# Thinking about 0 to the power of 0

Half by accident I came across What is 0 to the power of 0? on Youtube, an inspiring recording of a normal high school class session about the value of\$0^0\$. Half way through the video I thought: It would be interesting to see a graph of \$x^x\$ when \$x\$ is getting closer to \$0\$.

With Python’s matplotlib (and seaborn on top) that’s pretty easy, so I plotted \$f(x) = x^x\$ for \$x \$: This graph suggests that for \$x\$ approaching \$0\$, \$f(x)\$ approaches \$1\$. But lets plot a bit closer to 0: Same story. And if we look very close to \$0\$: Same story again. So let’s propose:

\$_{x ^+} x^x = 1\$

From this it makes sense to set \$x^x = 0\$. This is indeed how many mathematicians1 (but not all of them2) define it. Actually a good overview of the issue is given on Wikipedia.

1. See for example Donald Benson, The Moment Of Proof: Mathematical Epiphanies, 1999, p.29: The consensus is to use the definition \$0^0 = 1\$.”↩︎

2. See Charles Henry Edwards & David Penney, Calculus With Analytic Geometry, 1994, p.471: Although \$a^0 = 1\$ for any nonzero constant \$a\$, form \$0^0\$ is indeterminate[…]“↩︎