Pre-Calculus For Dummies, 2nd Edition (2012)
Part III. Analytic Geometry and System Solving
In this part . . .
The term analytic geometry usually means drawing out a shape or an equation to study it more deeply. This part starts with the set of complex numbers and explains how to perform operations with them and, yes, how to graph them. It then moves on to the new system of graphing known as polar coordinates. Conic sections finish up analytic geometry as we show you how to graph and examine the parts of circles, parabolas, ellipses, and hyperbolas.
Next we move on to solving systems of equations. We cover the old favorites of graphing, substitution, and elimination. Then we introduce the idea of a matrix and explain several ways to solve a system using matrices.
After that, we move on to sequences and series: how to find the term in any sequence as well as how to find the sum of certain types of series. Lastly, we bridge to calculus with the study of limits and continuity of functions.
Chapter 11. Plane Thinking: Complex Numbers and Polar Coordinates
In This Chapter
Pitting real versus imaginary
Exploring the complex number system
Plotting complex numbers on a plane
Picturing polar coordinates
Complex numbers and polar coordinates are some of the most interesting but often neglected topics in a standard pre-calculus course. Both of these concepts have very basic explanations, and they can vastly simplify a difficult problem or even allow you to solve a problem that you couldn’t solve before.
In previous math courses, you were told that you can’t find the square root of a negative number. If somewhere in your calculations you stumbled on an answer that required you to take the square root of a negative number, you simply threw that answer out the window. That changes here in Pre-Calculus For Dummies, however. As you advance in math, you need complex numbers to explain natural phenomena that real numbers are incapable of. In fact, entire math courses are dedicated to the study of complex numbers and their applications. We don’t go into that kind of depth here; we simply introduce you to the topics gradually.
In this chapter, we cover the concepts of complex numbers and polar coordinates. We show you where they come from and how you use them (as well as graph them).
Understanding Real versus Imaginary (According to Mathematicians)
Algebra I and II introduce you to the real number system. Pre-calculus is here to expand your horizons by adding complex numbers to your repertoire, including imaginary numbers. Complex numbers are numbers that include both a real and an imaginary part; they’re widely used for complex analysis, which theorizes functions by using complex numbers as variables (see the following section for more on this number system).
If you were previously taught to disregard negative roots whenever you found them, here’s a quick explanation: You can actually take the square root of a negative number, but the square root isn’t a real number. However, it does exist! It takes the form of an imaginary number.
Imaginary numbers have the form Bi, where B is a real number and i is an imaginary number — defined as .
Luckily, you’re already familiar with the x-y coordinate plane, which you use to graph functions (such as in Chapter 3). You also can use a complex coordinate plane to graph imaginary numbers. Although these two planes are constructed the same way — two axes perpendicular to one another at the origin — they’re very different. For numbers graphed on the x-y plane, the coordinate pairs represent real numbers in the form of variables (x and y). You can show the relationships between these two variables as points on the plane. On the other hand, you use the complex plane simply to plot complex numbers. If you want to graph a real number, all you really need is a real number line. However, if you want to graph a complex number, you need an entire plane so that you can graph both the real and imaginary part.
Enter the Gauss or Argand coordinate plane. In this plane, pure real numbers in the form a + 0i exist completely on the real axis (the horizontal axis), and pure imaginary numbers in the form 0 + Bi exist completely on the imaginary axis (the vertical axis). Figure 11-1a shows the graph of a real number, and Figure 11-1b shows that of an imaginary number.
Figure 11-1:Comparing the graphs of a real and an imaginary number.
Combining Real and Imaginary: The Complex Number System
The complex number system is more complete than the real number system or the pure imaginary numbers in their separate forms. You can use this system to represent real numbers, imaginary numbers, and numbers that have both a real and an imaginary part. In fact, the complex number system is the most comprehensive set of numbers you deal with in pre-calculus.
