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Zero to the zero power at google, wolfram alpha and others

Yesterday I was trying some features of the google search engine such as the built-in calculator. After trying some simple functions that it supports, I wanted to see its limitations.

First, I tried searching for 1 / 0 [1] or ln(0) [2] in order to see if it has support for infinity. The calculator didn’t even show up to return results, even if searches with a similar format that don’t return infinity such as 4 / 2 [3] and ln(e) [4] returned the correct result. So, google calculator supports infinity but doesn’t inform you about it when it is the result of a calculation.

Then I tried searching for something that is an indeterminate form, such as 0^0 . And the result given by google when searching for 0 ^ 0 [5] was 1! I then tried the same query at Wolfram Alpha [6] which uses the mathematica engine and I got the correct result, indeterminate. EDIT: I made a HUGE error trying to fool the mathematica engine and I fooled myself!!! Thanks to my readers I had the chance to fix it! Still, I wasn’t satisfied and I wanted to see if I could fool it. First I had to find something that is equal to 0 but doesn’t look like this. I decided to use e^{-\frac{1}{x}} which is equal to 0 when x equals 0. Then I tried to evaluate (e^{-\frac{1}{x}})^x , i.e. 0^0 for x = 0. The query I used was (e^(-1/x))^x [7] which returned a lot of information for this function. One thing that I noticed is that it stated «Alternate form assuming all variables are real: \frac{1}{e} ». Since 0 is real, by substituting in the function we get 0^0 = \frac{1}{e} ! To be honest I didn’t believe that the mathematica engine would fail here and it would be difficult to fool it but it seems I was wrong!

After all these I made some tests to see what real calculator programs return when computing 0^0 . Some results are given below: