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Topic: A number-theoretic rationale for leaving 0^0 undefined
Replies: 6   Last Post: Jun 5, 2015 9:10 AM

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Dan Christensen

Posts: 6,760
Registered: 7/9/08
A number-theoretic rationale for leaving 0^0 undefined
Posted: Sep 26, 2013 8:17 PM
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As taught in high schools and many university courses, and as has been
the practice for centuries, 0^0 is left undefined.

In this school of thought, ^ is defined recursively as:

(1) x^0 = 1 for x=/=0
(2) x^(y+1) = x^y * x

Many mathematicians and designers of programming languages assume that
0^0=1. They define ^ recursively as:

(1) x^0 = 1
(2) x^(y+1) = x^y * x

Rather than 0^0=1, consider the following definition:

(1) 0^0=0
(2) x^0 = 1 for x=/=0
(3) x^(y+1) = x^y * x

Note that this 0^0=0 definition agrees with the 0^0=1 definition
everywhere except where the base and exponent are both 0.

It can also be shown that all the usual Laws of Exponents for the
natural numbers can be derived from the this 0^0=0 definition:

(1) x^1 = x
(2) x^(y+z) = x^y * x^z
(3) (x^y)^z = x^(y*z)
(4) (x*y)^z = x^z * y^z

This being the case, it seems there are at least two distinct binary
functions on the natural numbers that satisfy the Laws of Exponents. As
a result of this seeming ambiguity, there may be some justification, in
purely number-theoretic terms, for leaving 0^0 undefined.

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