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What does Infinity to the Negative One Power Equal to?


I was recently asked does infinity to the negative one power (∞ ^ -1) equal to 0. Additionally, I was asked does zero to the negative one power (0 ^ -1) equal to infinity.

At first, I tried plugging in both equations in my calculator. I could not enter infinity into my calculator, so I entered one million to the negative one power (1,000,000 ^ -1), and that resulted with a very small number that was almost equal to zero. Also, when I entered zero to the negative one power (0 ^ -1), this resulted with an undefined error message.

So, I reasoned that I should collect a whole slew of data points to see if a trend would become obvious. Here is what I got:



X X ^ -1
-1,000,000
-0.000001
-100,000
-0.00001
-10,000
-0.0001
-1,000
-0.001
-100
-0.01
-10
-0.1
-1
-1
-0.1
-10
-0.01
-100
-0.001
-1,000
-0.0001
-10,000
-0.00001
-100,000
0
Undefined
                    
X X ^ -1
0
Undefined
0.00001
100,000
0.0001
10,000
0.001
1,000
0.01
100
0.1
10
1
1
10
0.1
100
0.01
1,000
0.001
10,000
0.0001
100,000
0.00001
1,000,000
0.000001


I think it is safe to assume that infinity to the negative one power (∞ ^ -1) does approach zero. Similarly, negative infinity to the negative one power (-∞ ^ -1) also approaches zero.

Unfortunately, I was not able to prove what zero to the negative one power (0 ^ -1) equals. It looks like from the positive data set (from the table on the right) that zero to the negative one power (0 ^ -1) approaches positive infinity. However, it looks like from the negative data set (from the table on the left) that zero to the negative one power (0 ^ -1) approaches negative infinity.

Obviously, more investigation is necessary.




So, let us try proving what infinity to the negative one power (∞ ^ -1) equals.

y = ∞ ^ -1

This can be re-written as:

y = 1 / (∞ ^ 1)

This can be simplified to:

y = 1 / ∞

And this evaluates to:

y = 0

Therefore, we just proved that:

∞ ^ -1 = 0




Conversively,

y = - ∞ ^ -1

This can be re-written as:

y = -1 / (∞ ^ 1)

This can be simplified to:

y = -1 / ∞

And this evaluates to:

y = 0

Therefore, we also proved that:

-∞ ^ -1 = 0




Unfortunately, I do not know of any method of calculating zero to the negative one power (0 ^ -1). However, I do suspect that it equals to both infinity and negative infinity at the same time, and this is basically undefined.

Keep in mind that:

0 ^ -1 = 1 / 0

You can read this article to read my failed attempt at dividing by zero.

by Phil for Humanity
on 03/30/2009

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