}\sqrt{-12}\end{align*} c. 17\begin{align*}\sqrt{-17}\end{align*} d. 140108\begin{align*}\sqrt{140-108}\end{align*}
2. Solve each equation and express it as a complex number. (Note: If the imaginary part is 0, you can still express the solution as a + bi, but you will have b = 0 a. x2 + 24 = 0 b. 2x2 - 4x + 7 = 0
3. Plot each of the following complex numbers: a. (4 + 2i) b. (-3 + i) c. (3 - 4i) d. 3i

1. a. 3i\begin{align*}3i\end{align*} b. 2i3\begin{align*}2i\sqrt{3}\end{align*} c. i17\begin{align*}i\sqrt{17}\end{align*} d. 32=62\begin{align*}\sqrt{32} = 6\sqrt{2}\end{align*}
2. 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*}

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