Extending the Definition

When discussing the zero index we began the process of extending the definition of indices to include zero, negative and fractional indices. Please look at that page before continuing. We will again look at definitions, patterns and properties to help us decide on a suitable meaning for negative indices.

We will try to find a meaning for .

First Definition

Our first definition says that is an expression with negative four factors all of which are two. But can you have negative four of something? We even found that having zero factors was a bit confusing, but negative four factors is worse.

However it is possible to have negative four of something in any situation in which you can gain and lose things, and it is possible to lose more than you gain. With indices you gain factors by multiplying and you lose factors by dividing. This is clearer if we use the second definition.

Second Definition

Our second definition says to start with 1 and multiply by 2 negative four times. Can you do something a negative number of times? This seems to be just as bad as having a negative number of things.

However it is possible to do something a negative number of times, if the action has an inverse or undoing or cancelling action, and it is possible to do the cancelling operation more than you do the original operation. With indices the operation that cancels multiplying by 2 is dividing by 2. So to work out we should start with 1 and divide by 2 four times.

While looking at the rule for dividing indices with the same base we learnt that this could be written in a number of different ways using fractions. Here is are some of them.