Zero to the power of zero
It is determined by definition that 00 = 0! ≝ 1.
Explanation
We start with the power series for the exponential function
If you write it very carefully you will find
For x = 0 this gives then
As e = 2.7182… applies e0 = 1. So we can write the power series as
There are infinitely many terms, but these are all zero, except for one. That first term must therefor be 1. Thus the numerator and the denominator are equal, so
because per definition applies 00 ≝ 1 and 0! ≝ 1.
HistoryThe French mathematician Augustin-Louis Cauchy (1789 - 1857) used this calculation to justify the definition. |