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Indices - what happens if the index is 0?
By Pete Golding (P4302) on Tuesday, April 24, 2001 - 05:42 pm:

For my maths homework I have to find out what happens when the index is 0: what would the answer for this be?
eg. 30

Cheers

By Mark Adams (Maa25) on Tuesday, April 24, 2001 - 06:40 pm:

Do you know the rule for adding powers? That is, do you know what 3a × 3b is? You are really going to be making a new definition for what 30 means, and it needs to be consistent with this rule.

By Michael Doré (Michael) on Wednesday, April 25, 2001 - 02:05 pm:

If you look at the following pattern:

35 = 3×3×3×3×3 = 243
34 = 3×3×3×3 = 81
33 = 3×3×3 = 27
32 = 3×3 = 9
31 = 3
30 = ?

You can see that each time you go down one, you divide the result by 3.

Therefore it would be sensible to define 30 so that this pattern is continued. So it would be nice if we could define:
30 = 3 ÷ 3 = 1

If you follow similar reasoning for other positive numbers, you get the same result: a0 = 1 is a very useful convention, because it allows the property an = a×an-1 to be preserved for all values of n.

By The Editor:

You can use the same reasoning to work what happens with negative indices:
3-1 = 30 ÷ 3 = 1/3
3-2 = 3-1 ÷ 3 = 1/9
etc.