Grasping the usefulness of complex numbers
You may be asking two important questions right now: When are complex numbers useful, and where will I stumble across them? Imaginary numbers are as important to the real world as real numbers, but their applications are hidden among some pretty heavy concepts, such as chaos theory and quantum mechanics. In addition, forms of mathematical art, called fractals, use complex numbers. Perhaps the most famous fractal is called the Mandelbrot Set. However, you don’t have to worry about that stuff in pre-calculus. For you, imaginary numbers can be used as solutions to equations that don’t have real solutions (such as quadratic equations).
For example, consider the quadratic equation x2 + x + 1 = 0. This equation isn’t factorable (see Chapter 4). Using the quadratic formula, you get the following solution:
Notice that the discriminant (the b2 – 4ac part) is a negative number, which you can’t solve with only real numbers. When you first discovered the quadratic formula in algebra, you most likely used it to find real roots only. But because of complex numbers, you don’t have to discard this solution. The previous answer is a legitimate complex solution, or a complex root. (Perhaps you remember encountering complex roots of quadratics in Algebra II. Check out Algebra II For Dummies, by Mary Jane Sterling [Wiley], for a refresher.)
Performing operations with complex numbers
Sometimes you come across situations where you need to operate on real and imaginary numbers together, so you want to write both numbers as complex numbers in order to be able to add, subtract, multiply, or divide them.
Consider the following three types of complex numbers:
A real number as a complex number: 3 + 0i
Notice that the imaginary part of the expression is 0.
An imaginary number as a complex number: 0 + 2i
Notice that the real portion of the expression is 0.
A complex number with both a real and an imaginary part: 1 + 4i
This number can’t be described as solely real or solely imaginary — hence the term complex.
You can manipulate complex numbers arithmetically just like real numbers to carry out operations. You just have to be careful to keep all the i’s straight. You can’t combine real parts with imaginary parts by using addition or subtraction, because they’re not like terms, so you have to keep them separate. Also, when multiplying complex numbers, the product of two imaginary numbers is a real number; the product of a real and an imaginary number is still imaginary; and the product of two real numbers is real. Many people get confused with this topic.
The following list presents the possible operations involving complex numbers:
To add and subtract complex numbers: Simply combine like terms. For example, (3 – 2i) – (2 – 6i) = 3 – 2i – 2 + 6i = 1 + 4i.
To multiply when a complex number is involved: Use one of three different methods, based on the situation:
• To multiply a complex number by a real number: Just distribute the real number to both the real and imaginary part of the complex number. For example, here’s how you handle a scalar (a constant) multiplying a complex number in parentheses: 2(3 + 2i) = 6 + 4i.
• To multiply a complex number by an imaginary number: First, realize that the real part of the complex number becomes imaginary and that the imaginary part becomes real. When you express your final answer, however, you still express the real part first followed by the imaginary part, in the form A + Bi.
For example, here’s how 2i multiplies into the same parenthetical number: 2i(3 + 2i) = 6i + 4i2. Note: You define i as , so that i2 = –1! Therefore, you really have 6i + 4(–1), so your answer becomes –4 + 6i.
• To multiply two complex numbers: Simply follow the FOIL process (see Chapter 4). For example, (3 – 2i)(9 + 4i) = 27 + 12i – 18i – 8i2, which is the same as 27 – 6i – 8(–1), or 35 – 6i.
To divide complex numbers: Multiply both the numerator and the denominator by the conjugate of the denominator, FOIL the numerator and denominator separately, and then combine like terms. This process is necessary because the imaginary part in the denominator is really a square root (of –1, remember?), and the denominator of the fraction must not contain an imaginary part.
For example, say you’re asked to divide . The complex conjugate of
3 – 4i is 3 + 4i. Follow these steps to finish the problem:
1. Multiply the numerator and the denominator by the conjugate.
2. FOIL the numerator.
You go with (1 + 2i)(3 + 4i) = 3 + 4i + 6i + 8i2, which simplifies to (3 – 8) + (4i + 6i), or –5 + 10i.
3. FOIL the denominator.
You have (3 – 4i)(3 + 4i), which FOILs to 9 + 12i – 12i – 16i2. Because i2 = –1 and 12i – 12i = 0, you’re left with the real number 9 + 16 = 25 in the denominator (which is why you multiply by 3 + 4i in the first place).
