## 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 or ln(0) 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 and ln(e) 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 . And the result given by google when searching for 0 ^ 0 was 1! I then tried the same query at Wolfram Alpha 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 which is equal to 0 when x equals 0. Then I tried to evaluate , i.e. for x = 0. The query I used was (e^(-1/x))^x 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: ». Since 0 is real, by substituting in the function we get ! 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 . Some results are given below:

- Libc 2.9

pow, powf, powl return 1

Since this returns 1 many of the following will return 1 too - Perl 5.10.0

0**0 returns 1 - Python 2.5.4

0**0 returns 1 - Bash 3.2.48

$((0**0)) returns 1 - XCalc from X.Org 7.4

0 0 returns 1 - Kcalc 2.5 using kde 4.3.0

0 0 returns nan! Well done! - Windows calculator 5.1 from windows xp with sp3

0 x^y 0 returns 1 - Mac OS X calculator version 4.0

0 0 returns 1

(e^(-1/x))^x with x = 0 is 1/e!!!. wolframalpha (mathematica) is correct – you have to use l’Hopital’s rule here.

Easy way:

(e ^ (-1/x)) ^ x = (e^(-1/x * x)) = e^-1. (here 1/x * x = 1 when lim x->0).