00 - Münchausen's Syndrome
Type 00 into wolframalpha to observe an uncharacteristic one-liner, "indeterminate". And it is hard not to agree with such a designation. The function f(x, y) = xy is discontinuous at (0, 0) however the value at the origin is defined. Indeed, x0 = 1, while also 0x = 0, for x > 0.
On the other hand, type 0^0 into google and see the unequivocal
The controversy of what value 00 - if at all - must be ascribed is far from being new. L. Euler believed that 00 = 1, and it seems that at least in discrete mathematics this definition became common.
This is why I was surprised to come across a different usage.
Depending on how you look at it, there are either 2 or 4 integers that possess a certain property associated for some reason with the name of Baron Münchausen.
Integer abc...d is said to be a Münchausen number if it equals the sum of its digits raised to the power which is that digit:
abc...d = aa + bb + cc + ... + dd
Unquestionably, 1 is a Münchausen number since 11 = 1. There is also a non-trivial example:
3435 = 33 + 44 + 33 + 55.
And that's it - there are just two of them. However, if one allows for the uncommon
438579088 = 44 + 33 + 88 + 55 + 77 + 99 + 00 + 88 + 88.
The assignment 00 = 0 is so unusual in mathematics, that it may explain the association with the name of Baron Münchausen. Be as it may, the Encyclopedia of Integer Squences lists the 4-term sequence 0, 1, 3435, 438579088 as #A046253.
No related posts.
I'm a bit surprised that Wolfram Alpha says "indeterminate" for 0^0, because, the truth is, it's fairly well accepted (at least among people who are even consciously aware of the debate) as 1. To obtain a monstrous compendium of support for this by people who don't even realize they're supporting it, simply gather up textbooks which define polynomials/power series/etc. as
f(x)=sum_{i=0}^{whatever} a_i x^i.
Then, when you plug in x=0, the result is supposed to be a_0, which implicitly presumes 0^0=1.
In fact, your post inspired me to discover a flaw in Wolfram Alpha. Enter in:
April 17th, 2011 at 12:21 pmsum_{i=0}^{infinity} (1/i!)x^i
and Wolfram Alpha correctly identifies this as e^x.
Now enter in:
sum_{i=0}^{infinity} (1/i!)0^i
and Wolfram Alpha says the value is 0. Wolfram Alpha thinks e^0=0. OMG!
And Wolfram Alpha is not even being *consistent* in its wrongness here. Instead of "0", to be consistent, it ought to output "indeterminate".
[...] [1] Anonymous Blogger, 16 Apr 2011, “00 – Münchausen’s Syndrome” [...]
April 17th, 2011 at 12:57 pmThis wolframalpha's behavior is a real curiosity. If you know how, you should inform them.
April 19th, 2011 at 1:14 am[...] 00 – Münchausen’s Syndrome, √(-1) does not exist – A. De [...]
April 23rd, 2011 at 12:21 pm