4. Rewrite the numerator and the denominator.
This answer still isn’t in the right form for a complex number, however.
5. Separate and divide both parts by the constant denominator.
Notice that the answer is finally in the form A + Bi.
Graphing Complex Numbers
To graph complex numbers, you simply combine the ideas of the real-number coordinate plane and the Gauss or Argand coordinate plane (which we explain in the “Understanding Real versus Imaginary [According to Mathematicians]” section earlier in this chapter) to create the complex coordinate plane. In other words, you take the real portion of the complex number (A) to represent the x-coordinate, and you take the imaginary portion (B) to represent the y-coordinate.
Although you graph complex numbers much like any point in the real-number coordinate plane, complex numbers aren’t real! The x-coordinate is the only real part of a complex number, so you call the x-axis the real axis and the y-axis the imaginary axis when graphing in the complex coordinate plane.
Graphing complex numbers gives you a way to visualize them, but a graphed complex number doesn’t have the same physical significance as a real-number coordinate pair. For an (x, y) coordinate, the position of the point on the plane is represented by two numbers. In the complex plane, the value of a single complex number is represented by the position of the point, so each complex number A + Bi can be expressed as the ordered pair (A, B).
You can see several examples of graphed complex numbers in Figure 11-2:
Point A: The real part is 2 and the imaginary part is 3, so the complex coordinate is (2, 3) where 2 is on the real (or horizontal) axis and 3 is on the imaginary (or vertical) axis. This point is 2 + 3i.
Point B: The real part is –1 and the imaginary part is –4; you can draw the point on the complex plane as (–1, –4). This point is –1 – 4i.
Point C: The real part is 1/2 and the imaginary part is –3, so the complex coordinate is (1/2, –3). This point is 1/2 – 3i.
Point D: The real part is –2 and the imaginary part is 1, which means that on the complex plane, the point is (–2, 1). This coordinate is –2 + i.
Figure 11-2:Complex numbers plotted on the complex coordinate plane.
Plotting Around a Pole: Polar Coordinates
Polar coordinates are an extremely useful addition to your mathematics toolkit because they allow you to solve problems that would be extremely ugly if you were to rely on standard x- and y-coordinates (for example, problems when the relationship between two quantities is most easily described in terms of the angle and distance between them, such as navigation or antenna signals). Instead of relying on the x- and y-axes as reference points, polar coordinates use only the positive x-axis (the line starting at the origin and continuing in the positive horizontal direction forever). From this line, you measure an angle (which you call theta, or θ) and a length (or radius) along the terminal side of the angle (which you call r). These coordinates replace x- and y-coordinates.
In polar coordinates, you always write the ordered pair as (r, θ). For instance, a polar coordinate could be (5, π/6) or (–3, π).
In the following sections, we show you how to graph points in polar coordinates and how to graph equations as well. You also discover how to change back and forth between Cartesian coordinates and polar coordinates.
Wrapping your brain around the polar coordinate plane
In order to fully grasp how to plot polar coordinates, you need to see what a polar coordinate plane looks like. In Figure 11-3, you can see that the plane is no longer a grid of rectangular coordinates; instead, it’s a series of concentric circles around a central point, called the pole. The plane appears this way because the polar coordinates are a given radius and a given angle in standard position from the pole. Each circle represents one radius unit, and each line represents the special angles from the unit circle (to make finding the angles easier; see Chapter 6).
Although θ and r may seem strange as plotting points at first, they’re really no more or less strange or useful than x and y. In fact, when you consider a sphere such as Earth, describing points on, above, or below its surface is much more straightforward with polar coordinates on a round coordinate plane.
Figure 11-3: A blank polar coordinate plane (not a dartboard).
Because you write all points on the plane as (r, θ), in order to graph a point on the polar plane, we recommend that you find θ first and then locate r on that line. This approach allows you to narrow the location of a point to somewhere on one of the lines representing the angles. From there, you can simply count out from the pole the radial distance. If you go the other way and start with r, you may find yourself in a pickle when the problems get more complicated.
