Welcome back to Intermediate Algebra. As you recall from last
time, we discussed the four basic arithmetic operations for real
numbers. With the exception of division, we were able to
define these operations for all numbers. Yet when it came to
division, we had to restrict the domain – the set of inputs – to
non-zero real numbers. In less formal terms, we concluded,
“You can’t divide by zero!” But we didn’t say or attempt to
prove why that is the case. Typically, algebra teachers just
state, “Division by zero is undefined” and move on.
That is not an acceptable way to teach mathematics that is
accessible to students. It makes the subject appear
unnecessarily mysterious and may serve to make students feel
unable to think about or ask why something is logically true.
The standard explanation is straightforward. We assume that
the equation a/0 = x, where a is a non-zero real number has a
real solution, x. We then multiply both sides of this equation by
0 to obtain the equivalent equation, 0*x = a. By assumption, a is
not equal to 0. However, by the 0 property of multiplication,
0*x must equal 0 for all real numbers. This is a logical
contradiction, so the assumption that there existed such an x
that was the quotient of a and 0 must be false. We conclude
that any non-zero number divided by 0 is undefined (it should
be mentioned that 0 divided by 0 DOES require mathematics
beyond the scope of this course).
Here is another way to think about the issue: a loose definition
of multiplication a * b = c is that c is the sum of a addends of b,
or in other words that we can obtain the product ab via
repeated addition (this is not strictly accurate for all real
numbers but works as a way to COMPUTE product of natural
numbers, integers, and rational numbers).
By analogy, we can argue that division – the inverse operation
for multiplication – can be considered as repeated subtraction –
the inverse operation for addition. We thus can say, for
instance, that 6 / 2 = 3 because we can take 2 away from 6
three times before reaching 0 or having less than 2 remaining
to subtract another time.
This works well for many sorts of numbers, though it takes a
little thinking to make it work for negative numbers and
becomes particularly tricky if both the dividend and divisor are
non-integer rational numbers, e.g., 2/3 divided by ¾.
What does this have to do with division by zero? Well, consider
the quotient of 6 / 0: if we take 0 away from 6, we have 6 left. If
we do that again, we still have 6 left. If we do it ten times, a
hundred times, a thousand times, a trillion times, we still have 6
remaining. In fact, there is no real number of times we can do
this and get any closer to a remainder that is less than 6, let
alone that is 0. We might argue that the answer is infinity
(again, an issue somewhat beyond the scope of intermediate
algebra), but it suffices to mention that infinity is not a real
number. And we need the results of operations on real
numbers to be real numbers. That is essential to what we
generally mean by an operation on a given set: it has to be
closed under that operation.
This more intuitive explanation leads to a similar, if more
complicated, conclusion about division by zero in the real
numbers or any of its subsets: it must be undefined.