## Why is Division by Zero Forbidden?

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.

## Functions and Relations: Teaching the Concepts

“Why are functions and relations important?” students ask. “How will I ever use functions and relations? Who cares?”

The answer is that the concepts of function and relation are core concepts in mathematics, and almost every course a student will take in the future will use these concepts. What’s more, we use functions and relations every day without realizing it.

All functions and relations have three components:

Domain Component: The domain of a function or relation can be any collection of objects. Elements in the domain can be numbers, matrices, people, cars, food items on a menu, or anything one can imagine. The domain variable of a function or relation represents all possible input values and is said to be an independent variable.

Range Component: The range of a function or relation can be any collection of objects. Elements in the range can be numbers, matrices, people, cars, food items on a menu, or anything one can imagine. The range variable of a function or relation represents all possible output values. Range variables are called dependent variables, because their values depend on domain input values.

Matching Rule Component: The matching rule component tells us how to match domain objects with range objects. Matching rules are usually in the form of an equation, table, list of ordered pairs, catalog, or set of directions. Every function is a relation, but not every relation is a function. If a relation is a function, every domain object is matched with only one range object. If a function is a one-to-one function, then any two different input values always have different output values. If a and b are domain input values, the symbols f(a) and f(b) represent the corresponding output values. If f(x) is a one-to-one function and f(a) = f(b), then a = b.

To introduce functions and relations to students, I often use the following examples:

Restaurant Menu Function: When a person is handed a menu in a restaurant, they instinctively start matching food items on the menu with prices. The domain of a restaurant menu equals all the food items for sale, and the range equals the prices of the food items. Every food item of a specific type and size is matched with exactly one price. Since different food items may have the same price, a restaurant menu is not a one-to-one function. When a customer sees chateaubriand for two on the menu with a price of \$150, the customer probably thinks that chateaubriand is out of his price range. The matching rule is usually described on a sheet of heavy paper, sign on a wall, or on a chalkboard mounted on a wall.

Student Report Card Function: The domain of a student’s report card equals all courses in which the student was enrolled. The range of a report card equals all grades the student received for the courses enrolled. Students receive one grade for each course taken. Because students can receive the same grade in different courses, most report cards are not one-to-one functions. The matching rule for a report card is usually described on a sheet of paper or in a database that can be accessed at the school’s web site.

People and Social Security Number Function: The domain equals all people with a valid social security number, and the range equals all valid social security numbers. Individuals can have only one valid social security number. The matching rule is determined and maintained by the Social Security Administration. This function is one-to-one because two different people who have valid social security numbers always have different social security numbers.

Cars and VIN Number Function: The domain equals all legally manufactured cars, and the range equals all vehicle identification numbers (VINs) assigned to the cars. Every car has only one VIN, and no two cars have the same VIN. The one-to-one matching rule is described in some computer database.

People and Phone Number Relation: The domain equals all people who have a phone, and the range equals the phone numbers of all people who have a phone. Since an individual can have more than one phone number, the matching rule is a relation, but not a function.

Musical Notes and Sound Function: The domain of a piece of music is the collection of note symbols that appear on a sheet of paper and the range equals the sounds that correspond to the notes on the sheet of paper. The matching rule is one-to-one because different note symbols always result in different sounds. The music matching rule is a set of rules developed by musicians over the course of many centuries.

Mathematical Functions and Relations: In beginning algebra courses, both the domain and range of a function or relation are subsets of the real number system. The matching rule is usually a relatively simple equation such as y = -x + 3, y = 3x^2 + 1, y = (2x – 5)/(x + 3) or x^2 + y^2 = 36.

Free Functions and Relations Handout:

Math teachers and students can go to www.mathteachersresource.com/instructional-content.html to download the free Functions and Relations Introduction Handout. I use this handout with my developmental algebra, technical math, and college algebra students. This handout also contains graphs of mathematical functions and relations. I recently used this handout to tutor a developmental algebra student in the math lab at the junior college where I teach. She said, “When you explain it that way, it makes perfect sense.” By the way, all the 2D graphs in this handout were created with the program Basic Trig Functions.

Math Teacher’s Resource offers the program, Basic Trig Functions, which allows students and teachers to graph a wide variety of mathematical functions and relations. Go to www.mathteachersresource.com to view numerous screen shots of the program’s capabilities. Click the <get all the details> button in the TRIG FUNCTIONS AND GRAPHING section to get to the screen shots. At the bottom of each screen shot are comments that teachers will find helpful. The polar graphs demo has a cool animation of the trace cursor as it races around the edges of the leaves of a five petal rose.