For example, to plot point E at (2, π/3) — which has a positive value for both the radius and the angle — you simply move from the pole counterclockwise until you reach the appropriate angle (θ). You start there in the following list:
1. Locate the angle on the polar coordinate plane.
Refer to Figure 11-3 to find the angle: θ = π/3.
2. Determine where the radius intersects the angle.
Because the radius is 2 (r = 2), you start at the pole and move out 2 spots in the direction of the angle.
3. Plot the given point.
At the intersection of the radius and the angle on the polar coordinate plane, plot a dot and call it a day! Figure 11-4 shows point E on the plane.
Polar coordinate pairs can have positive angles or negative angles for values of θ. In addition, they can have positive and negative radii. This concept is new; in past classes you’ve always heard that a radius must be positive. When graphing polar coordinates, though, the radius can be negative, which means that you move in the opposite direction of the angle from the pole.
Because polar coordinates are based on angles, unlike Cartesian coordinates, polar coordinates have many different ordered pairs. Because infinitely many values of θ have the same angle in standard position (see Chapter 6), an infinite number of coordinate pairs describe the same point. Also, a positive and a negative co-terminal angle can describe the same point for the same radius, and because the radius can be either positive or negative, you can express the point with polar coordinates in many ways. Usually, providing four different representations of the same point is sufficient.
Figure 11-4:Visualizing simple and complex polar coordinates.
Graphing polar coordinates with negative values
Simple polar coordinates are points where both the radius and the angle are positive. You work on graphing these coordinates in the previous section. But you also must prepare yourself for when teachers spice it up a tiny bit with complicated polar coordinates — points with negative angles and/or radii. The following list shows you how to plot in three situations — when the angle is negative, when the radius is negative, and when both are negative:
When the angle is negative: Negative angles move in a clockwise direction (see Chapter 6 for more on these angles). Check out Figure 11-4 to see an example point, D. To locate the polar coordinate point D at (1, –π/4), first locate the angle π/4 and then find the location of the radius, 1, on that line.
When the radius is negative: When graphing a polar coordinate with a negative radius (essentially the x value), you move from the pole in the direction opposite the given positive angle (on the same line as the given angle but in the direction opposite to the angle from the pole). For example, check out point F at (–1/2, π/3) in Figure 11-4.
Some teachers prefer to teach their students to move right along the x- (polar) axis for positive numbers (radii) and left for negative. Then you do the rotation for the angle in a positive direction. You’ll get to the same spot with that method.
For example, take a look point F (–1/2, π/3) in Figure 11-4. Because the radius is negative, move along the left x-axis 1/2 of a unit. Then rotate the angle in the positive direction (counterclockwise) π/3 radians. You should arrive at your destination, point F.
When both the angle and radius are negative: To express a polar coordinate with a negative radius and a negative angle, locate the terminal side of the negative angle first and then move in the opposite direction to locate the radius. For example, point G in Figure 11-4 has these characteristics at (–2, –5π/3).
Indeed, except the origin, each given point can have the following four types of representations:
Positive radius, positive angle
Positive radius, negative angle
Negative radius, positive angle
Negative radius, negative angle
For example, point E in Figure 11-4 (2, π/3) can have three other polar coordinate representations with different combinations of signs for the radius and angle:
When polar graphing, you can change the coordinate of any point you’re given into polar coordinates that are easy to deal with (such as positive radius, positive angle).
Changing to and from polar coordinates
You can use both polar coordinates and (x, y) coordinates at any time to describe the same location on the coordinate plane. Sometimes you’ll have an easier time using one form, and for this reason we teach you how to navigate between the two. Cartesian coordinates are much better suited for graphs of straight lines or simple curves. Polar coordinates can yield you a variety of pretty, very complex graphs that you can’t plot with Cartesian coordinates.
When changing to and from polar coordinates, your work is often easier if you have all your angle measures in radians. You can make the change by using the conversion factor 180° = π radians. You may choose, however, to leave your angle measures in degrees, which is fine as long as your calculator is in the right mode.
