What is 5/0? When I ask my beginning algebra students that question, the most popular incorrect answer they give me is 0. The next most popular incorrect answer is 5. After repeated reminders by their math teachers, students eventually learn that 5/0 is undefined, has no value, or is meaningless. (I once told a class of 9th grade algebra students that if they use their calculator to divide a number by zero, the calculator will explode in their face. One student looked at me and said, “Really?” I forgot how literal 9th graders can be. At least I got the student’s attention.) When I ask college algebra, trigonometry, statistics, technical math or calculus students why a number divided by zero is undefined, I either get an answer that begs the question or students say it’s simply a mathematical fact that they learned in a previous course.
So how do you explain division by zero? There are two ways. The first depends on a basic understanding of division of two numbers. It goes something like this: Students learn that a / b = c if and only if a = b*c. Therefore 986 / 58 = 17 because 58*17 = 986. Is 5 / 0 = 0? No, because 0 * 0 ≠ 5. Is 5 / 0 = 5? No, because 0*5 ≠ 5. Since 0 times any number never equals 5, 5 / 0 is NOTHING or undefined. So what about 0 / 0? The problem here is that 0 times any number equals 0, and therefore 0 / 0 would have infinitely many answers, which in turn would be rather confusing. So we say that any number divided by zero is undefined.
The second explanation involves a deep mathematical insight from the 12th century Indian mathematician and astronomer, Bhāskara II, who developed the basic concepts of differential calculus. The 17th century European mathematicians, Newton and Leibniz, independently rediscovered differential calculus. This second explanation due to Bhāskara II goes something like this. Consider a single piece of fruit. If we divide 1 piece of fruit by ¼, we get 4 pieces of fruit. If we divide 1 piece of fruit by 1/10,000, we get 10,000 pieces of fruit. As 1 is divided by smaller and smaller numbers that approach zero, the number of pieces of fruit increases without bound. Therefore 1/0 = ∞ and, in general, n/0 = ±∞ if n does not equal 0.
Bhāskara II, Newton and Leibniz discovered the revolutionary concept of a limit of a function at a point, which enabled them to get around the problem of division by zero. Once that problem was solved, it was a relatively easy task to find methods to calculate a rate of change over a time interval of length zero, rate of change over a fleeting instant of time, or rate of change over a flux of time, as Newton would say. In The Ascent of Man, Dr. Bronowski tells the viewer, “In it, mathematics becomes a dynamic mode of thought, and that is a major mental step in the ascent of man.” Differential calculus is all about the mathematics of variable rates of change. I should mention that differential calculus students learn a slick technique for finding the limiting value of an x-variable expression as x approaches a constant k and the value of the expression when x = k is 0/0 or ∞/∞.
The graphic below shows the graphs of the functions y = 2Sin(x) and y = 2Csc(x) along with its vertical asymptotes. The graphs are color coded green, blue and red respectively. Because Csc(x) = 1 / Sin(x), the Csc(x) function is undefined at precisely those values of x where Sin(x) = 0. It’s interesting and fun to advance a trace mark cursor on the graphs of these functions. On both graphs, the horizontal velocity of the trace mark is constant, but the vertical velocity of the trace mark changes as the value of the x changes. As x approaches a vertical asymptote, the trace mark races towards ± ∞. Differential calculus gives us a complete understanding of the phenomena of the moving trace cursor.
The above graphic, created with the program Basic Trig Functions, is offered by Math Teacher’s Resource. The equations entered into the program were: y = 2Sin(x), y = 2Csc(x), and Sin(x) = 0. Go to www.mathteachersresource.com to view multiple screen shots of the program’s modules. Click the ‘learn more’ button in the TRIGONOMETRIC FUNCTIONS section. Teachers will find useful comments at the bottom of each screen shot.
Differential calculus is not only interesting and fun, but it can also be a stress reliever. At least it was for Omar Bradley, the famous American WWII general. He took a calculus book with him on battle campaigns, and when opportunity allowed, he worked differential calculus problems to relieve the stress of a battle campaign.