IN THIS CHAPTER

Sparking your math imagination with imaginary numbers

Performing operations with complex numbers

Dividing and conquering complex numbers and radicals

Facing complex solutions in quadratic equations

Identifying complex roots of polynomials

Imaginary numbers are the results of mathematicians’ imaginations. No, imaginary numbers aren’t real — although, is any number real? Can you touch a number (if you don’t count playing with a toddler’s educational toy set)? Can you feel it? Who decided a 9 should be shaped the way it is, and what makes that person right? Does your brain hurt yet?

Mathematicians define real numbers as all the whole numbers, negative and positive numbers, fractions and decimals, radicals — anything you can think of to use in counting, graphing, and comparing amounts. Mathematicians introduced imaginary numbers when they couldn’t finish problems without them. For example, when solving for roots of quadratic equations such as , you quickly discover that you can find no real answers. Using the quadratic formula (see Chapter 3), the solutions come out to be

The answers seem sort of final — like you have nowhere to go except to estimate some value for it. But you can’t find the square root of a negative number, because no real number multiplies itself and ends up with a negative result. So, rather than staying stuck there, with no final solution, mathematicians came up with a new rule: They let .

They made up a number to replace , and they called it i (didn’t take much imagination to come up with that). When you take the square root of each side of , you get or . Using this equivalence, you can complete the problems and write the answers with i’s in them.

In this chapter, you find out how to create, work with, and analyze imaginary numbers and the complex expressions they appear in. Just remember to use your imagination!

Using Your Imagination to Simplify Powers of i

The powers of x (representing real numbers) — , and so on — follow the rules of exponents, such as adding exponents when multiplying the powers of x together or subtracting exponents when dividing them (see Chapter 1). The powers of i (representing imaginary numbers) are rule-followers, too. The powers of i, however, have some neat features that set them apart from other numbers.

You can write all the powers of i as one of four different numbers: , and ; all it takes is some simplifying of products, using the properties of exponents, to rewrite the powers of i:

• : Just plain old i
• : From the definition of an imaginary number (see the introduction to this chapter)
• : Use the rule for exponents and then replace with ; so,
• : Because
• : Because
• : Because
• : Because
• : Because

Consider the two powers of i presented in the following list and how you determine the rewritten values; if you want to find a power of i, you make use of the rules for exponents and the first four powers of i:

• : Because
• : Because

The process of changing the powers of i seems like a lot of work — plus you need to figure out what to multiply four by to get a high power (you want to find a multiple of four — the biggest possible value that’s smaller than the exponent). But you really don’t need to go through all the raising of powers if you recognize a particular pattern in the powers of i.

Every power of i that’s a multiple of four is equal to 1. If the power is one value greater than a multiple of four, the power of i is equal to i. And so the process goes. Here’s the list in full:

So, all you need to do to change the powers of i is figure out where a power of i is in relation to some multiple of four. If you need the value of , for example, you know that 5,000 is a multiple of 4 (because it ends in 00), and 5,001 is 1 value greater than 5,000, so . It really simplifies expressions when you have a rule that reduces the powers of numbers.

Understanding the Complexity of Complex Numbers

An imaginary number, i, is a part of the numbers called complex numbers, which arose after mathematicians established imaginary numbers. The standard form of complex numbers is , where a and b are real numbers, and is . The fact that is equal to and i is equal to is the foundation of the complex numbers. If you didn’t have the imaginary numbers with the i’s, you’d have no need for complex numbers with imaginaries in them.

Some examples of complex numbers include , , and 7i. In the last number, 7i, the value of a is zero, so the number is pure imaginary. If the value of b is zero as well, you have a complex number that is a real number.

So, a is the real part of a complex number and the bi is the imaginary part (even though b is a real number). Is that complex enough for you?

Complex numbers have many applications, and mathematicians study them extensively. In fact, whole math courses and fields of study are devoted to complex numbers. And, imagine this, you get a glimpse of this ethereal world right here and now in this section.

Operating on complex numbers

You can add, subtract, multiply, and divide complex numbers — in a very careful manner. The rules used to perform operations on complex numbers look very much like the rules used for any algebraic expression, with two big exceptions:

• You simplify the powers of i, change them to their equivalent in the first four powers of i (see the section “Using Your Imagination to Simplify Powers of i” earlier in the chapter), and then combine like terms.
• You don’t really divide complex numbers; you multiply by the conjugate (I tell you all about this in the section “Multiplying by the conjugate to perform division” later in the chapter).