Devising the changing equations
Examine the point in Figure 11-5, which illustrates a point mapped out in both (x, y) and (r, θ) coordinates, allowing you to see the relationship between them.
Figure 11-5: A polar and (x, y)coordinate mapped in the same plane.
What exactly is the geometric relationship between r, θ, x, and y? Look at how they’re labeled on the graph — all parts of the same triangle!
Using right-triangle trigonometry (see Chapter 6), you know the following facts:
These equations simplify into two very important expressions for x and y in terms of r and θ:
y = r sin θ
x = r cos θ
Furthermore, you can use the Pythagorean theorem in the right triangle to find the radius of the triangle if given x and y:
x2 + y2 = r2
One final equation allows you to find the angle θ; it derives from the tangent of the angle:
So if you solve this equation for θ, you get the following expression:
With respect to the final equation, keep in mind that your calculator always returns a value of tangent that puts θ in the first or fourth quadrant. You need to look at your x- and y-coordinates and decide whether that placement is actually correct for the problem at hand. Your calculator doesn’t look for tangent possibilities in the second and third quadrants, but that doesn’t mean you don’t have to!
As with degrees (see the section “Wrapping your brain around the polar coordinate plane”), you can add or subtract 2π to any angle to get a co-terminal angle so you have more than one way to name every point in polar coordinates. In fact, there are infinite ways of naming the same point. For instance, (2, π/3), (2, –5π/3), (–2, 4π/3), (2, –2π/3), and (2, 7π/3) are several ways of naming the same point.
Putting the equations into action
Together, the four equations for r, θ, x, and y allow you to change (x, y) coordinates into polar (r, θ) coordinates and back again anytime. For example, to change the polar coordinate (2, π/6) to a rectangular coordinate, follow these steps:
1. Find the x value.
If x = r cos θ, substitute what you know (r = 2 and θ = π/6) to get
Use the unit circle to get
which means that .
2. Find the y value.
If y = r sin θ, substitute what you know to get
which means that y = 1.
3. Express the values from Steps 1 and 2 as a coordinate point.
You find that (, 1) is the answer as a point.
Time for an example in reverse. Given the point (–4, –4), find the equivalent polar coordinate:
1. Plot the (x, y) point first.
Figure 11-6 shows the location of the point in quadrant III.
Figure 11-6: An (x, y) coordinate changed to a polar coordinate.
2. Find the r value.
For this step, you use the Pythagorean theorem for polar coordinates: x2 + y2 = r2. Plug in what you know (x = –4 and y = –4) to get (–4)2 + (–4)2 = r2, or .
3. Find the value of θ.
Use the tangent ratio for polar coordinates: tan θ = –4/–4, or tan θ = 1. The reference angle for this value is θ' = π/4 (see Chapter 6). You know from Figure 11-6 that the point is in the third quadrant, so θ = 5π/4.
4. Express the values of Steps 2 and 3 as a polar coordinate.
You can say that (–4, –4) = (, 5π/4).
Picturing polar equations
Polar coordinates allow for the graphing of some strange and remarkable equations. We give some of the most common equations and their conditions and shapes in Table 11-1.
A graphing calculator is fully capable of plotting all these strange, new polar functions. However, it will spit out nothing but nonsense if you don’t change two things before entering a function into your graphing utility:
1. Make sure your calculator is in radians mode, not degrees.
2. Change your graphing mode to “polar.”
After you take these steps, your calculator’s graphing menu should change. Instead of displaying “y =,” it will display “r =.” It also should give you a fairly simple way of entering θ rather than x as your variable.
Make sure that you enter a maximum and minimum θ values for the calculator to graph to (found in your graph’s “window” settings), because the standard window usually is from 0 to 2π. This consideration is especially important for working polar functions like Archimedan spirals, which you want to follow out to large values of θ.
Note: Although the polar functions from Table 11-1 are much different from the types of functions you’ve seen before, and the graphing menu on your calculator has changed, you’re still graphing in the Cartesian plane.