The Zero Index
(Patterns, Definitions and Properties)
All the rules we have looked at so far work when the index is a
positive integer. We know this because we've worked them through,
using the definitions and the properties of multiplication, division
and fractions. However since we have assumed the index is positive
we don't yet know if there is a meaning for zero, negative and fractional
To extend the meaning of indices into these new areas we are going
to do three things. We are going to extend the definitions (if we
can), we are going to look at patterns (and hope they keep on working)
and we are going to look at the properties or rules (and hope they
don't change either). If we succeed we will have made something
very useful indeed.
Let's start with the zero index. Consider . What could this mean? Let's see if we can find a suitable
meaning for it.
Our first definition of indices says that
is an expression
with zero factors all of which are four. But what does it mean to
have zero factors? Is
Our second definition says
is an expression that begins with one. You then multiply by four zero
times. This is a bit clearer. It's easy to do something zero times.
You just don't do anything. So you start with one, do nothing, and
end up with one. According to this . Of course by the same argument
. Can this
To help us decide a suitable meaning let's look at some patterns.
The following table shows a pattern and your job is to decide what
the pattern is. A brief warning first though. Patterns can be described
in terms of the things written on the paper. For example you might
say "cross out a four". This may or may not be correct
but it is dangerous. Such patterns are about the way we write mathematics,
not the way things behave. (The two are related but not the same.)
Your patterns should be about mathematical operations such as "multiply
by four" or "factorize".
So how would you describe this pattern? And what does it tell you
So we think we know that
. But do our rules for indices
still work the same. In the following table I have rewritten some
of our rules to include
. Notice that everything else but
has positive integer indices. So you know what these things are
(if you remember the definitions). Are these rules still correct
if we accept that
? Click to simplify.
Notice that the last line shows us that
any number raised to the power zero is also equal to one.
The Special Case
Now what about ? Can we give this a meaning? is zero for any positive integer . Try it yourself using either definition. But we have
just seen that is one for any base . So is equal to zero or one? Whenever something seems to have
contradictory meanings like this it is usually safer to say that
it has no meaning, or that it is undefined.
We have not talked much about negative bases but you will find
that everything we have said so far works just the same when the
base is negative.