Adding complex numbers

When you add two complex numbers and , you find the sum of the real parts and the sum of the imaginary parts:

The result of the addition is now in the form of a complex number, where is the real part and is the imaginary part.

When you add , for example, you get .

Subtracting complex numbers

When you subtract the complex numbers and , you find the difference of the real parts and the difference of the imaginary parts:

The result of the subtraction is now in the form of a complex number, where is the real part and is the imaginary part.

When you subtract , for example, you get .

Multiplying complex numbers

When you multiply complex numbers, the operations get a little more exciting.

To multiply complex numbers, you can’t just multiply the real parts together and the complex parts together; you have to treat the numbers like binomials and distribute both of the terms of one complex number over the other. Another way to look at it is that you have to FOIL the terms (for the details on FOIL, refer to Chapter 1):

The result of the multiplication shown here is in the form of a complex number, with as the real part and as the imaginary part. You can see from the following distribution where the values in this rule come from. First, FOILing the binomials, you have . You combine the two middle terms by factoring out the . Replace the with . And now the first and last terms are both real numbers, and you can write them together: .

To find the product of , for example, you FOIL to get . You simplify the last term to and combine it with the first term; the two middle terms also combine. Your result is , a complex number.

You don’t really have to memorize the form of the rule. You can just as easily FOIL the complex binomials and simplify the terms.

Multiplying by the conjugate to perform division

The complex thing about dividing complex numbers is that you don’t really divide. Do you remember when you first found out how to multiply and divide fractions? You never really divided fractions; you changed the second fraction to its reciprocal, and then you changed the problem to multiplication. You found that the answer to the multiplication problem you created is the same as the answer to the original division problem. You avoid division in much the same way with complex numbers. You do a multiplication problem — one that has the same answer as the division problem. But before you tackle the “division,” you should know more about the conjugate of a complex number.

Defining the conjugate

A complex number and its conjugate have opposite signs between the two terms. The conjugate of the complex number is , for instance.

Here are a couple numerical examples: The conjugate of is , and the conjugate of is . Seems simple enough, because you don’t see the special trait attributed to the conjugate of a complex number until you multiply the complex number and its conjugate together.

The product of an imaginary number and its conjugate is a real number (no imaginary part) and takes the following form of the sum of two squares:

Here’s the product of the complex number and its conjugate, all worked out by using FOIL (see Chapter 1): . The middle terms are opposites of one another, and (the definition of an imaginary number) gets rid of the factor.

Using conjugates to divide complex numbers

When a problem calls for you to divide one complex number by another, you write the problem as a fraction and then multiply by the number one. You don’t actually multiply by one; you multiply by a fraction that has the conjugate of the denominator in both numerator and denominator (because the same value appears in the numerator and denominator, the fraction is equal to one):

To write the result of the division of complex numbers in a strictly real-and-imaginary-parts format, you break up the fraction:

The form of the fraction that results looks awfully complicated, and you don’t really want to have to memorize it. When doing a complex division problem, you can use the same process I use to get the previous form for dividing complex numbers. To divide by , for example, perform the following steps:

Simplifying radicals

Until mathematicians defined imaginary numbers, many problems had no answers because the answers involved square roots of negative numbers, or radicals. After the definition of an imaginary number, , came into being, doors opened; windows flung wide; parades were held; children danced in the streets; and problems were solved. Eureka!

To simplify the square root of a negative number, you write the square root as the product of square roots and simplify: . If you want to simplify , for example, you first split up the radical into the square root of and the square root of the rest of the number, and then you do any simplifying by factoring out perfect squares:

By convention, you write the previous solution as . This form isn’t at all different in value from the other form. Most mathematicians just want to put the numerical part of the coefficient first, the variables or other letters next (in alphabetical order), and the radicals last. They think it looks better that way.

Technically, in complex form, you write a number with the i at the end, after all the other numbers — even after the radical. That’s the strict form. If you write the number this way, be sure not to tuck the i under the radical — keep it clearly to the right. For instance, . To avoid confusion, write it as .

Solving Quadratic Equations with Complex Solutions

You can always solve quadratic equations with the quadratic formula. It may be easier to solve quadratic equations by factoring, but when you can’t factor, the formula comes in handy. (Refer to Chapter 3 if you need a refresher on the quadratic formula.)

