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

August 23rd, 2009 1 comment

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 $0^0$. 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 $e^{-\frac{1}{x}}$ 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 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:

• 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 $x^y$ 0 returns 1
• Kcalc 2.5 using kde 4.3.0
0 $x^y$ 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 $y^x$ 0 returns 1
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