AP CALCULUS BCAppreciating Mathematics

beginning of content:

Authored by

  • Mike Breen
    American Mathematical Society
    ,

Math Is Good for Something, in Fact Many Things

"When will I ever use this?"

"Is math good for anything?"

Teachers hear these questions often, and although many of us love mathematics for its logic and elegance alone, other people are more interested in functionality than in aesthetics. The aim of this article is to show some applications of mathematics and point you to some sources where you can find more, which may help you answer the questions above so that others will appreciate mathematics. We can't necessarily make mathematics easy for everyone, but when people know that it is useful, indeed ubiquitous, with extremely contemporary uses, then they may appreciate mathematics as much as we do.

Check Digits

Here's an application people see every day, and it's an application that anyone who knows multiplication and addition can do, even though higher algebra is used to justify why this application works. Numbers on credit cards, driver's licenses, and products all have a digit to verify that the identifying number is correct. This digit is called a check digit. What follows is an explanation about how the check digit in a bar code works.

You and your students have been to stores where no matter how many times the checkout person waves the product over the scanner, the price isn't recorded: the checkout person has to enter the number by hand. You also know how easy it is to make a mistake entering a long number, for example, making a single-digit error or interchanging two digits. The check digit is there to catch such a mistake and make sure that your 89-cent bottle of soda doesn't wind up as an $18.27 charge for caviar.

Counting from left to right, the first digit (in the example above, a 0) in a Universal Product Code (UPC) indicates the kind of product, the next five digits identify the manufacturer, the following five identify the product, and the last digit on the right (a 6 in the example) is the check digit. This is how that last digit is obtained. Add up the digits in the odd positions (digits 1, 3, 5, ... , 11) and multiply that sum by 3. In this example, we get 3 x (0 + 4 + 0 + 1 + 5 + 9), which gives us 57. Now add to this the digits in the even positions (but not the check digit), that is, digits 2, 4, ..., 10—in this case (6 + 2 + 0 + 1 + 8) + 57 = 74. The check digit will be the number that brings that sum to the next (larger) multiple of 10. Since that multiple of 10 is 80, the check digit is 6.

What would happen if the clerk entered the UPC as 064300115896 (that is, the fourth digit is incorrect)? Then the result without the check digit would be 75, added with the check digit gives 81, which is not a multiple of 10. The number would not be accepted and would have to be reentered.

Similarly, if a clerk entered 064200115986—that is, if the last two digits before the check digit were transposed—then the sum would be 78, again not a multiple of 10, so that type of error is detected as well.

Humans aren't the only fallible part of the process—the machine scanner can also read the number incorrectly. The check digit will detect the same kinds of errors that the scanner could make.

Why not just add the digits in the odd positions instead of multiplying them by 3? (Hint: think about the transpose example.) Why not multiply by 2 or 5? (Hint: would you multiply by 10?) Are there other numbers you could multiply by, besides 3, and still catch the kinds of errors indicated? What if two errors are made?

The Web site UPC Check Digit has a good explanation about the UPC format. The first references at the end of this article are about check digits.
http://www.cs.queensu.ca/home/bradbury/checkdigit/upccheck.htm

The UPC check digit and others are simple examples of using codes to catch errors and ensure validity. In the next application, a more sophisticated example—an error-correcting code—is used in compact discs so that music is reproduced accurately.

Digital Music (and Movies)

Compact Discs

Almost all music is digital, despite the fact that sound is analog. Most people know that CDs use 0s and 1s to produce sound, and so they might be prodded into believing that mathematics is involved, but math puts much more than just two bits' worth into creating CDs. Below are some of the steps involved in making a compact disc.

First the sound (the original music) has to be sampled. Some of the points from the infinite number of points that make up a sound wave are selected to approximate the original sound. This sampling is similar to what might be done when a Riemann sum is used to approximate the area under a curve. To ensure an accurate representation of the original sound, at least 40,000 samples per second are taken. Results from signal processing (including Shannon's Sampling Theorem) are used to choose a good sample and eventually arrive at a good approximation to the original sound. Fourier analysis provides a basis for the validity of this procedure.

The amplitudes of the sampled points are converted to binary form: 16-digit strings of 0s and 1s. Error-correcting codes are incorporated into the strings to compensate for imperfections (for example, a small scratch) that can arise once the CD is created and for errors made by the laser reader when the CD plays. Think of how a small scratch in a record or a piece of dust on a needle could ruin the sound of an album (provided you can think back that far). The error-correcting codes, based on linear algebra, prevent that from happening.

Later in the manufacturing process, the 0s and 1s are translated into bumps and flat portions on the disc. The subject of fluid dynamics is part of this stage in the process as extremely thin layers of acrylic and aluminum are formed onto the plastic disc.

When the CD plays, a laser reads the disc along a spiral track at a constant rate, so that the flow of bits is steady. If the disc were to spin at a constant rate, then the speed of the data stream would change from the center of the disc to the edge. Thus, a motor continually moves the CD at a slower rate as the laser moves across the disc. Since this involves angular velocity, calculus and trigonometry come into play here.

The laser reads the bumps and flat portions, which reflect light differently. These are translated back to the 16-digit binary number, which contains the properties of the original sampled sound. Thus the output is only an approximation of the original wave (if graphed, it would look more like a histogram than a smooth curve), but it is such a good approximation, that most of us don't notice the difference.

