Checking the Properties

As a final check on our new meaning for let's see if it fits in with the rules that we have learnt for zero and positive indices. The idea is to make up an equation using the rules in which everything in the equation has a positive or zero index, except for one thing only which will have a negative index. You then have to see whether the equation is still true. If this works out OK then we will be very happy with our new definition.

Because negative indices involve division and fractions which can be written in many different ways, there are many ways to look at these properties too. I have put quite a few in the following table to give you an idea of what is involved, but this is only the beginning.

As you can see all the properties check out fine. Thus we can go ahead with a formal definition of negative indices.

Formal Definition of Negative Indices

If is any real number, , and is any integer, then