Until mathematicians began recognizing imaginary numbers, however, they couldn’t complete many results of the quadratic formula. Whenever a negative value appeared under a radical, the equation stumped the mathematicians.

The modern world of imaginary numbers, to the rescue! To solve the quadratic equation , for example, you can use the quadratic formula to get the following:

Along with the quadratic formula that produces a complex solution, it helps to look at the curve that corresponds to the equation on a graph (see Chapter 7).

A parabola that opens upward or downward always has a y-intercept; these parabolas are functions and have domains that contain all real numbers. However, a parabola that opens upward or downward doesn’t necessarily have any x-intercepts.

To solve a quadratic function for its x-intercepts, you set the equation equal to zero. When no real solution for the equation exists, you find no x-intercepts. You still have a y-intercept, because all quadratic functions cross the y-axis somewhere, but its graph may stay above or below the x-axis without crossing it.

The quadratic equation corresponds to the parabola . Substituting zero for the y variable allows you to solve for the x-intercepts of the parabola. The fact that you find no real solutions for the equation tells you that the parabola has no x-intercepts. (Figure 14-1 shows you a graph of this situation.)

Parabolas that open to the left or right aren’t functions. A function has only one y-value for every x-value, so these curves violate that requirement. A parabola opening to the left or right may never cross the y-axis if its vertex is to the left or right of that axis, so you find no real solution for the y-intercept. A parabola that opens to the left or right always has an x-intercept, however.

To solve for any y-intercept, you set x equal to zero and then solve the quadratic equation. If the equation has no solution, it has no y-intercepts. Solving the equation for its y-intercept, for example, you let . The equation doesn’t factor, so you use the quadratic formula:

The parabola opens to the left and never crosses the y-axis. The graph of this parabola is shown in Figure 14-2.

Working Polynomials with Complex Solutions

Polynomials are functions whose graphs are nice, smooth curves that may or may not cross the x-axis. If the degree (or highest power) of a polynomial is an odd number, its graph must cross the x-axis, and it must have a real root or solution. (In Chapter 8, you discover how to find the real solutions or roots of polynomial equations.) When solving equations formed by setting polynomials equal to zero, you plan ahead as to how many solutions you can expect to find. The highest power tells you the maximum number of solutions you can find. In the case of polynomial equations where the highest powers are even numbers, you may find that the equations have absolutely no real solutions, whereas if the highest powers are odd, you’re guaranteed at least one solution in each equation.

Interpreting complex zeros

The polynomial function has two real roots and two complex roots. According to Descartes’ Rule of Signs (see Chapter 8), the function could’ve contained as many as four real roots (you can tell by trying the roots suggested by the Rational Root Theorem; see Chapter 8). You can determine the number of complex roots in two different ways: by factoring the polynomial or by looking at the graph of the function.

The example function factors into . The first two factors give you real roots, or x-intercepts. When you set equal to 0, you get the intercept (2, 0). When you set equal to 0, you get the intercept . Setting the last factor, , equal to 0 doesn’t give you a real root, as you see here when you use the quadratic formula:

You can also tell that a polynomial function has complex roots by looking at its graph. You can’t tell what the complex roots are, but you can see that the graph has some. If you need the values of the roots, you can resort to using algebra to solve for them. Figure 14-3 shows the graph of the example function, . You can see the two x-intercepts, which represent the two real zeros. You also see the graph flattening on the left.

A flattening-out behavior can represent a change of direction, or a point of inflection, where the curvature of the graph changes. Areas like this on a graph indicate that complex zeros exist for the polynomial.

Figure 14-4 can tell you plenty about the number of real zeros and complex zeros the graph of the polynomial has … before you ever see the equation it represents.

The polynomial in Figure 14-4 appears to have one real zero and several complex zeros. Do you see how it changes direction all over the place under the x-axis? It goes up, down, up, down, and then up. These changes indicate the presence of complex zeros. The graph represents the polynomial function . The function has four complex zeros — two complex (conjugate) pairs — and one real zero (when ).

You need the efforts and capabilities of good algebra; the Rational Root Theorem and Descartes’ Rule of Signs combined with a graphing calculator or computer; and some luck and common sense to solve these high-order equations. When the zeros are nice integers, life is good. When the zeros are irrational or complex, life is still good, but it’s a bit more complicated. Assume the best, and work through the challenges if they arise.