The Place of Mathematical Paradoxes in the Instructional Program

By AlFred Posamentier, Ph.D.

A paradox or fallacy in mathematics generally results from a violation of some rule or law of mathematics. This makes these paradoxes excellent vehicles for presenting these rules, for their violation leads to some rather “curious” results, such as 1=2, or 1=0, just absurd! They are clearly entertaining since they very subtly lead the student to an impossible result. Often the student becomes frustrated by the fact that every step to this weird result seemed correct. So where did he/she go wrong? This is quite motivating and will make the conclusion that much more impressionable.

Again, it is a fine source for investigating the mathematical borders. Why isn’t division by zero permissible? Why isn’t the product of the radicals always equal to the radical of the product? These are just a few of the questions that this article entertainingly investigates. The “funny” results are entertaining to expose. But most important, beyond the enjoyment of the points made in this article is the instructional value. Students are apt never to violate rules that lead to some of these fallacies. They usually make a lasting impression on students.

Are All Numbers Equal?

This statement is clearly preposterous! But as you will see from the demonstration below, such may not be the case. Present this demonstration line-by-line and let students draw their own conclusions. We shall begin with the easily accepted equation: ((x – 1) / (x – 1)) = 1. Each succeeding row can be easily justified with elementary algebra. There is nothing wrong with the algebra. See if your students can find the flaw.

When x = 1,the numbers 1, 2, 3, 4, … , n are each equal to 0/0, which would make them all equal to each other. Of course, this cannot be true. For this reason we define 0/0 to be meaningless. To define something to make things meaningful or consistent is what we do in mathematics to avoid ridiculous statements, as was the case here. Be sure to stress this point with your students before leaving this unit.

Negative One is Not Equal to Positive One

Your students should be aware of the notion that __6 = __2 • __3  and then they might conclude that __ab = __a • __b .

From this, have your students multiply and simplify: __–1 • __–1 .

Some students will do the following to simplify this expression:

__–1 • __–1 = —(–1)(–1) = __+1 = 1.

Other students may do the following with the same request: __–1 • __–1 = (__–1)2 = –1

If both groups of students were correct, then this would imply that 1 = -1, since both are equal to __–1 • __–1. Clearly this can’t be true!

What could be wrong? Once again a “fallacy” appears when we violate a mathematics rule. Here (for obvious reasons) we define that __ab = __a • ___b  is only valid when at least one of a or b is non-negative. This would indicate that the first group of students that got

__–1 • __–1 = —(–1)(–1) = __+1 = 1 was wrong.

Thou Shalt Not Divide by Zero

Every math teacher knows that division by zero is forbidden. As a matter of fact, on the list of commandments in mathematics, this is at the top. But why is division by zero not permissible? We in mathematics pride ourselves in the order and beauty in which everything in the realm of mathematics falls neatly into place. When something arises that could spoil that order, we simple define it to suit our needs. This is precisely what happens with division by zero. You give students a much greater insight into the nature of mathematics by explaining why “rules” are set forth. So let’s give this “commandment” some meaning.

Consider the quotient n/0, with n­0. Without acknowledging the division-by-zero commandment, let us speculate (i.e., guess) what the quotient might be. Let us say it is p. In that case, we could check by multiplying 0•p to see if it equals n, as would have to be the case for the division to be correct. We know that 0•p­n, since 0•p= 0. So there is no number p that can take on the quotient to this division. For that reason, we define division by zero to be invalid.

A more convincing case for defining away division by zero is to show students how it can lead to a contradiction of an accepted fact, namely, that 1 ­ 2. We will show them that were division by zero acceptable, then 1 = 2, clearly an absurdity!

Here is the “proof” that 1 = 2:

In the step where we divided by (a – b), we actually divided by zero, because a=b, so a – b = 0. That ultimately led us to an absurd result, leaving us with no option other than to prohibit division by zero. By taking the time to explain this rule about division by zero to your students, they will have a much better appreciation for mathematics.

Mathematics teachers must realize that they can achieve a lasting impression by presenting important facts in a dramatic fashion. That is precisely what we have done here. Enjoy! #

Dr. Alfred Posamentier is Dean of the School of Education at City College of NY, author of over 40 Mathematics books including: Math Wonders to Inspire Teachers and Students (ASCD, 2003) and The Fabulous Fibonacci Numbers (Prometheus, 2007), and member of the NYS Mathematics Standards Committee.