Movies

Of course, some of the same technology goes into the making of DVDs, but mathematics is also used in the filming itself when the film contains animated or computer-generated (CG) characters. The two techniques used in CG characters are key-frame animation and motion capture. Computer drawings are used in the former technique, while an actor's motions are used in the second. In both techniques, computers record the movement of an approximation to a character (a wire-frame figure made of polygons or simple curves), and when the movement has been achieved, the animators use computers to add the details to the approximation.

Software does much of this, but it is software based on mathematics—especially geometry and multivariable calculus (normal vectors, for example). Said one animator, a math major in college, when explaining that knowing the mathematics behind the software made her a better animator, "It's all controlled by math... all those little x, y, and zs that you had in school—oh my gosh, suddenly they all apply!"

Dipstick Calculus

In December 2002 a trucker named Rich called auto mechanics (and NPR radio personalities) Click and Clack on their radio call-in show. Rich's gas gauge was broken, but he still needed to know how much gas was in the tank, so he had been using a dowel that he inserted in the tank to estimate how much gas was left. Rich's tank was a right circular cylinder of radius 10 inches and lay on its side on his truck. When the gas reached 10 inches up the dowel, the tank was half full, but Rich wanted to know when the tank was one-quarter full. A quick estimate was when the gas reached five inches up the dowel. Would that be correct? An overestimate of the gas? An underestimate? Why? Rich wanted a better estimate than 5 inches (why?).

Initially, MIT graduates Click and Clack thought that this could be done quickly with pencil and paper, but they soon realized that the problem was not so easy and asked their listeners to submit solutions to their Web site. The problem can be solved with calculus or trigonometry. Below is a solution using calculus.

Since the tank is a right circular cylinder, the problem can be brought down to two dimensions, and we can find where to draw a horizontal line (y = d) so that one-quarter of the area of the circle (centered at the origin) is below the line. That amounts to finding d so that:

Note that:

  1. The 2 appears in the integral either using symmetry or from subtracting the equation for the left half of the circle from the right half.
  2. The integral is easier in terms of y than in terms of x or polar coordinates.

To two decimal places, d = -4.04 (so that the gas should be about 5.96 inches up the dowel). A solution with more detail can be found at Eric Weisstein's World of Mathematics.
http://mathworld.wolfram.com/CircularSegment.html

Preston Nichols at Cornell College in Iowa also has a nice solution at his Web site.
http://people.cornellcollege.edu/pnichols/miscmath/trucktank/trucktank.html

The problem is actually a simpler version of Kepler's problem involving wine casks, where the measuring pole has to fit at an angle into a bung-hole. There is a short description of this problem in the Marriage and Wine Barrels section of the Web site.
http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Kepler.html

The original dipstick problem is a real, down-to-earth problem for which mathematics was able to provide a solution. An interesting side note is that Rich's problem prompted thousands of responses on the Car Talk Web site (unfortunately, the lively discussion thread is no longer there), which shows that people do like to solve problems and are interested in mathematics. The debate on the Web site also provided a hint of the kind of discussion that goes on among mathematicians about solving problems.

Mathematics: Behind-the-Scenes Science

These are just a few applications of mathematics, and there are many more: encrypting e-mail messages and Internet transactions, sequencing the human genome, the Global Positioning System, numerical models of weather, etc. By its nature, however, mathematics' use in our lives is often not apparent. Once a problem is solved, or a method is developed, then others can just use the result without knowing the math behind it, or indeed that there is any math behind it. The more effective the solution, the less apparent the mathematics. It is similar to integration—the more integrals students do, presumably the more often the Fundamental Theorem of Calculus is used, the less aware they are that the Fundamental Theorem is being used.

It could be conjectured that mathematics' usefulness is rivaled only by its modesty.

Other Web Sites of Interest

Mathematical Moments: Short descriptions of applications of mathematics.
http://www.ams.org/ams/mathmoments.html

What's New in Mathematics: A monthly Feature Column along with summaries of articles with mathematics in them from newspapers and national magazines.
http://www.ams.org/new-in-math

MAA Online Columns: Monthly columns by Keith Devlin and Ivars Peterson.
http://www.maa.org/news/columns.html

Plus: Monthly articles and a profile of someone who uses mathematics on the job.
http://plus.maths.org/

References

Gallian, Joseph A. "Assigning Driver's License Numbers." Mathematics Magazine 64, no. 1 (1991): 13-22.

— — —, "The Mathematics of Identification Numbers." The College Mathematics Journal 22, no. 3 (1991): 194-202.

Littleton, Pam and David A. Sánchez. "Dipsticks for Cylindrical Storage Tanks—Exact and Approximate." The College Mathematics Journal 32, no. 5 (2001): 352-358.

Mike Breen became certified to teach secondary school mathematics and earned his Ph.D. in mathematics in the state of Arkansas. He taught at Alfred University and Tennessee Technological University before becoming a public awareness officer of the American Mathematical Society in 2000. Part of Mike's job is to show the public the beauty and usefulness of mathematics. Mike has been an AP Calculus Reader since 1993. Mike has also been a contestant on Jeopardy! and Wheel of Fortune. (If you want to know if he won lots of money on either show, note that he is still working for a living.) Mike thinks that he is the only person ever to cut his hand on the wheel.