Math Education Concepts

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Division by Zero

Many students ask why we cannot divide by zero.  Some students do not ask at all and just accept it as a fact.  How many of you have actually thought about why we cannot divide by zero?  Hopefully, I can shed some light on the topic of division by zero.

What is zero?  Is it a number?  What does it represent?  Zero is an even, natural number, as well as a digit.  The number zero is the smallest non-negative number.  It represents the empty set (nothing).  For example, if you do not have any money, you have zero dollars.  As a digit, zero is a place holder in numbers where there is a gap in a place value.  For example, the number one hundred two is written 102.  The zero is filling the space of the tens place which does not have a value.  For all intents and purposes, zero represents emptiness, nothingness, lack of value, etc.

Let’s take a brief look at division.  Division is the act of dividing a number by some other number (4 ÷ 2).  Put another way, division is the act of distributing a number of items evenly among other things or people (4 apples distributed evenly between 2 people).  In both cases, you have something to distribute and something/someone to distribute between or among.

What happens when you divide by zero?  Suppose I have $25.00 to give to students in even amounts.  If there are five students, then each student will receive $5.00.  If there are ten students, then each student will receive $2.50.  What will happen if there are no students to divide the money between?  The problem does not exist because I can not divide the money evenly since there is no one to divide the money between.  Therefore, the whole idea of dividing $25.00 between zero students does not exist.

Let’s look at division by zero in terms of fractions.  Fractions represent parts of a whole.  For example, 2/8 (two-eighths) represent 2 parts of a whole cut into eight parts.  If I have a whole pizza cut into eight slices and eat two slices, how many slices did I eat?  How many parts of the whole pizza did I eat?  I ate 2/8 (two-eighths) or two parts of the eight slices of pizza.  Now take the fraction 2/0.  That would represent 2 parts of zero parts.  Is this possible?  Can I take two parts from something that does not exist?  No.  Therefore, the fraction 2/0 does not exist.

The nonexistence of division by zero can be explained in various scenarios.  However, the result will be the same.  The idea, concept or representation of any number or value divided by zero simply does not exist.  Of course there are formal proofs about division by zero, but I’ll leave that for the classroom.