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:


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.
This can be rewritten as:
This can be simplified to:
And this evaluates to:
Therefore, we just proved that:
Conversively,
This can be rewritten as:
This can be simplified to:
And this evaluates to:
Therefore, we also proved that:
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:
You can read this article to read my failed attempt at dividing by zero.
by Phil for Humanity
on 03/30/2009